The modes in circle of fifths order
I mostly won’t be analyzing the modes in their traditional order: I’m analyzing how lowering a regular pattern of notes by a semitone each walks us through every mode on every key. I call this the circle of fifths order for reasons that will soon become clear, and I refer to scales with single-note transformations between all their modes as mutable or mutant scales, and the process as scale mutation, to distinguish it from scale transformations more generally. A few notes (pun intended):
- Most of this analysis focuses on twelve-tone equal temperament (12-TET for short), the tuning system the vast majority of Western music has used for hundreds of years. I’ll try to note exceptions in other temperaments where they exist, but given my at-best spotty understanding of other tuning systems, I can’t promise to have gotten them all.
- For the most part, I’ll attempt to avoid bogging this analysis down in heavy mathematics, but I must begin this analysis by noting that in 12-TET, every note on the chromatic scale vibrates at exactly 2¹⁄₁₂ its lower neighbor’s frequency – i.e., the note spacing is exactly even. This is an important precondition for the analysis I’ve conducted here: much of it wouldn’t apply otherwise.
- In 12-TET, A♯ and B♭ are enharmonically equivalent: they signify the same pitch. Likewise B and C♭, C♯ and D♭, and so on. However, in music theory and notational terms, A♯ and B♭ are semantically quite different – be careful. (And in tunings with more than 12 notes per octave, they often won’t be the same pitch.)
- I largely omit seven-sharp and seven-flat key signatures: using both six-flat and six-sharp key signatures already felt extravagant enough. (There are even monstrosities with double-flats [𝄫] and double-sharps [𝄪], which I’ll ignore entirely until later. At least the double-flat symbol makes sense; the double-sharp is an abomination unto Nuggan that doesn’t even slightly resemble what it’s meant to double.)
- If we begin at Lydian mode, lowering the correct note sequence by a semitone each gives us the next mode in the cycle. I’ll use C Lydian as an example to demonstrate the principle:
| Modes descending from Lydian | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root & mode | Pitch lowered | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 | |
| 4 | C♮ | – | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | C♮ | |
| 1 | C♮ | 4 | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | C♮ | |
| 5 | C♮ | 7 | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | C♮ | |
| 2 | C♮ | 3 | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | C♮ | |
| 6 | C♮ | 6 | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | |
| 3 | C♮ | 2 | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | |
| 7 | C♮ | 5 | C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | C♮ | |
| 4 | C♭ | 1 | C♭ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | C♭ | |
- In other words:
- Lowering C Lydian’s F♯ to F gives us C Ionian (i.e., C major).
- Lowering C Ionian’s B to B♭ gives us C Mixolydian.
- And so on, until we reach C Locrian – whence the pattern repeats for C♭/B, the key below C on the chromatic scale.
- Lowering Locrian’s root gives us Lydian mode in the key below it on the chromatic scale.
- In other words, lowering C Locrian’s C to C♭/B gives us C♭/B Lydian.
- The sequence repeats from there:
- B Ionian is B Lydian with a lowered E♯.
- B Mixolydian is B Ionian with a lowered A♯.
- This sequence repeats for every note on the chromatic scale: C, C♭/B, B♭/A♯, A, A♭/G♯, G, G♭/F♯, F, F♭/E, E♭/D♯, D, D♭/C♯, and back to C.
- Here are the mode transformations in circle of fifths order, with the lowered notes in boxes:
| Mode transformations revisited | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Mode | Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |||
| 4 | ♯4 | |||||||||||
| 1 | ||||||||||||
| 5 | ♭7 | |||||||||||
| 2 | ♭3 | ♭7 | ||||||||||
| 6 | ♭3 | ♭6 | ♭7 | |||||||||
| 3 | ♭2 | ♭3 | ♭6 | ♭7 | ||||||||
| 7 | ♭2 | ♭3 | ♭5 | ♭6 | ♭7 | |||||||
The significance of that note sequence may not be immediately obvious, but if we reshuffle the above table back into linear order, it becomes easier to understand its inverse relationship to the modes themselves:
| Mode transformations re-revisited | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Mode | Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |||
| 1 | ||||||||||||
| 2 | ♭3 | ♭7 | ||||||||||
| 3 | ♭2 | ♭3 | ♭6 | ♭7 | ||||||||
| 4 | ♯4 | |||||||||||
| 5 | ♭7 | |||||||||||
| 6 | ♭3 | ♭6 | ♭7 | |||||||||
| 7 | ♭2 | ♭3 | ♭5 | ♭6 | ♭7 | |||||||
The Russian author Anton Chekhov (1860-1904) might have a few words to say about this table. (This is my roundabout way of advising you to remember it.)
The principles of inverse operations
- Now, note that we’ve effectively been subtracting 1 from a repeating sequence of notes:
- 4, 7, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 1…
- As a result, the mathematical properties of inverse operations tell us that the converse of everything we’ve just done will apply if we add 1 to the same notes in the reverse order:
- 1, 5, 2, 6, 3, 7, 4, 1, 5, 2, 6, 3, 7, 4…
- In other words:
- C Locrian is C♭ Lydian with a raised C♭.
- C Phrygian is C Locrian with a raised G♭.
- C Aeolian is C Phrygian with a raised D♭.
- And so on. Our inverted transformations are:
| Mode transformations inverted | Mode transformations inverted | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| circle of fifths order | circle of fifths order | linear order | ||||||||||||||||||||||||||
| # | Mode | Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | # | Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||||||||||
| 7 | ♭2 | ♭3 | ♭5 | ♭6 | ♭7 | 7 | ♭2 | ♭3 | ♭5 | ♭6 | ♭7 | |||||||||||||||||
| 3 | ♭2 | ♭3 | ♭6 | ♭7 | 6 | ♭3 | ♭6 | ♭7 | ||||||||||||||||||||
| 6 | ♭3 | ♭6 | ♭7 | 5 | ♭7 | |||||||||||||||||||||||
| 2 | ♭3 | ♭7 | 4 | ♯4 | ||||||||||||||||||||||||
| 5 | ♭7 | 3 | ♭2 | ♭3 | ♭6 | ♭7 | ||||||||||||||||||||||
| 1 | 2 | ♭3 | ♭7 | |||||||||||||||||||||||||
| 4 | ♯4 | 1 | ||||||||||||||||||||||||||
| linear order | ||||||||||||||||||||||||||||
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |||||||||||||||||||||
| ♭2 | ♭3 | ♭5 | ♭6 | ♭7 | ||||||||||||||||||||||||
| ♭3 | ♭6 | ♭7 | ||||||||||||||||||||||||||
| ♭7 | ||||||||||||||||||||||||||||
| ♯4 | ||||||||||||||||||||||||||||
| ♭2 | ♭3 | ♭6 | ♭7 | |||||||||||||||||||||||||
| ♭3 | ♭7 | |||||||||||||||||||||||||||
An audio demonstration
-
None of this will mean much to readers in isolation, so I put together a ⟨aaronfreed
.github .io /c_ lydian_ to_ b_ lydian .flac⟩ in Logic Pro. For the following modes, in order: - C Lydian
- C Ionian
- C Mixolydian
- C Dorian
- C Aeolian
- C Phrygian
- C Locrian
- C♭/B Lydian
you’ll hear, in order:
- the scale, in ascending order
- the root or tonic (I, i, or iᵒ) chord, in arpeggiated and block forms
- the subdominant (IV, iv, or ivᵒ) chord, in arpeggiated and block forms
- the dominant (V, v, or vᵒ) chord, in arpeggiated and block forms
- the root (I, i, or iᵒ) chord, in arpeggiated and block forms
In case any of the above terminology confuses you:
- An arpeggiated chord is played one note at a time; a block chord’s notes are all played simultaneously.
- Roman numerals refer to the chord based on its root note’s position in the scale.
Major chords are uppercase, minor chords are lowercase, and diminished chords are lowercase followed by a superscript “o”. Thus, on C:
- Lydian, Ionian, and Mixolydian have I root chords (C major).
- Dorian, Aeolian, and Phrygian have i root chords (C minor).
- Locrian has a iᵒ root chord (C diminished).
- Ionian, Dorian, and Mixolydian have IV subdominant chords (F major).
- Phrygian, Aeolian, and Locrian have iv subdominant chords (F minor).
- Lydian has a ivᵒ subdominant chord (F♯ diminished).
- Ionian, Lydian, and Locrian have V dominant chords (respectively, G major, G major, and G♯ major).
- Dorian, Mixolydian, and Aeolian have v dominant chords (G minor).
- Phrygian has a vᵒ dominant chord (G diminished).
- “Chord analysis by mode” below lists all seven chords for each mode.
(I considered going down the entire chromatic scale, but that would take thirteen and a half minutes. Even I think that’s overkill, and for context on my definition of “overkill”, I submit this exact book as Exhibit A.)
Further notes
- It might help solidify your conception of this principle to understand that while we’re lowering (or raising) a pitch on our scale, we’re also, in a sense, jumping down (or up) a fifth. F Lydian is in the same key signature as C Ionian, not F Ionian. This is why so many of these patterns mirror the circle of fifths.
- In ‘Modes descending from Lydian’, the mode numbers and ‘Pitch Lowered’ reverse each other’s order. This isn’t coincidental: it’s an inevitable mathematical result of the scale’s interval distribution.
- The modes are also always either three steps on the scale below their predecessors (usually, but not always, a perfect fourth), or four steps above (usually, but not always, a perfect fifth). In this context, they’re equivalent, since the scale repeats every octave.
- Although this is only somewhat relevant here, one pair of intervals always constitutes an exception to the “perfect fourths” and “perfect fifths” rules: one fourth is always augmented, and one fifth is always diminished. These are actually the same interval, a tritone. The following steps on each scale are tritones in either direction:
Note the two identical columns in ‘Modes descending from Lydian’ and ‘Diabolus in mūsicā’. These are also not coincidental. Incidentally, Ionian is one of 12-TET’s only two seven-note scales in which only one note pair forms a tritone. The other is the chromatic heptatonic scale (C, C♯, D, D♯, E, F, F♯). Perhaps coincidentally, perhaps not, these are also the only heptatonic scales in 12-TET where transforming a single note can cycle us through all seven modes of the scale and all twelve notes of the chromatic scale. However, as we’ll see below, the transformations are quite different.Diabolus in mūsicā # Mode Tritone 4 4 1 1 7 4 5 3 7 2 6 3 6 2 6 3 5 2 7 1 5
- Although this is only somewhat relevant here, one pair of intervals always constitutes an exception to the “perfect fourths” and “perfect fifths” rules: one fourth is always augmented, and one fifth is always diminished. These are actually the same interval, a tritone. The following steps on each scale are tritones in either direction:
The major scale’s modes & the circle of fifths
12 major scales × 7 modes = 84 permutations
- I’ve printed the C major scale’s modes in bold, blue type to make them stand out. These are the only scales played using only the white keys of the piano (this is also why music notation can express these specific keys without accidentals, either in the key signature or after it). Their numbers always differ by multiples of twelve; this is also not a coincidence.
- Halfway between C major’s modes, I always print a mode twice, in orange, with its key signature first using six flats (G♭ major), then six sharps (F♯ major). I don’t really like either option (E♯ is F♮! C♭ is B♮!), but both avoid a repeated letter in a scale, and thus a rash of accidentals in the notation of any piece that uses them. G♭/F♯ major’s modes are also always separated by multiples of twelve, for the same reason: this progression separates all seven modes of each scale from each other by multiples of twelve.
- I’ve listed these descending by pitch so higher pitches will be, well, higher, which may confuse some people since we read from top to bottom and are used to thinking of harmony in ascending order. Someday, I plan to write a JavaScript add-on to give readers an option to reverse the order – and eventually, to give them then option to create similar tables for different scales (at the bare minimum, melodic minor, harmonic minor, and chromatic Hypolydian; possibly others as well).
- “RM” is an initialism for “Relative Major”, and “KS” is for “Key Signature”. A cheat sheet for what modes use what key signatures can be found in the section immediately following this one.
- I’ve used zero-based indexing for these tables, so they’re indexed from zero to eighty-three. This is partly because I prefer zero-based indexing, but to be honest, it was mostly to give C Ionian an index of one.
| C (B♯) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 0 | C | G | 1♯ | C♮ | D♮ | E♮ | F♯ | G | A♮ | B♮ | C | |
| 1 | C | C | ♮ | C♮ | D♮ | E♮ | F♮ | G | A♮ | B♮ | C | |
| 2 | C | F | 1♭ | C♮ | D♮ | E♮ | F♮ | G | A♮ | B♭ | C | |
| 3 | C | B♭ | 2♭ | C♮ | D♮ | E♭ | F♮ | G | A♮ | B♭ | C | |
| 4 | C | E♭ | 3♭ | C♮ | D♮ | E♭ | F♮ | G | A♭ | B♭ | C | |
| 5 | C | A♭ | 4♭ | C♮ | D♭ | E♭ | F♮ | G | A♭ | B♭ | C | |
| 6 | C | D♭ | 5♭ | C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | C | |
| B (C♭) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 7 | C♭ | G♭ | 6♭ | C♭ | D♭ | E♭ | F | G♭ | A♭ | B♭ | C♭ | |
| 7 | B | F♯ | 6♯ | B | C♯ | D♯ | E♯ | F♯ | G♯ | A♯ | B | |
| 8 | B | B | 5♯ | B | C♯ | D♯ | E | F♯ | G♯ | A♯ | B | |
| 9 | B | E | 4♯ | B | C♯ | D♯ | E | F♯ | G♯ | A | B | |
| 10 | B | A | 3♯ | B | C♯ | D | E | F♯ | G♯ | A | B | |
| 11 | B | D | 2♯ | B | C♯ | D | E | F♯ | G | A | B | |
| 12 | B | G | 1♯ | B | C | D | E | F♯ | G | A | B | |
| 13 | B | C | ♮ | B | C | D | E | F | G | A | B | |
| A♯ / B♭ | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 14 | B♭ | F♮ | 1♭ | B♭ | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | |
| 15 | B♭ | B♭ | 2♭ | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | |
| 16 | B♭ | E♭ | 3♭ | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | |
| 17 | B♭ | A♭ | 4♭ | B♭ | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | |
| 18 | B♭ | E♭ | 5♭ | B♭ | C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | |
| 19 | B♭ | G♭ | 6♭ | B♭ | C♭ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | |
| 19 | A♯ | F♯ | 6♯ | A♯ | B♮ | C♯ | D♯ | E♯ | F♯ | G♯ | A♯ | |
| 20 | A♯ | B♮ | 5♯ | A♯ | B♮ | C♯ | D♯ | E♮ | F♯ | G♯ | A♯ | |
| A | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 21 | A♮ | E | 4♯ | A♮ | B♮ | C♯ | D♯ | E♮ | F♯ | G♯ | A♮ | |
| 22 | A♮ | A | 3♯ | A♮ | B♮ | C♯ | D♮ | E♮ | F♯ | G♯ | A♮ | |
| 23 | A♮ | D | 2♯ | A♮ | B♮ | C♯ | D♮ | E♮ | F♯ | G♮ | A♮ | |
| 24 | A♮ | G | 1♯ | A♮ | B♮ | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | |
| 25 | A♮ | C | ♮ | A♮ | B♮ | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | |
| 26 | A♮ | F | 1♭ | A♮ | B♭ | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | |
| 27 | A♮ | B♭ | 2♭ | A♮ | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | |
| G♯ / A♭ | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 28 | A♭ | E♭ | 3♭ | A♭ | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | |
| 29 | A♭ | A♭ | 4♭ | A♭ | B♭ | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | |
| 30 | A♭ | D♭ | 5♭ | A♭ | B♭ | C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | |
| 31 | A♭ | G♭ | 6♭ | A♭ | B♭ | C♭ | D♭ | E♭ | F♮ | G♭ | A♭ | |
| 31 | G♯ | F♯ | 6♯ | G♯ | A♯ | B♮ | C♯ | D♯ | E♯ | F♯ | G♯ | |
| 32 | G♯ | B | 5♯ | G♯ | A♯ | B♮ | C♯ | D♯ | E♮ | F♯ | G♯ | |
| 33 | G♯ | E | 4♯ | G♯ | A♮ | B♮ | C♯ | D♯ | E♮ | F♯ | G♯ | |
| 34 | G♯ | A | 3♯ | G♯ | A♮ | B♮ | C♯ | D♮ | E♮ | F♯ | G♯ | |
| G | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 35 | G♮ | D | 2♯ | G♮ | A♮ | B♮ | C♯ | D♮ | E♮ | F♯ | G♮ | |
| 36 | G♮ | G | 1♯ | G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | F♯ | G♮ | |
| 37 | G♮ | C | ♮ | G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | F♮ | G♮ | |
| 38 | G♮ | F | 1♭ | G♮ | A♮ | B♭ | C♮ | D♮ | E♮ | F♮ | G♮ | |
| 39 | G♮ | B♭ | 2♭ | G♮ | A♮ | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | |
| 40 | G♮ | E♭ | 3♭ | G♮ | A♭ | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | |
| 41 | G♮ | A♭ | 4♭ | G♮ | A♭ | B♭ | C♮ | D♭ | E♭ | F♮ | G♮ | |
| F♯ / G♭ | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 42 | G♭ | D♭ | 5♭ | G♭ | A♭ | B♭ | C♮ | D♭ | E♭ | F♮ | G♭ | |
| 43 | G♭ | G♭ | 6♭ | G♭ | A♭ | B♭ | C♭ | D♭ | E♭ | F♮ | G♭ | |
| 43 | F♯ | F♯ | 6♯ | F♯ | G♯ | A♯ | B♮ | C♯ | D♯ | E♯ | F♯ | |
| 44 | F♯ | B | 5♯ | F♯ | G♯ | A♯ | B♮ | C♯ | D♯ | E♮ | F♯ | |
| 45 | F♯ | E | 4♯ | F♯ | G♯ | A♮ | B♮ | C♯ | D♯ | E♮ | F♯ | |
| 46 | F♯ | A | 3♯ | F♯ | G♯ | A♮ | B♮ | C♯ | D♮ | E♮ | F♯ | |
| 47 | F♯ | D | 2♯ | F♯ | G♮ | A♮ | B♮ | C♯ | D♮ | E♮ | F♯ | |
| 48 | F♯ | G | 1♯ | F♯ | G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | F♯ | |
| F (E♯) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 49 | F♮ | C | ♮ | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | F♮ | |
| 50 | F♮ | F | 1♭ | F♮ | G♮ | A♮ | B♭ | C♮ | D♮ | E♮ | F♮ | |
| 51 | F♮ | B♭ | 2♭ | F♮ | G♮ | A♮ | B♭ | C♮ | D♮ | E♭ | F♮ | |
| 52 | F♮ | E♭ | 3♭ | F♮ | G♮ | A♭ | B♭ | C♮ | D♮ | E♭ | F♮ | |
| 53 | F♮ | A♭ | 4♭ | F♮ | G♮ | A♭ | B♭ | C♮ | D♭ | E♭ | F♮ | |
| 54 | F♮ | D♭ | 5♭ | F♮ | G♭ | A♭ | B♭ | C♮ | D♭ | E♭ | F♮ | |
| 55 | F♮ | G♭ | 6♭ | F♮ | G♭ | A♭ | B♭ | C♭ | D♭ | E♭ | F♮ | |
| 55 | E♯ | F♯ | 6♯ | E♯ | F♯ | G♯ | A♯ | B♮ | C♯ | D♯ | E♯ | |
| E (F♭) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 56 | E♮ | B | 5♯ | E♮ | F♯ | G♯ | A♯ | B♮ | C♯ | D♯ | E♮ | |
| 57 | E♮ | E | 4♯ | E♮ | F♯ | G♯ | A♮ | B♮ | C♯ | D♯ | E♮ | |
| 58 | E♮ | A | 3♯ | E♮ | F♯ | G♯ | A♮ | B♮ | C♯ | D♮ | E♮ | |
| 59 | E♮ | D | 2♯ | E♮ | F♯ | G♮ | A♮ | B♮ | C♯ | D♮ | E♮ | |
| 60 | E♮ | G | 1♯ | E♮ | F♯ | G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | |
| 61 | E♮ | C | ♮ | E♮ | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | |
| 62 | E♮ | F | 1♭ | E♮ | F♮ | G♮ | A♮ | B♭ | C♮ | D♮ | E♮ | |
| D♯ / E♭ | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 63 | E♭ | B♭ | 2♭ | E♭ | F♮ | G♮ | A♮ | B♭ | C♮ | D♮ | E♭ | |
| 64 | E♭ | E♭ | 3♭ | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | D♮ | E♭ | |
| 65 | E♭ | A♭ | 4♭ | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | D♭ | E♭ | |
| 66 | E♭ | D♭ | 5♭ | E♭ | F♮ | G♭ | A♭ | B♭ | C♮ | D♭ | E♭ | |
| 67 | E♭ | G♭ | 6♭ | E♭ | F♮ | G♭ | A♭ | B♭ | C♭ | D♭ | E♭ | |
| 67 | D♯ | F♯ | 6♯ | D♯ | E♯ | F♯ | G♯ | A♯ | B♮ | C♯ | D♯ | |
| 68 | D♯ | B | 5♯ | D♯ | E♮ | F♯ | G♯ | A♯ | B♮ | C♯ | D♯ | |
| 69 | D♯ | E | 4♯ | D♯ | E♮ | F♯ | G♯ | A♮ | B♮ | C♯ | D♯ | |
| D | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 70 | D♮ | A | 3♯ | D♮ | E♮ | F♯ | G♯ | A♮ | B♮ | C♯ | D♮ | |
| 71 | D♮ | D | 2♯ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | C♯ | D♮ | |
| 72 | D♮ | G | 1♯ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | C♮ | D♮ | |
| 73 | D♮ | C | ♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | |
| 74 | D♮ | F | 1♭ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | C♮ | D♮ | |
| 75 | D♮ | B♭ | 2♭ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | C♮ | D♮ | |
| 76 | D♮ | E♭ | 3♭ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | D♮ | |
| C♯ / D♭ | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 77 | D♭ | A♭ | 4♭ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | D♭ | |
| 78 | D♭ | D♭ | 5♭ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | C♮ | D♭ | |
| 79 | D♭ | G♭ | 6♭ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | C♭ | D♭ | |
| 79 | C♯ | F♯ | 6♯ | C♯ | D♯ | E♯ | F♯ | G♯ | A♯ | B♮ | C♯ | |
| 80 | C♯ | B | 5♯ | C♯ | D♯ | E♮ | F♯ | G♯ | A♯ | B♮ | C♯ | |
| 81 | C♯ | E | 4♯ | C♯ | D♯ | E♮ | F♯ | G♯ | A♮ | B♮ | C♯ | |
| 82 | C♯ | A | 3♯ | C♯ | D♮ | E♮ | F♯ | G♯ | A♮ | B♮ | C♯ | |
| 83 | C♯ | D | 2♯ | C♯ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | C♯ | |
Key signature cheat sheet
- Twelve-tone equal temperament has fifteen key signatures, 𝄪/𝄫 atrocities notwithstanding:
| Key signatures of the Ionian scale’s seven modes in twelve-tone equal temperament | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Lyd | Maj | Mix | Dor | Min | Phr | Loc | KS | A | B | C | D | E | F | G |
| F♯ | C♯ | G♯ | D♯ | A♯ | E♯ | B♯ | 7♯ | ♯ | ♯ | ♯ | ♯ | ♯ | ♯ | ♯ |
| B | F♯ | C♯ | G♯ | D♯ | A♯ | E♯ | 6♯ | ♯ | ♯ | ♯ | ♯ | ♯ | ♯ | |
| E | B | F♯ | C♯ | G♯ | D♯ | A♯ | 5♯ | ♯ | ♯ | ♯ | ♯ | ♯ | ||
| A | E | B | F♯ | C♯ | G♯ | D♯ | 4♯ | ♯ | ♯ | ♯ | ♯ | |||
| D | A | E | B | F♯ | C♯ | G♯ | 3♯ | ♯ | ♯ | ♯ | ||||
| G | D | A | E | B | F♯ | C♯ | 2♯ | ♯ | ♯ | |||||
| C | G | D | A | E | B | F♯ | 1♯ | ♯ | ||||||
| F | C | G | D | A | E | B | ♮ | |||||||
| B♭ | F | C | G | D | A | E | 1♭ | ♭ | ||||||
| E♭ | B♭ | F | C | G | D | A | 2♭ | ♭ | ♭ | |||||
| A♭ | E♭ | B♭ | F | C | G | D | 3♭ | ♭ | ♭ | ♭ | ||||
| D♭ | A♭ | E♭ | B♭ | F | C | G | 4♭ | ♭ | ♭ | ♭ | ♭ | |||
| G♭ | D♭ | A♭ | E♭ | B♭ | F | C | 5♭ | ♭ | ♭ | ♭ | ♭ | ♭ | ||
| C♭ | G♭ | D♭ | A♭ | E♭ | B♭ | F | 6♭ | ♭ | ♭ | ♭ | ♭ | ♭ | ♭ | |
| F♭ | C♭ | G♭ | D♭ | A♭ | E♭ | B♭ | 7♭ | ♭ | ♭ | ♭ | ♭ | ♭ | ♭ | ♭ |
- Legend for the above:
- Lyd = Lydian
- Maj = major (Ionian)
- Mix = Mixolydian
- Dor = Dorian (if you’re not sure why it’s grey, ask your English teacher about the picture)
- Min = natural minor (Aeolian)
- Phr = Phrygian
- Loc = Locrian
- KS = key signature (highlighted because it’s the key to the table – pun coincidental, though I definitely didn’t even try to avoid it)
- Additional notes:
- 5♯ is enharmonically equivalent to 7♭.
- 6♯ is enharmonically equivalent to 6♭.
- 7♯ is enharmonically equivalent to 5♭.
- I’ve again printed C major’s modes in blue and F♯/G♭ major’s modes in orange.
- Accidentals also fall onto the circle of fifths (e.g., B♭ to E♭ and E♭ to A♭ are both perfect fifths).
Why is this happening?
Traversing the circle of fifths
Whether we realized it or not, we’ve been traversing the circle of fifths this entire time. My introduction notes that traveling from C Lydian to C Ionian is, in a sense, traveling from G major to C major. Here’s the C table again. Note Relative Major’s traversal down the circle of fifths:
| C++ | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 0 | C♮ | G | 1♯ | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | C♮ | |
| 1 | C♮ | C | ♮ | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | C♮ | |
| 2 | C♮ | F | 1♭ | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | C♮ | |
| 3 | C♮ | B♭ | 2♭ | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | C♮ | |
| 4 | C♮ | E♭ | 3♭ | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | |
| 5 | C♮ | A♭ | 4♭ | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | |
| 6 | C♮ | D♭ | 5♭ | C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | C♮ | |
Apart from C Ionian, these modes each rearrange different major scales, as we can see by reshuffling them back to Ionian:
| You were expecting modes, but it was me, | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Original Mode | Root | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 0 | G | 1♯ | G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | F♯ | G♮ | |
| 1 | C | ♮ | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | C♮ | |
| 2 | F | 1♭ | F♮ | G♮ | A♮ | B♭ | C♮ | D♮ | E♮ | F♮ | |
| 3 | B♭ | 2♭ | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | |
| 4 | E♭ | 3♭ | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | D♮ | E♭ | |
| 5 | A♭ | 4♭ | A♭ | B♭ | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | |
| 6 | D♭ | 5♭ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | C♮ | D♭ | |
Every note of every scale in this table is, in fact, a perfect fifth below its counterpart in its predecessor. That’s the first part of the explanation.
Scale generators: A brief introduction
The second part of the explanation explains why the modes’ circle of fifths order is what it is. Reminder: octave numbering starts at C, so the note above B0 is C1, which in turn is eleven semitones below B1. (I’m sure this must’ve been done purely to annoy people on the obsessive-compulsive spectrum. Other obsessive-compulsive programmers will understand what an atrocity this is without my having to explain it, and for everyone else, I don’t believe the English language possesses adequate expressiveness to explain.)
Anyhow, let’s go to F1 near the bottom of the piano. (The grand piano spans A0 to C8.) What’s a perfect fifth above that? C2. A perfect fifth above C2? G2. Move up another perfect fifth. We’re at D3. Up another perfect fifth. A3. Another perfect fifth. E4. Another. B4. So, to recap, we have the notes:
- F1
- C2
- G2
- D3
- A3
- E4
- B4
F, C, G, D, A, E, B: the fourth, first, fifth, second, sixth, third, and seventh degrees of the C major scale. There’s our circle of fifths order. The entire scale is literally just 7/12 of the circle of fifths, rearranged into linear order. The mode depends merely on which note in the sequence you use as the base:
- Lydian (in our example, F)
- Ionian (in our example, C)
- Mixolydian (in our example, G)
- Dorian (in our example, D)
- Aeolian (in our example, A)
- Phrygian (in our example, E)
- Locrian (in our example, B)
So, start on the desired mode’s note, put the other six in linear order, and voilà, there’s your scale.
The notes are deterministic: the lowest note in the sequence of perfect fifths determines the other six. The note selected as scale root, meanwhile, determines the mode. Not coincidentally, these correspond exactly to the key signature table: if we start our perfect fifths on C, then a root of C yields Lydian, a root of G yields Ionian, a root of D yields Mixolydian, and so on. This is also why flats are ordered B♭, E♭, A♭, D♭, G♭, C♭, F♭, and sharps are ordered F♯, C♯, G♯, D♯, A♯, E♯, B♯.
It’s worth noting that we can also construct the major scale by starting on B0 and moving up a perfect fourth six times. In scale theory, a perfect fifth is the same pitch class as a perfect fourth, because a perfect fourth up and a perfect fifth down are the same note when we disregard octaves. (However, swapping the intervals will reverse the order of the modes’ correspondence to the interval stack, thus putting Locrian’s root on the bottom and Lydian’s on top. This holds true for the same reason that making the notes into a scale rearranges them to occur within the same octave: a perfect fourth up equates to a perfect fifth down.)
In short, the diatonic major scale consists of seven stacked and flattened perfect fifths. This is known as a scale generator. Only a small fraction of scales have these, and only two mutate in exactly the same way as diatonic major. One is the pentatonic scale (as we will see in §5.1-8), which consists of four stacked perfect fifths or perfect fourths; the other is the hendecatonic scale (§5.9.1), which can be expressed as ten stacked perfect fourths or perfect fiths (or minor seconds or major sevenths, for that matter).
No other scale sizes can mutate exactly like this: the one-note scale has no modes to mutate into, and no other numbers are coprime with 12 (the parent tonality): that is, they share no prime factors with 12. We’ll examine why the length must be coprime with 12 below (§5.9.8). That said, the heptatonic chromatic scale, consisting of six stacked minor seconds (or major sevenths), exhibits similar behavior to diatonic major in several important ways, which we’ll also explore below (§5.9.2).
Only minor seconds, perfect fourths, perfect fifths, and major sevenths work as heptatonic scale generators, for two reasons: the scale’s length, and (again) only 1, 5, 7, and 11 (those intervals’ sizes in semitones) being coprime with 12. Generators have upper size limits for intervals that aren’t coprime with 12. For instance, stacking six major thirds (a four-semitone interval) duplicates the first note twice and the others once, giving us not a heptatonic scale but a three-note scale consisting solely of an augmented chord.
Scale generators can produce the following scale sizes:
| All scale generators in twelve-tone equal temperament | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Interval(s) | # Notes | ||||||||||||
| Minor second | Major seventh | ||||||||||||
| Perfect fourth | Perfect fifth | ||||||||||||
| Major second | Minor seventh | ||||||||||||
| Minor third | Major sixth | ||||||||||||
| Major third | Minor sixth | ||||||||||||
| Tritone | |||||||||||||
| Octave | |||||||||||||
A few additional notes:
- Intervals on the same line are reflections that create identical scales from inverted pitch orders (e.g., C major’s perfect fourth generator goes from B up to F; its perfect fifth generator goes from F up to B).
- The perfect fourth and minor second generators create the same hendecatonic and dodecatonic scales for a simple reason: only one scale of each size exists in our tuning system.
Interval distribution analysis
The last part of the explanation has to do with interval distributions:
| Ionian interval spacing | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| 4 | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |
| 1 | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |
| 5 | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | |
| 2 | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |
| 6 | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |
| 3 | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |
| 7 | C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | |
A brief explanation of the above table format is in order, since you’ll be seeing it a lot. The first interval is the number of whole steps between the first and second notes. The second interval is the number of whole steps between the second and third notes. And so on, until the final interval, which, for an l-note scale, is the number of whole steps between notes l and l + 1. “How can an l-note scale have a note l + 1?”, I hear you object. Simple: A scale is a pattern that repeats every octave. Thus, note l + 1 is an octave above note 1. Note (2 × l) + 1 is two octaves above note 1. And so on. Such a scale’s interval i(l), expressed in whole steps, should always equal 6 minus the sum of intervals i(1) through i(l − 1); also, for n = the number of whole steps between notes 1 and l, it should equal 6 − n. This is a mathematical property of how scales work; if any scale’s intervals ever sum up to anything but six whole steps, I made a mistake.
The Ionian scale is virtually unique among 12-TET’s seven-note scales in that, for every mode of the scale, it is possible to swap two notes (or two consecutive intervals) and produce a different mode of the same scale, and it is possible to cycle through the entire chromatic scale by doing these transformations. The other seven-note scale that most unambiguously displays this trait is actually, for various reasons, its polar opposite in virtually every important way. In §5, particularly §5.9, I will go over more about why, precisely, this is. For now, the important point is that each step down the circle of fifths order swaps only one pair of consecutive intervals, and therefore moves only one note. If more intervals changed, the pattern would break.
Chord analysis by mode
| Chord tonalities by scale position & mode (linear order) | ||||||||
|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
| 1 | Maj | min | min | Maj | Maj | min | dim | |
| 2 | min | min | Maj | Maj | min | dim | Maj | |
| 3 | min | Maj | Maj | min | dim | Maj | min | |
| 4 | Maj | Maj | min | dim | Maj | min | min | |
| 5 | Maj | min | dim | Maj | min | min | Maj | |
| 6 | min | dim | Maj | min | min | Maj | Maj | |
| 7 | dim | Maj | min | min | Maj | Maj | min | |
| Chord tonalities by scale position & mode (circle of fifths order) | ||||||||
|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
| 4 | Maj | Maj | min | dim | Maj | min | min | |
| 1 | Maj | min | min | Maj | Maj | min | dim | |
| 5 | Maj | min | dim | Maj | min | min | Maj | |
| 2 | min | min | Maj | Maj | min | dim | Maj | |
| 6 | min | dim | Maj | min | min | Maj | Maj | |
| 3 | min | Maj | Maj | min | dim | Maj | min | |
| 7 | dim | Maj | min | min | Maj | Maj | min | |
- I’ve printed each mode’s root (I) subdominant (IV), and dominant (V) chords in bold – they’re especially important to establishing its tonality.
- I’ve printed the seventh chord in fainter text because it’s diminished in the major scale.
- Minor chords are blue; diminished chords are orange.
- This analysis clarifies why Ionian became the standard major scale and Aeolian became the natural minor scale: Ionian’s root, subdominant, and dominant are all major; Aeolian’s are all minor. This makes them feel, respectively, especially major and especially minor.
- Since their root chords are major, Lydian and Mixolydian still sound more “major” than the other modes (except Ionian), and Dorian and Phrygian still sound more “minor” than the others (except Aeolian). Locrian, as mentioned, is the oddball because it’s the only mode of the scale whose root chord is diminished.
C major’s parallel modes have the following chords:
Chords for C major’s parallel modes Mode 1 2 3 4 5 6 7 1 C maj D min E min F maj G maj A min B dim 2 C min D min E♭ maj F maj G min A dim B♭ maj 3 C min D♭ maj E♭ maj F min G dim A♭ maj B♭ min 4 C maj D maj E min F♯ dim G maj A min B min 5 C maj D min E dim F maj G min A min B♭ maj 6 C min D dim E♭ maj F min G min A♭ maj B♭ maj 7 C dim D♭ maj E♭ min F min G♭ maj A♭ maj B♭ min Its relative modes have the following chords:
Chords for C major’s relative modes Mode 1 2 3 4 5 6 7 1 C maj D min E min F maj G maj A min B dim 2 D min E min F maj G maj A min B dim C maj 3 E min F maj G maj A min B dim C maj D min 4 F maj G maj A min B dim C maj D min E min 5 G maj A min B dim C maj D min E min F maj 6 A min B dim C maj D min E min F maj G maj 7 B dim C maj D min E min F maj G maj A min
Beyond the Ionian scale
Other heptatonic scales & tonalities
While the above analysis focuses exclusively on the Ionian scale’s modes, numerous possible scales (and modes thereof) don’t fit its pattern, many of which I’ll now analyze. Some examples:
- Hundreds of possible heptatonic (seven-note) scales aren’t modes of Ionian. I’ve rooted the following examples (many of which are common in jazz) in C to clarify their differences:
Other heptatonic scales Mode 1 2 3 4 5 6 7 Intervals C♮ D♭ E𝄫 F♮ G♮ A♭ B𝄫 ½ ½ 1½ 1 ½ ½ 1½ C♮ D♭ E♭ F♭ G♭ A♭ B𝄫 ½ 1 ½ 1 1 ½ 1½ C♮ D♭ E♭ F♭ G♭ A♭ B♭ ½ 1 ½ 1 1 1 1 C♮ D♭ E♭ F♮ G♭ A♮ B♭ ½ 1 1 ½ 1½ ½ 1 C♮ D♭ E♭ F♮ G♮ A♮ B♭ ½ 1 1 1 1 ½ 1 C♮ D♭ E♮ F♮ G♮ A♭ B♭ ½ 1½ ½ 1 ½ 1 1 C♮ D♭ E♮ F♮ G♮ A♭ B♮ ½ 1½ ½ 1 ½ 1½ ½ C♮ D♮ E♭ F♮ G♭ A♭ B♭ 1 ½ 1 ½ 1 1 1 C♮ D♮ E♭ F♮ G♭ A♭ B♮ 1 ½ 1 ½ 1 1½ ½ C♮ D♮ E♭ F♮ G♮ A♭ B♮ 1 ½ 1 1 ½ 1½ ½ C♮ D♮ E♭ F♮ G♮ A♮ B♮ 1 ½ 1 1 1 1 ½ C♮ D♮ E♭ F♯ G♮ A♭ B♮ 1 ½ 1½ ½ ½ 1½ ½ C♮ D♮ E♭ F♯ G♮ A♮ B♭ 1 ½ 1½ ½ 1 ½ 1 C♮ D♮ E♮ F♮ G♮ A♭ B♭ 1 1 ½ 1 ½ 1 1 C♮ D♮ E♮ F♮ G♯ A♮ B♮ 1 1 ½ 1½ ½ 1 ½ C♮ D♮ E♮ F♯ G♮ A♮ B♭ 1 1 1 ½ 1 ½ 1 C♮ D♮ E♮ F♯ G♯ A♮ B♮ 1 1 1 1 ½ 1 ½ C♮ D♯ E♮ F♮ G♮ A♯ B♮ 1½ ½ ½ 1 1½ ½ ½ C♮ D♯ E♮ F♯ G♮ A♮ B♮ 1½ ½ 1 ½ 1 1 ½ C♮ D♯ E♯ F♯ G♮ A♯ B♮ 1½ 1 ½ ½ 1½ ½ ½ C♮ D♯ E♯ F♯ G♯ A♯ B♮ 1½ 1 ½ 1 1 ½ ½ Since many scales have up to dozens of names (if you’re on a computer, hover over dotted scale names for more information), I’ve instead used the sorting methods of Southern India’s , which effectively alphabetize the scales themselves: sort intervals in ascending order, with the lowest intervals counting as most significant. (§9.2 lists the mēḷakartā in full; note that they categorically exclude scales with sharp thirds or fifths, or flat fourths or fifths.) Bold, blue text denotes raised notes and minor thirds; thin, orange text denotes lowered notes and minor seconds (doubly lowered notes are also fainter).
This list is far from exhaustive. Note that none of these scales use B♯ because, while it’s semantically different from C, the two notes represent the same pitch in twelve-tone equal temperament. Likewise, none of these scales use D𝄫 because it’s the same pitch as C.
In any case, one thing distinguishes these scales from the Ionian scale’s modes: we can print the latter with no accidentals outside the key signature, but every single scale in the above table requires accidentals for at least some notes. Key signatures are strictly based on the Ionian scale and the circle of fifths; using three flats that weren’t B♭, E♭, and A♭ in a key signature would just confuse readers.
The names “Chromatic Dorian” and “Chromatic Hypophrygian” will reappear below in the section on ancient Greek harmony below. (Don’t waste time trying to discern their connections to the similarly named diatonic modes: Chromatic Dorian actually relates to Phrygian, and Chromatic Hypophrygian to Mixolydian. The section on Greek harmony explains the relationships, and why they aren’t the ones their names suggest.)
-
Note that several sets of the above scales are modes of each other. In particular:
Harmonic minor & melodic minor’s modes at a glance Mode 1 2 3 4 5 6 7 Intervals C♮ D♮ E♭ F♮ G♮ A♭ B♮ 1 ½ 1 1 ½ 1½ ½ C♮ D♭ E♭ F♮ G♭ A♮ B♭ ½ 1 1 ½ 1½ ½ 1 C♮ D♮ E♮ F♮ G♯ A♮ B♮ 1 1 ½ 1½ ½ 1 ½ C♮ D♮ E♭ F♯ G♮ A♮ B♭ 1 ½ 1½ ½ 1 ½ 1 C♮ D♭ E♮ F♮ G♮ A♭ B♭ ½ 1½ ½ 1 ½ 1 1 C♮ D♯ E♮ F♯ G♮ A♮ B♮ 1½ ½ 1 ½ 1 1 ½ C♮ D♭ E♭ F♭ G♭ A♭ B𝄫 ½ 1 ½ 1 1 ½ 1½ C♮ D♮ E♭ F♮ G♮ A♮ B♮ 1 ½ 1 1 1 1 ½ C♮ D♭ E♭ F♮ G♮ A♮ B♭ ½ 1 1 1 1 ½ 1 C♮ D♮ E♮ F♯ G♯ A♮ B♮ 1 1 1 1 ½ 1 ½ C♮ D♮ E♮ F♯ G♮ A♮ B♭ 1 1 1 ½ 1 ½ 1 C♮ D♮ E♮ F♮ G♮ A♭ B♭ 1 1 ½ 1 ½ 1 1 C♮ D♮ E♭ F♮ G♭ A♭ B♭ 1 ½ 1 ½ 1 1 1 C♮ D♭ E♭ F♭ G♭ A♭ B♭ ½ 1 ½ 1 1 1 1 We can apply similar principles to them as we applied to the Ionian scale’s modes, though certain things may not line up as neatly with some of the scales. I leave ascertaining which other sets of scales are modes of each other as an exercise for the reader. (I didn’t list every mode of most other sets of scales.) The next two sections analyze these modes in much greater depth.
Note that two competing shorthands exist for transformations like “Aeolian ♯7”, relative to either:
- The current note: “♯7” always means “raise the base scale’s seventh note half a step”. If it’s currently A♭, make it A♮. If it’s currently A, make it A♯. If it’s currently C♯, make it (ugh) C𝄪 (and Jesus wept, for there were no more worlds to conquer… wait, that’s not how it goes).
- The C scale’s accidental: C Aeolian 7 is B♭, so ‘Aeolian ♮7’ = ‘raise Aeolian’s seventh note half a step’.
Not gonna front: I despise the latter with the burning passion of a thousand suns. “♯7” applies independently of the root, while in any key except C, “♮7” requires asking “wait, what accidental is on C Aeolian 7?” C♯ Aeolian 7 is already B♮, so what does “C♯ Aeolian ♮7” even tell us? F♯ Aeolian 7 is C♯, so if we read “F♯ Aeolian ♮7” literally, we’d lower the seventh note by a half-step – exactly the opposite of what it intends!
Then again, “Lydian ♮4” (“lower Lydian’s fourth note a half-step”) and “Mixolydian ♮7” (“raise Mixolydian’s seventh note a half-step”) make for the most pretentious possible ways to say “major” this side of «διατονικός Λῡ́δῐος τόνος» (“diatonic Lydian tonos”; see the section on Ancient Greek harmony below), so I guess there is one actual upside to the C-based convention.
Beyond pentatonic and heptatonic scales
Of course, scales needn’t contain seven notes. Pentatonic (five-note) scales are so complex to unpack that they’ll need their own section (mostly because both five and seven are coprime with twelve: neither share any prime factors with it). Here’s a brief overview of other scale sizes in ascending order:
- Hexatonic (six-note) scales are fairly common. Likely the most familiar example: the whole-tone scale.
- Scale size also isn’t limited to seven; octatonic (eight-note) and enneatonic (nine-note) scales occur in blues and jazz often. Decatonic (ten-note) scales are slightly rarer, but still occur.
- ⟨en
.wikipedia .org /wiki /12_ equal_ temperament⟩ only allows for one possible hendecatonic (eleven-note) scale, though it does have eleven separate modes. Do you understand why? (Hint: There’s only a single dodecatonic scale in 12-TET for exactly the same reason.) - The dodecatonic (twelve-note) chromatic scale. (Works that use the entire twelve-note scale for non-colorative purposes are quite rare and are mostly considered avant-garde.)
So far, we’ve exclusively been considering 12-TET, but of course, plenty of other tunings have been and still are used; nothing even constrains octaves to twelve notes. For instance:
- A ⟨en
.wikipedia .org /wiki /Quarter_ tone⟩ is common in Arabic and Turkic music. - Indian ⟨en
.wikipedia .org /wiki /Raga⟩ (similar to scales) typically use four to seven svaras (similar to scale tones) of the ⟨en .wikipedia .org /wiki /Shruti_ (music)⟩ (which collectively aren’t exactly 22-TET, but not entirely unlike it). - Experimental Western musicians also occasionally use microtonal tuning. A few recent rock and metal examples, in increasing order of heaviness:
- ⟨kinggizzard
.bandcamp .com /album /flying-microtonal-banana⟩ by King Gizzard & the Lizard Wizard (24-TET) - ⟨votsband
.bandcamp .com /album /nowherer⟩ by Victory over the Sun (17-TET) - Every metal album by ⟨jutegyte
.bandcamp .com /music⟩ from Discontinuities onward (24-TET)
- ⟨kinggizzard
- Ancient Greece’s enharmonic genus also used microtonality, as we’ll see below. (Note: in different contexts, enharmonic varies wildly in meaning. In 12-TET, two enharmonic notes have the same pitch. In microtonal tunings, they’re about as dissonant as you can get.)
For the record:
| Ἐπίθετᾰ πρός ᾰ̓ρῐθμούς τόνων Epíthetă prós ărĭthmoús tónōn Adjectives for numbers of notes |
|||||
|---|---|---|---|---|---|
| # | Ἐπίθετον Epítheton Adjective | Ἑλληνική «–τονος» Hellēnĭke «–tonos» Greek “–toned” | Ῥωμᾰῐ̈σμένη Rhṓmăĭ̈sméni Romanized | Ἑλληνική «–τονικός» Hellēnĭke «–tonikós» Greek “–tonic” | Ῥωμᾰῐ̈σμένη Rhṓmăĭ̈sméni Romanized |
| 1 | monotonic | μονότονος | monótonos | μονότονικός | monótonikós |
| 2 | διατονος | diatonos | διατονικός | diatonikós | |
| 3 | τρίτονος | trítonos | τρίτονικός | trítonikos | |
| 4 | tetratonic | τετράτονος | tetrátonos | τετράτονικός | tetrátonikós |
| 5 | pentatonic | πέντατονος | péntatonos | πέντατονικός | péntatonikós |
| 6 | hexatonic | ἑξατονος | hexatonos | ἑξατονικός | hexatonikós |
| 7 | heptatonic | ἑπτάτονος | heptátonos | ἑπτάτονικός | heptátonikós |
| 8 | octatonic | ὀκτάτονος | oktátonos | ὀκτάτονικός | oktátonikós |
| 9 | enneatonic | ἐννεάτονος | enneátonos | ἐννεάτονικός | enneátonikós |
| 10 | decatonic | δέκατονος | dékatonos | δέκατονικός | dékatonikós |
| 11 | hendecatonic | ἕνδεκάτονος | hendekátonos | ἕνδεκάτονικός | hendekátonikós |
| 12 | dodecatonic | δωδεκάτονος | dōdekátonos | δωδεκάτονικός | dōdekátonikós |
The suffix -tonic is Greek. Friends don’t let friends mix Latin prefixes and Greek suffixes. (Unless Latin already did so, that is. It did borrow τόνος as tonus, but only as a noun, never an adjective; it did not borrow τονικός.)
I only managed to find attestations of some of the Greek forms in this list, but it seems likely they all must have existed at some time. The ones I found are in bold; the ones I was unable to find are in fainter text.
Bolded English words, meanwhile, have attested usages for scale size in music theory contexts. Monotonic, diatonic, and tritonic are printed more faintly because they have completely different meanings that have nothing to do with the number of pitches in a scale, so using them to mean that will likely just confuse readers. The latter two are also struck through because their alternate meanings are ubiquitous in music theory contexts. You might be technically correct to use them to refer to scale size, but is that really the hill you want to die on?
- Monotonic mostly refers to the modern Greek accent system (cf. the old polytonic system with markers for word pitch and breathing). It has no widely established meaning in music theory, so this case is less clear-cut than the others.
- Diatonic means of two interval sizes. In English and Ancient Greek alike, it most often means what became our diatonic major scale: it’s at least 2,500 years old.
- Tritonic means spanning an interval of three whole tones, i.e., a tritone.
The pentatonic scale
As complement of Ionian
Pentatonic (“5t” for short) scales are ubiquitous in rock, blues, and jazz, though they’re much older than that and exist in many cultures. The most common pentatonic scale is literally Ionian’s scale complement.
What exactly is a scale complement? It’s the equivalent of a binary XOR. Say we represent a scale as a set of twelve 1s (“this tone is part of the scale”) or 0s (“this tone is not part of the scale”). Now, flip all the bits. Tones that had notes are no longer part of the scale; tones that didn’t now are. Since scales must start with a note, we now must rotate the scale so that the first bit is a 1. Once we’ve done so, we have a mode of the complement.
This means we can play the pentatonic scale using all the piano keys we didn’t use to play Ionian. Whenever I write “the pentatonic scale”, preceded by the definite article, I mean this pentatonic scale. To wit:
- C Ionian only uses the white keys on the piano: C, D, E, F, G, A, B, C.
- F♯ pentatonic major only uses the black keys: F♯, G♯, A♯, C♯, D♯ (or G♭, A♭, B♭, D♭, E♭, if you
’re nastyprefer flats).
To a huge extent, the pentatonic and Ionian scales’ relationship even extends to their modes. For instance:
- Starting C Ionian three semitones early creates its relative minor, A Aeolian.
- Starting F♯ pentatonic major three semitones early creates its relative minor, D♯/E♭ pentatonic minor.
I must clarify, however, that scale complements apply on a scale-wide basis, not on a modal basis. That is, because the complement of any non-hexatonic scale in 12-TET will have a different number of notes, its complement will also have a different number of modes (barring a few exceptions, known as modes of limited transposition, which are covered in §7’s discussion of symmetry). As a result, it’s not possible to make 1:1 comparisons between scale complements’ modes. I feel the need to emphasize this because we’re to compare the pentatonic and Ionian scales through a second lens, and 1:1 comparisons do apply through the second one.
As truncation of Ionian
Conveniently, though, it’s not just a complement, though: it’s also a truncation. We can get the pentatonic scale simply by deleting two notes of Ionian. As a result, the two scales’ modes correspond in countless ways.
Since the pentatonic scale has two fewer notes than Ionian, our analysis must delete two modes. But which two? We can derive the pentatonic scale from Ionian in at least three different ways.
Quick warning before we proceed further: we’re taking a quick detour into “right for the wrong reasons” land. After the third table, I’ll explain how, why, and where the first two tables go wrong.
Let’s try disregarding Phrygian and Locrian, the lowest modes in the circle of fifths progression. In this analysis:
- deletes ’s fourth and seventh notes: F♯ G♯ A♯ C♯ D♯ (1, 1, 1½, 1, 1½).
- deletes ’s third and seventh notes: C♯ D♯ F♯ G♯ A♯ (1, 1½, 1, 1, 1½).
- deletes ’s third and sixth notes: G♯ A♯ C♯ D♯ F♯ (1, 1½, 1, 1½, 1). Neutral pentatonic is comparable to Dorian mode in two ways: it is a symmetrical scale, and it’s the midpoint of the pentatonic circle of fifths order (which is complex enough to merit its own section below).
- deletes ’s second and sixth notes: D♯ F♯ G♯ A♯ C♯ (1½, 1, 1, 1½, 1).
- deletes ’s second and fifth notes: A♯ C♯ D♯ F♯ G♯ (1½, 1, 1½, 1, 1).
Got all that? Let’s recap. (Note: “H” = half-tone, ”W” = whole tone, “M” = minor third)
| The incorrect “upshift” hypothesis | ||||||
|---|---|---|---|---|---|---|
| [7t] Mode [5t] | [7t] Piano keys [5t] | [7t] Intervals [5t] | Del | |||
| F♮G♮A♮B♮C♮D♮E | F♯ G♯ A♯ C♯ D♯ | W W W H W W H | W W M W M | 4 7 | ||
| C♮D♮E♮F♮G♮A♮B | C♯ D♯ F♯ G♯ A♯ | W W H W W W H | W M W W M | 3 7 | ||
| G♮A♮B♮C♮D♮E♮F | G♯ A♯ C♯ D♯ F♯ | W W H W W H W | W M W M W | 3 6 | ||
| D♮E♮F♮G♮A♮B♮C | D♯ F♯ G♯ A♯ C♯ | W H W W W H W | M W W M W | 2 6 | ||
| A♮B♮C♮D♮E♮F♮G | A♯ C♯ D♯ F♯ G♯ | W H W W H W W | M W M W W | 2 5 | ||
I probably don’t even need to point out how many patterns recur in both scales.
We just analyzed the pentatonic modes based on notes a half-step above them, but we could just as easily have used the notes a half-step above. This means instead disregarding Lydian and Ionian. Oddly enough, we delete the same scale degrees either way:
| The incorrect “downshift” hypothesis | ||||||
|---|---|---|---|---|---|---|
| [7t] Mode [5t] | [7t] Piano keys [5t] | [7t] Intervals [5t] | Del | |||
| G♮A♮B♮C♮D♮E♮F | G♭ A♭ B♭ D♭ E♭ | W W H W W H W | W W M W M | 4 7 | ||
| D♮E♮F♮G♮A♮B♮C | D♭ E♭ G♭ A♭ B♭ | W H W W W H W | W M W W M | 3 7 | ||
| A♮B♮C♮D♮E♮F♮G | A♭ B♭ D♭ E♭ G♭ | W H W W H W W | W M W M W | 3 6 | ||
| E♮F♮G♮A♮B♮C♮D | E♭ G♭ A♭ B♭ D♭ | H W W W H W W | M W W M W | 2 6 | ||
| B♮C♮D♮E♮F♮G♮A | B♭ D♭ E♭ G♭ A♭ | H W W H W W W | M W M W W | 2 5 | ||
(End warning.)
We have to go deeper (Scaleception)
But wait, there’s more! Consulting the heptatonic circle of fifths progression enables us to “average” the above two tables. Eerily, averaging them doesn’t invalidate their results in any way except a trivial one: this is the first analysis that’s actually correct. But, as the famed philosopher Nigel Tufnel put it, that’s nitpicking, innit?
| How pentatonic transforms Ionian: The correct “tritone deletion” explanation | ||||||
|---|---|---|---|---|---|---|
| [7t] Mode [5t] | [7t] Piano keys [5t] | [7t] Intervals [5t] | Del | |||
| C♮D♮E♮F♮G♮A♮B | C♮D♮E♮G♮A | W W H W W W H | W W M W M | 4 7 | ||
| G♮A♮B♮C♮D♮E♮F | G♮A♮C♮D♮E | W W H W W H W | W M W W M | 3 7 | ||
| D♮E♮F♮G♮A♮B♮C | D♮E♮G♮A♮C | W H W W W H W | W M W M W | 3 6 | ||
| A♮B♮C♮D♮E♮F♮G | A♮C♮D♮E♮G | W H W W H W W | M W W M W | 2 6 | ||
| E♮F♮G♮A♮B♮C♮D | E♮G♮A♮C♮D | H W W W H W W | M W M W W | 2 5 | ||
So, to reiterate: Only the tritone deletion table is completely accurate; its two predecessors are filthy half-truths, owing to a mathematical pattern they don’t account for. Can you figure out why?
-
If both scales have symmetrical modes (Dorian and neutral pentatonic), then:
- We should compare the symmetrical modes to each other.
- Symmetrical modes should be their circle-order comparisons’ center data rows.
- Rows whose intervals are mirrors in the base should remain mirrors in the transformation.
- Both circle-order interval comparisons should possess 180° rotational symmetry.
All of these are false in the first two tables and true in the third.
- Since we didn’t delete the root, we must compare the base scales. Only table three does so. Not comparing the base scales is a surefire recipe for confusion.
- If we delete a note, we must delete its mode. We deleted notes four and seven. Only the third table deletes both notes’ modes (Lydian and Locrian).
-
Our analysis must compare the same notes within each scale. We didn’t move notes, only remove them, so our analysis can’t either. Examining the first table closely reveals why this is a problem: Mixolydian four, Dorian seven, Aeolian three, Phrygian six, and Locrian two are Ionian’s root!
The problem is less obvious in the second table, but it tells us to remove Ionian’s third note. Major pentatonic, like Ionian, opens with two whole steps, so it hasn’t removed Ionian’s third note! This is why we should only shift notes in our analysis if we delete the root. (And if so, good luck – you’ll need it.)
“Tritone Deletion”, “Tritone Substitution”, or “Tritone Shift” could all fit for the third table. The notes it removes correspond exactly to the Ionian scale’s sole tritone; it also lists the pentatonic scales exactly a tritone from where its two predecessors had them. I chose “Tritone Deletion” in the end because it’s a more accurate description of what we’re actually doing, and the difference in pitch is a direct consequence of removing the tritone instead of removing other notes, then shifting the scale.
Since pentatonic’s mode nomenclature isn’t as well established as Ionian’s, my brain’s cutesy part wants to rename them Nianoi, Niadyloxim, Niarod, Nialoea, and Niagyrhp. I’m afraid that even after the above explanation, that might confuse people, but that won’t stop me from using them as alternate names.
Analysis of modes in linear order
Viewing the modes in linear order, with the Ionian scale’s missing modes included, may help further clarify why the “upshift” and “downshift” tables are wrong:
| How pentatonic transforms Ionian: The correct “root note” explanation | ||||||
|---|---|---|---|---|---|---|
| [7t] Mode [5t] | [7t] Piano keys [5t] | [7t] Intervals [5t] | Del | |||
| C♮D♮E♮F♮G♮A♮B | C♮D♮E♮G♮A | W W H W W W H | W W M W M | 7 4 | ||
| D♮E♮F♮G♮A♮B♮C | D♮E♮G♮A♮C | W H W W W H W | W M W M W | 6 3 | ||
| E♮F♮G♮A♮B♮C♮D | E♮G♮A♮C♮D | H W W W H W W | M W M W W | 5 2 | ||
| F♮G♮A♮B♮C♮D♮E | W W W H W W H | 4 1 | ||||
| G♮A♮B♮C♮D♮E♮F | G♮A♮C♮D♮E | W W H W W H W | W M W W M | 3 7 | ||
| A♮B♮C♮D♮E♮F♮G | A♮C♮D♮E♮G | W H W W H W W | M W W M W | 2 6 | ||
| B♮C♮D♮E♮F♮G♮A | H W W H W W W | 1 5 | ||||
(Note: I reordered a few “deleted notes” entries in this table to clarify its pattern.)
Modes and roots are inextricably linked. We must compare the same notes in each scale:
- We didn’t delete this note; its mode, , becomes .
- We didn’t delete this note; its mode, , becomes .
- We didn’t delete this note; its mode, , becomes .
- Deleting this note deletes its mode; has no pentatonic equivalent.
- We didn’t delete this note; its mode, , becomes .
- We didn’t delete this note; its mode, , becomes .
- Deleting this note deletes its mode; has no pentatonic equivalent.
Put another way, recall how Ionian’s modes got their numbering:
- starts on its first note.
- starts on its second note.
- starts on its third note.
- starts on its fourth note.
- starts on its fifth note.
- starts on its sixth note.
- starts on its seventh note.
So, applying the same principle to the pentatonic scale:
- starts on its first note.
- starts on its second note.
- starts on its third note.
- starts on its fourth note.
- starts on its fifth note.
Comparison of interval spacing
Why pentatonic and Ionian’s intervals are out of sync may not be obvious. Let’s revisit the above table, this time with deleted tones in red and combined intervals in purple:
| How pentatonic transforms Ionian: Interval analysis (circle order) | ||||||
|---|---|---|---|---|---|---|
| [7t] Mode [5t] | [7t] Piano keys [5t] | [7t] Intervals [5t] | Del | |||
| F♮G♮A♮B♮C♮D♮E | W W W H W W H | 4 1 | ||||
| C♮D♮E♮F♮G♮A♮B | C♮D♮E♮G♮A | W W H W W W H | W W M W M | 4 7 | ||
| G♮A♮B♮C♮D♮E♮F | G♮A♮C♮D♮E | W W H W W H W | W M W W M | 3 7 | ||
| D♮E♮F♮G♮A♮B♮C | D♮E♮G♮A♮C | W H W W W H W | W M W M W | 3 6 | ||
| A♮B♮C♮D♮E♮F♮G | A♮C♮D♮E♮G | W H W W H W W | M W W M W | 2 6 | ||
| E♮F♮G♮A♮B♮C♮D | E♮G♮A♮C♮D | H W W W H W W | M W M W W | 2 5 | ||
| B♮C♮D♮E♮F♮G♮A | H W W H W W W | 1 5 | ||||
It may help to emphasize that we aren’t deleting intervals; we’re deleting notes and combining intervals. For instance, deleting a scale’s second note combines its first two intervals. Furthermore:
- Ionian has only semitones and whole tones
-
We only delete notes that:
- follow semitones and precede whole tones
- follow whole tones and precede semitones
Thus, the two intervals around every deleted note turn become a single minor third aligning exactly to the pattern of deletions. Since this pattern is out of phase with the original one, it changes, but if the deleted notes were surrounded by different intervals, the new interval pattern wouldn’t map so precisely to the deletions.
Interestingly, that relationship may be less obvious in linear order, even when highlighted:
| How pentatonic transforms Ionian: Interval analysis (linear order) | ||||||
|---|---|---|---|---|---|---|
| [7t] Mode [5t] | [7t] Piano keys [5t] | [7t] Intervals [5t] | Del | |||
| C♮D♮E♮F♮G♮A♮B | C♮D♮E♮G♮A | W W H W W W H | W W M W M | 7 4 | ||
| D♮E♮F♮G♮A♮B♮C | D♮E♮G♮A♮C | W H W W W H W | W M W M W | 6 3 | ||
| E♮F♮G♮A♮B♮C♮D | E♮G♮A♮C♮D | H W W W H W W | M W M W W | 5 2 | ||
| F♮G♮A♮B♮C♮D♮E | W W W H W W H | 4 1 | ||||
| G♮A♮B♮C♮D♮E♮F | G♮A♮C♮D♮E | W W H W W H W | W M W W M | 3 7 | ||
| A♮B♮C♮D♮E♮F♮G | A♮C♮D♮E♮G | W H W W H W W | M W W M W | 2 6 | ||
| B♮C♮D♮E♮F♮G♮A | H W W H W W W | 1 5 | ||||
So, let’s correct our original analysis, shall we?
- deletes ’s fourth and seventh notes: C D E G A (1, 1, 1½, 1, 1½). It’s the root form of the scale and the pentatonic circle of fourths’ lowest mode.
- deletes ’s third and seventh notes: G A C D E (1, 1½, 1, 1, 1½). It’s the pentatonic circle of fourths’ second-lowest mode.
- deletes ’s third and sixth notes: D E G A C (1, 1½, 1, 1½, 1). Like Dorian, it’s symmetrical and the midpoint of its own circle of fourths.
- deletes ’s second and sixth notes: A C D E G (1½, 1, 1, 1½, 1). It’s the pentatonic circle of fourths’ second-highest mode.
- deletes ’s second and fifth notes: E G A C D (1½, 1, 1½, 1, 1). It’s the pentatonic circle of fourths’ highest mode.
Oh, right. I haven’t explained why I call it a circle of fourths, which in turn means I need to explain scale rotation.
A brief explanation of scale rotation
Scale rotation is the practice of forming a mode by moving intervals of a scale from its start to its end, or vice versa. When I refer to moving a scale’s intervals “left”, I’m referring both to the piano keyboard and to the scale interval tables I use. Rotating a scale left means moving most of its intervals to a lower point in the scale. Since scales repeat every octave, the rest move to the top.
“It might not always be ‘most’,” I hear you object. True, you could rotate a seven-note scale six degrees to the right, but why would you, when that’s the same as rotating it one degree to the left?
…Which means I need to explain rotation by degrees, too. No, we’re not talking angles here. Rotating a scale n degrees left moves its first n intervals to the end; rotating it n degrees right moves its last n intervals to the start. Dorian is one degree left of Ionian; Lydian is three degrees left of Ionian. And so on.
I may also refer to rotation by semitones. Rotating a scale by five semitones means the intervals moved sum up to five semitones. Thus, rotating Ionian five semitones to the left also takes you to Lydian.
A scale rotation’s size, measured either by degrees or by interval sum, has nothing to do with how far it moves the scale’s notes. Lydian is a five-semitone leftward rotation from Ionian, but it only moves one note (the fourth degree of the scale) by a semitone (F to F♯, when rooted on C).
I may also refer to scale rotations by how many notes they move. A single-note rotation only moves one note. This does not signify anything about how many intervals it moves forward or backward, the size of those intervals, or even about the interval by which the note is moved.
So, to summarize: Ionian to Lydian is a single-note rotation; it moves the note by one semitone, but it rotates the scale by five semitones (and three degrees).
One final note: A parallel rotation preserves the original mode’s root while changing its interval order and note composition; a relative rotation preserves the original mode’s key signature while changing its interval order and root. With rare exceptions (modes of limited transposition, discussed in §7.1), rotation by anything other than exact multiples of an octave cannot preseve both the root and the note composition.
I’ll try to keep this terminology from being ambiguous, but words are an imperfect medium for discussing music at the best of times, and when we throw mathematics, geometry, and set theory into the mix, forget it. If anything feels confusingly worded, please let me know, and I’ll try to clarify.
The pentatonic circle of fourths, or, contrary motion explained
The pentatonic and Ionian circle of fifths orders move their parent scales in exact opposite directions. In fact, to emphasize this, I’m not even gonna call it the pentatonic circle of fifths anymore. I’ma call it monkeydude Josh the pentatonic circle of fourths. I find this fitting for at least two reasons:
- In scale analysis, octaves don’t matter: we get to the same note whether we go a perfect fifth down or a perfect fourth up. Altering the nomenclature helps call attention to their contrary directions.
- I’ve analyzed heptatonic scales through a seven-semitone lens. Analyzing pentatonic scales through anything but a five-semitone lens would feel wrong.
But why do they move in opposite directions? It’s probably easiest to analyze in terms of interval spacing.
Pentatonic changes two of Ionian’s interval pairs from “tone, semitone” to “minor third”. It so happens that one of Ionian’s two semitones closes out the scale. Thus, compare what happens when we shift pentatonic major’s intervals to the left to what happens when we shift Ionian’s.
| Pentatonic interval spacing (root order) | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Mode | 1 | 2 | 3 | 4 | 5 | Intervals | |||||||
| 1 | C♮ | D♮ | E♮ | G♮ | A♮ | 1 | 1 | 1½ | 1 | 1½ | ||||
| 2 | C♮ | D♮ | F♮ | G♮ | A♯ | 1 | 1½ | 1 | 1½ | 1 | ||||
| 3 | C♮ | D♯ | F♮ | G♯ | A♯ | 1½ | 1 | 1½ | 1 | 1 | ||||
| 4 | C♮ | D♮ | F♮ | G♮ | A♮ | 1 | 1½ | 1 | 1 | 1½ | ||||
| 5 | C♮ | D♯ | F♮ | G♮ | A♯ | 1½ | 1 | 1 | 1½ | 1 | ||||
| Ionian interval spacing (root order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| 1 | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |
| 2 | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |
| 3 | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |
| 4 | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |
| 5 | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | |
| 6 | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |
| 7 | C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | |
That’s a lot to unpack. Some analysis:
Both scales contain a repeated interval set (called a tetrachord in Ionian and a trichord in pentatonic) and a spare whole step, or synaphe. I’ve marked one possible reading within the interval list. (Two such readings exist of pentatonic, and three of Ionian; in this example, I avoided splitting tetrachords or trichords in the base scale.)
Synaphe (plural: synaphai or synaphes) comes from the Attic Greek word σῠνᾰφή (sŭnăphḗ, literally connection, union, junction; point or line of junction; conjunction of two tetrachords). Its Attic pronunciation was roughly suh-nup-HEY pre-φ shift and suh-nuh-FAY (so, basically how a drunk person would say Santa Fe) afterward, but I think English speakers, mistakenly assuming it to be French, might say sy-NAFF.
(Pro tip: If a word contains ph and doesn’t split it across two syllables, it’s almost always transliterated Greek. Also, pronouncing foreign words using the wrong orthography is a great way to make a linguistics nerd’s blood boil. Speaking of which, orthography descends from ορθο- (ortho-, correct) and -γραφίᾱ (-graphíā, writing).)
Ionian and pentatonic share similar structures and five notes… but for this analysis’ purposes, that’s almost where their similarities stop.
| Ionian interval spacing (circle of fifths order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| 4 | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |
| 1 | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |
| 5 | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | |
| 2 | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |
| 6 | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |
| 3 | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |
| 7 | C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | |
I’m analyzing Ionian’s tetrachord as whole tone, whole tone, semitone, placing the synaphe mid-scale. Swapping the synaphe with the tetrachord above shifts the scale down, note by note. (Remember, a scale is a repeating note pattern, so in Mixolydian, the tetrachord above is intervals 1-3; in Aeolian and Locrian, it’s split across the start and end of the scale.) Only Lydian starts with larger intervals than Ionian – it swaps Ionian’s first semitone and third whole tone, with the following results:
- Lydian’s fourth note is the only sharp in the entire table.
- Each mode after Ionian adds flats to its notes, one by one.
- Locrian’s first and fourth notes are the only ones not lowered from Ionian.
- C♭ Lydian lowers C Locrian’s first note.
- Every mode lowers the note five semitones above or seven below the note its predecessor lowered.
- Each mode’s sole tritone contains the notes it and its successor lower.
| Pentatonic interval spacing (circle of fourths order) | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Mode | 1 | 2 | 3 | 4 | 5 | Intervals | |||||||
| 3 | C♮ | D♯ | F♮ | G♯ | A♯ | 1½ | 1 | 1½ | 1 | 1 | ||||
| 5 | C♮ | D♯ | F♮ | G♮ | A♯ | 1½ | 1 | 1 | 1½ | 1 | ||||
| 2 | C♮ | D♮ | F♮ | G♮ | A♯ | 1 | 1½ | 1 | 1½ | 1 | ||||
| 4 | C♮ | D♮ | F♮ | G♮ | A♮ | 1 | 1½ | 1 | 1 | 1½ | ||||
| 1 | C♮ | D♮ | E♮ | G♮ | A♮ | 1 | 1 | 1½ | 1 | 1½ | ||||
I’m analyzing pentatonic with the synaphe up front, making its trichord whole tone, minor third. Since major pentatonic’s intervals are as back-loaded as possible, its other C-rooted modes move at least one note up a semitone, and its rotations must lower the root before any other notes.
Thus, while Ionian is second in its circle of fifths, major pentatonic is last in its own circle of fourths. The next transposition in the above sequence yields C♭ blues minor (or B blues minor, whichever you please).
A few additional observations about both scales:
-
Note the notes in common to multiple modes of each scale:
Land of Confusion Pentatonic mode Notes Ionian modes Notes C♮ D♯ F♮ G♯ A♯ , , C♮ E♭ F♮ A♭ B♭ C♮ D♯ F♮ G♮ A♯ , , C♮ E♭ F♮ G♮ B♭ C♮ D♮ F♮ G♮ A♯ , , C♮ D♮ F♮ G♮ B♭ C♮ D♮ F♮ G♮ A♮ , , C♮ D♮ F♮ G♮ A♮ C♮ D♮ E♮ G♮ A♮ , , C♮ D♮ E♮ G♮ A♮ This may explain a possible source of confusion about how their modes correspond.
- Ionian’s last semitone is why C Locrian and C Phrygian contain D♭. That semitone doesn’t exist in pentatonic.
- Shifting major pentatonic one degree to the right changes D to D♯, E to F, and A to A♯.
- As a result, the scales have effectively opposite directions of motion.
A final note: Don’t read too much into my decision to highlight the synaphai. Their positions are only part of why the scales move in different directions. In fact, since both scales are, apart from two outliers, made entirely of whole steps, multiple intervals can be read as their synaphai; the choice depends entirely on the arbitrary choice of trichord or tetrachord pattern. Two such patterns can fit for the pentatonic scale and three for Ionian; each result in different synaphai and n-chord divisions. I settled on divisions that wouldn’t split the base scale’s n-chords. (I’ve highlighted my approaches below.)
| Ionian tetrachords & pentatonic trichords & synaphai, oh my | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Scale | Pattern | n-chord 1 | n-chord 2 | Synaphe | |||||||||
| Major 5t | 1 | 1½ | 2 | 3 | 4 | 5 | 1 | ||||||
| Major 5t | 1½ | 1 | 3 | 4 | 5 | 1 | 2 | ||||||
| Ionian | 1 | 1 | ½ | 5 | 6 | 7 | 1 | 2 | 3 | 4 | |||
| Ionian | 1 | ½ | 1 | 6 | 7 | 1 | 2 | 3 | 4 | 5 | |||
| Ionian | ½ | 1 | 1 | 7 | 1 | 2 | 3 | 4 | 5 | 6 | |||
Note that the Ancient Greeks used the final analysis for their diatonic genus, which was nearly identical to our Ionian scale in all but name. When in Greece, I shall do as the Greeks did, but in this section, I figured it was better to defer the added complexity until this part of my explanation.
In short, the extra whole step’s position per se doesn’t affect the scale’s direction; their different directions are mostly due to Ionian ending with a semitone and major pentatonic with a minor third. But the circle orders are direct results of the pentatonic and Ionian scales’ atypically even note distribution. Transforming one mode of most other scales into another requires far more work. Let’s explore why.
An analysis of five-semitone scale rotation
The pentatonic and Ionian scales are, respectively, the only pentatonic and heptatonic scales in 12-TET for which moving a single note by a semitone amounts to a scale rotation. There’s a relatively simple mathematical explanation for why, too:
-
Both scales are primarily composed of two identical interval collections, summing to five semitones each.
Ionian & pentatonic building blocks Scale n-chord Notes Intervals Outlier Ionian Tetrachord 4 3 Semitone Pentatonic Trichord 3 2 Minor third - A synaphe (extra whole tone) separates the two n-chords.
- All but one of the n-chord’s intervals are whole tones. The outlier differs in size by a semitone.
- Rotating the scale by n degrees swaps the n-chord and the synaphe.
- This has a single effect on the interval composition: a whole tone and an outlier swap places.
-
Since the outliers differ by a semitone, the sole result on the scale’s harmonic composition is to move a single note by a semitone:
Outcome of swapping n-chord with synaphe Outlier Outlier moves Note moves Minor third Down Up Semitone Down Down Minor third Up Down Semitone Up Up - This is why the pattern requires almost, but not quite, completely even note distribution. Rotating a scale with less uniform interval spacing changes its interval composition substantially more, while rotating a scale with entirely uniform intervals does not change its interval composition at all – thus, as we’ll see when we examine modes of limited transposition, an entirely uniform scale has no other modes.
- This is also one reason this pattern is so linked to the circles of fourths and fifths: swapping the synaphe and n-chord rotates the scale by either five or seven semitones.
- Five and seven are magic numbers for this kind of rotation in 12-TET because 12 modulo 7 = 5. In other words, after a note has moved seven semitones, five semitones of the octave remain.
-
Let’s examine Ionian (1, 1, ½, 1, 1, 1, ½) and Mixolydian (1, 1, ½, 1, 1, ½, 1) as a case study.
- Changing Ionian to Mixolydian has a single result: intervals 6 and 7 swap places.
-
These sets of intervals are identical:
- Ionian’s first three and last three
- Mixolydian’s first three and fourth through sixth
- Ionian and Mixolydian’s first three
- Ionian’s fifth through seventh and Mixolydian’s fourth through sixth
- Ionian and Mixolydian’s fourth and fifth
- Ionian’s fourth and Mixolydian’s seventh
- Ionian’s sixth interval is a semitone larger than Mixolydian’s.
- Mixolydian’s seventh interval is a semitone larger than Ionian’s.
If any of this weren’t true, swapping those two intervals wouldn’t rotate the scale by five semitones.
In short, for moving a single note of any five- or seven-note scale in 12-TET by a semitone to rotate the scale, its intervals must be almost completely uniform, with only two identical outliers that:
- Differ in size from the remainder by only a semitone
- Are separated by five semitones
Readers may still have one final question: why is the number of outliers so important? Actually, it isn’t; it’s just important that the outliers be identical. If a seven-note scale could be completely uniform apart from one outlier, moving that interval would also rotate the scale. And, as it turns out, it can, but not, ironically, by making its note distribution more uniform.
- Take the temperament modulo the scale size to get the number of extra semitones to distribute: 12 modulo 7 is 5. We have five extra semitones to distribute.
- Take the note count modulo the extra semitones to figure out the most uniform note distribution possible: 7 modulo 5 is 2.
In short, two intervals must be outliers in the most uniform heptatonic note distribution possible. The way to get a single outlier, therefore, is to go in the exact opposite direction and make the outlier as big as possible. Which brings us to our next point of analysis.
Other single-note scale rotations
The hendecatonic scale
I’ve focused most of my analysis so far on pentatonic and heptatonic scales, but now that I better understand the mathematical principles explaining why this happens, I’ve expanded my scope somewhat to see if I can uncover other examples of similar patterns with scale rotations of various interval sizes. I’ve uncovered a few, which I’ll explain in this section.
Other equal temperaments certainly have similar examples (for instance, in 24-TET, rotation by 11 or 13 quarter-tones should produce similar results for similar 11- and 13-note scales), but I haven’t finished developing tools for scale analysis outside 12-TET, so they’ll have to wait.
Do other scales exist in 12-TET that don’t contain the above composition for which moving a single note by a semitone will qualify as a scale rotation? As it happens, yes: I can say with complete confidence, without even having to think about it, that the hendecatonic scale must demonstrate the same principle. And I say the hendecatonic scale for a simple reason: 12-TET contains only a single hendecatonic scale. The reason may be self-explanatory, but if it isn’t, I’ll give you a hint: It’s the same reason there’s only one dodecatonic scale.
In 12-TET, hendecatonic scales must contain all but one note of the chromatic scale. Thus, it must contain ten semitones and one whole tone, and swapping its whole tone with any of its semitones qualifies as a scale rotation by default. There are only eleven ways to remove notes that aren’t the root; thus, a single scale with eleven modes, which displays similar patterns not just for the circle of fifths but for every possible interval in 12-TET.
| Transforming the hendecatonic scale | |||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | Intervals | |||||||||||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | 1 | ||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | B♮ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | 1 | ½ | ||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♯ | B♮ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | 1 | ½ | ½ | ||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | A♮ | A♯ | B♮ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | 1 | ½ | ½ | ½ | ||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | ½ | ½ | ½ | 1 | ½ | ½ | ½ | ½ | ||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | G♮ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | ½ | ½ | 1 | ½ | ½ | ½ | ½ | ½ | ||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | ½ | 1 | ½ | ½ | ½ | ½ | ½ | ½ | ||
| C♮ | C♯ | D♮ | D♯ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | 1 | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ||
| C♮ | C♯ | D♮ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | 1 | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ||
| C♮ | C♯ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | ½ | 1 | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ||
| C♮ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | 1 | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ||
| C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | 1 | ||
Of course, the very fact that only one hendecatonic scale exists in 12-TET somehow makes this fact feel vastly less impressive, even though it has exactly the same cause as Ionian’s circle of fifths pattern. Funny how that works. (In fact, the hendecatonic scale can be generated using the exact same generator as Ionian; it just runs for four more notes.)
The heptatonic chromatic scale
Applying this same principle, we can determine that one other heptatonic scale exists in 12-TET for which moving a series of single notes by a constant interval size will rotate the scale. However, you don’t move its notes by a semitone; you move them by a perfect fourth. And, ironically, this doesn’t rotate it by five semitones; it rotates it by one. And its root won’t progress through the chromatic scale by semitones: it’ll progress through it by perfect fourths. (This will still take it all the way around the chromatic scale, just in a different order.) It’s the , which has the following interval spacing:
semitone, semitone, semitone, semitone, semitone, semitone, tritone
Trying to restrict ourselves to using every letter of the scale gives us an absolutely cursed set of notations. (For this set of tables and only this set of tables, I’ve used chromatic coloring rather than Doppler-shift coloring.)
| Modes of the heptatonic chromatic scale | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♭ | E𝄫 | F𝄫 | G𝄫♭ | A𝄫𝄫 | B𝄫𝄫♭ | ½ | ½ | ½ | ½ | ½ | ½ | 3 | ||
| C♮ | D♭ | E𝄫 | F𝄫 | G𝄫♭ | A𝄫𝄫 | B♮ | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ||
| C♮ | D♭ | E𝄫 | F𝄫 | G𝄫♭ | A♯ | B♮ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ||
| C♮ | D♭ | E𝄫 | F𝄫 | G𝄪 | A♯ | B♮ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||
| C♮ | D♭ | E𝄫 | F𝄪♯ | G𝄪 | A♯ | B♮ | ½ | ½ | 3 | ½ | ½ | ½ | ½ | ||
| C♮ | D♭ | E𝄪♯ | F𝄪♯ | G𝄪 | A♯ | B♮ | ½ | 3 | ½ | ½ | ½ | ½ | ½ | ||
| C♮ | D𝄪𝄪 | E𝄪♯ | F𝄪♯ | G𝄪 | A♯ | B♮ | 3 | ½ | ½ | ½ | ½ | ½ | ½ | ||
“D𝄪𝄪” and “B𝄫𝄫♭” use the same color because they represent the same note. You probably know it as F♯ or G♭.
Sorry. Like I said, it’s an absolutely cursed set of notations. Perhaps it’ll be more legible if we abandon any pretense of using every note name once. In fact, let’s observe the entire set of transformations:
| The heptatonic chromatic scale (complete transformation) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | ½ | ½ | ½ | ½ | ½ | ½ | 3 | ||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | B♮ | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ||
| C♮ | C♯ | D♮ | D♯ | E♮ | A♯ | B♮ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ||
| C♮ | C♯ | D♮ | D♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||
| C♮ | C♯ | D♮ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | 3 | ½ | ½ | ½ | ½ | ||
| C♮ | C♯ | G♮ | G♯ | A♮ | A♯ | B♮ | ½ | 3 | ½ | ½ | ½ | ½ | ½ | ||
| C♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | 3 | ½ | ½ | ½ | ½ | ½ | ½ | ||
| F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | ½ | ½ | ½ | 3 | ||
| F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | E♮ | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ||
| F♮ | F♯ | G♮ | G♯ | A♮ | D♯ | E♮ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ||
| F♮ | F♯ | G♮ | G♯ | D♮ | D♯ | E♮ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||
| F♮ | F♯ | G♮ | C♯ | D♮ | D♯ | E♮ | ½ | ½ | 3 | ½ | ½ | ½ | ½ | ||
| F♮ | F♯ | C♮ | C♯ | D♮ | D♯ | E♮ | ½ | 3 | ½ | ½ | ½ | ½ | ½ | ||
| F♮ | B♮ | C♮ | C♯ | D♮ | D♯ | E♮ | 3 | ½ | ½ | ½ | ½ | ½ | ½ | ||
| A♯ | B♮ | C♮ | C♯ | D♮ | D♯ | E♮ | ½ | ½ | ½ | ½ | ½ | ½ | 3 | ||
| A♯ | B♮ | C♮ | C♯ | D♮ | D♯ | A♮ | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ||
| A♯ | B♮ | C♮ | C♯ | D♮ | G♯ | A♮ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ||
| A♯ | B♮ | C♮ | C♯ | G♮ | G♯ | A♮ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||
| A♯ | B♮ | C♮ | F♯ | G♮ | G♯ | A♮ | ½ | ½ | 3 | ½ | ½ | ½ | ½ | ||
| A♯ | B♮ | F♮ | F♯ | G♮ | G♯ | A♮ | ½ | 3 | ½ | ½ | ½ | ½ | ½ | ||
| A♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | 3 | ½ | ½ | ½ | ½ | ½ | ½ | ||
| D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | ½ | ½ | ½ | ½ | ½ | ½ | 3 | ||
| D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | D♮ | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ||
| D♯ | E♮ | F♮ | F♯ | G♮ | C♯ | D♮ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ||
| D♯ | E♮ | F♮ | F♯ | C♮ | C♯ | D♮ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||
| D♯ | E♮ | F♮ | B♮ | C♮ | C♯ | D♮ | ½ | ½ | 3 | ½ | ½ | ½ | ½ | ||
| D♯ | E♮ | A♯ | B♮ | C♮ | C♯ | D♮ | ½ | 3 | ½ | ½ | ½ | ½ | ½ | ||
| D♯ | A♮ | A♯ | B♮ | C♮ | C♯ | D♮ | 3 | ½ | ½ | ½ | ½ | ½ | ½ | ||
| G♯ | A♮ | A♯ | B♮ | C♮ | C♯ | D♮ | ½ | ½ | ½ | ½ | ½ | ½ | 3 | ||
| G♯ | A♮ | A♯ | B♮ | C♮ | C♯ | G♮ | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ||
| G♯ | A♮ | A♯ | B♮ | C♮ | F♯ | G♮ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ||
| G♯ | A♮ | A♯ | B♮ | F♮ | F♯ | G♮ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||
| G♯ | A♮ | A♯ | E♮ | F♮ | F♯ | G♮ | ½ | ½ | 3 | ½ | ½ | ½ | ½ | ||
| G♯ | A♮ | D♯ | E♮ | F♮ | F♯ | G♮ | ½ | 3 | ½ | ½ | ½ | ½ | ½ | ||
| G♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | 3 | ½ | ½ | ½ | ½ | ½ | ½ | ||
| C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | ½ | ½ | ½ | ½ | ½ | ½ | 3 | ||
| C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | C♮ | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ||
| C♯ | D♮ | D♯ | E♮ | F♮ | B♮ | C♮ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ||
| C♯ | D♮ | D♯ | E♮ | A♯ | B♮ | C♮ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||
| C♯ | D♮ | D♯ | A♮ | A♯ | B♮ | C♮ | ½ | ½ | 3 | ½ | ½ | ½ | ½ | ||
| C♯ | D♮ | G♯ | A♮ | A♯ | B♮ | C♮ | ½ | 3 | ½ | ½ | ½ | ½ | ½ | ||
| C♯ | G♮ | G♯ | A♮ | A♯ | B♮ | C♮ | 3 | ½ | ½ | ½ | ½ | ½ | ½ | ||
| F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | C♮ | ½ | ½ | ½ | ½ | ½ | ½ | 3 | ||
| F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | F♮ | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ||
| F♯ | G♮ | G♯ | A♮ | A♯ | E♮ | F♮ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ||
| F♯ | G♮ | G♯ | A♮ | D♯ | E♮ | F♮ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||
| F♯ | G♮ | G♯ | D♮ | D♯ | E♮ | F♮ | ½ | ½ | 3 | ½ | ½ | ½ | ½ | ||
| F♯ | G♮ | C♯ | D♮ | D♯ | E♮ | F♮ | ½ | 3 | ½ | ½ | ½ | ½ | ½ | ||
| F♯ | C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | 3 | ½ | ½ | ½ | ½ | ½ | ½ | ||
| B♮ | C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | ½ | ½ | ½ | ½ | ½ | ½ | 3 | ||
| B♮ | C♮ | C♯ | D♮ | D♯ | E♮ | A♯ | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ||
| B♮ | C♮ | C♯ | D♮ | D♯ | A♮ | A♯ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ||
| B♮ | C♮ | C♯ | D♮ | G♯ | A♮ | A♯ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||
| B♮ | C♮ | C♯ | G♮ | G♯ | A♮ | A♯ | ½ | ½ | 3 | ½ | ½ | ½ | ½ | ||
| B♮ | C♮ | F♯ | G♮ | G♯ | A♮ | A♯ | ½ | 3 | ½ | ½ | ½ | ½ | ½ | ||
| B♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | 3 | ½ | ½ | ½ | ½ | ½ | ½ | ||
| E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | ½ | ½ | ½ | ½ | ½ | ½ | 3 | ||
| E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | D♯ | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ||
| E♮ | F♮ | F♯ | G♮ | G♯ | D♮ | D♯ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ||
| E♮ | F♮ | F♯ | G♮ | C♯ | D♮ | D♯ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||
| E♮ | F♮ | F♯ | C♮ | C♯ | D♮ | D♯ | ½ | ½ | 3 | ½ | ½ | ½ | ½ | ||
| E♮ | F♮ | B♮ | C♮ | C♯ | D♮ | D♯ | ½ | 3 | ½ | ½ | ½ | ½ | ½ | ||
| E♮ | A♯ | B♮ | C♮ | C♯ | D♮ | D♯ | 3 | ½ | ½ | ½ | ½ | ½ | ½ | ||
| A♮ | A♯ | B♮ | C♮ | C♯ | D♮ | D♯ | ½ | ½ | ½ | ½ | ½ | ½ | 3 | ||
| A♮ | A♯ | B♮ | C♮ | C♯ | D♮ | G♯ | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ||
| A♮ | A♯ | B♮ | C♮ | C♯ | G♮ | G♯ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ||
| A♮ | A♯ | B♮ | C♮ | F♯ | G♮ | G♯ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||
| A♮ | A♯ | B♮ | F♮ | F♯ | G♮ | G♯ | ½ | ½ | 3 | ½ | ½ | ½ | ½ | ||
| A♮ | A♯ | E♮ | F♮ | F♯ | G♮ | G♯ | ½ | 3 | ½ | ½ | ½ | ½ | ½ | ||
| A♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | 3 | ½ | ½ | ½ | ½ | ½ | ½ | ||
| D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | ½ | ½ | ½ | ½ | ½ | ½ | 3 | ||
| D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | C♯ | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ||
| D♮ | D♯ | E♮ | F♮ | F♯ | C♮ | C♯ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ||
| D♮ | D♯ | E♮ | F♮ | B♮ | C♮ | C♯ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||
| D♮ | D♯ | E♮ | A♯ | B♮ | C♮ | C♯ | ½ | ½ | 3 | ½ | ½ | ½ | ½ | ||
| D♮ | D♯ | A♮ | A♯ | B♮ | C♮ | C♯ | ½ | 3 | ½ | ½ | ½ | ½ | ½ | ||
| D♮ | G♯ | A♮ | A♯ | B♮ | C♮ | C♯ | 3 | ½ | ½ | ½ | ½ | ½ | ½ | ||
| G♮ | G♯ | A♮ | A♯ | B♮ | C♮ | C♯ | ½ | ½ | ½ | ½ | ½ | ½ | 3 | ||
| G♮ | G♯ | A♮ | A♯ | B♮ | C♮ | F♯ | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ||
| G♮ | G♯ | A♮ | A♯ | B♮ | F♮ | F♯ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ||
| G♮ | G♯ | A♮ | A♯ | E♮ | F♮ | F♯ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||
| G♮ | G♯ | A♮ | D♯ | E♮ | F♮ | F♯ | ½ | ½ | 3 | ½ | ½ | ½ | ½ | ||
| G♮ | G♯ | D♮ | D♯ | E♮ | F♮ | F♯ | ½ | 3 | ½ | ½ | ½ | ½ | ½ | ||
| G♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | 3 | ½ | ½ | ½ | ½ | ½ | ½ | ||
Other truncations of the chromatic scale
Similar corollaries apply to certain other scale sizes: moving successive notes of the (½ ½ ½ ½ 4, or semitone, semitone, semitone, semitone, minor sixth) by seven semitones will inevitably take the scale through all its modes, moving the scale root up by a perfect fifth each time. (The pentatonic chromatic scale is heptatonic chromatic’s scale complement.)
However, this only works for a few scale sizes. Why? Rotating the (½ ½ ½ ½ ½ 3½, or semitone, semitone, semitone, semitone, semitone, perfect fifth) moves each note by a tritone. Two tritones add up to an octave, so we skip five-sixths of the chromatic scale.
The only non-semitone interval of any such truncation of the chromatic scale is (13 - n) semitones, where n is the scale’s note count. Thus, scale rotation moves such scales’ notes (12 - n) semitones. For such a rotation to take us through every mode across the entire chromatic scale, this interval must be coprime with 12. In fact, in any n-TET, for scale rotations that move single notes by more than a chromatic step to cover the entire chromatic scale, the interval size (in units of 1/n octave) must be coprime with n. Thus, in 12-TET:
| Note movements and the 12-TET chromatic scale | |||||
|---|---|---|---|---|---|
| Semitones | Notes used | Pattern | |||
| 1 | 11 | all notes | linear order | ||
| 2 | 10 | six notes | major second | ||
| 3 | 9 | four notes | minor third | ||
| 4 | 8 | three notes | major third | ||
| 5 | 7 | all notes | circle of fourths | ||
| 6 | two notes | tritone | |||
Why do most rows list two interval sizes? Moving a note up by seven semitones equates to moving it down by five. Thus, in 12-TET, the only truncations of the chromatic scale for which this method of scale rotation will work contain 1, 5, 7, and 11 notes, and their rotations will respectively move single notes by a semitone, a perfect fourth, a perfect fourth, and a semitone.
Alternating heptamode and alternating heptamode inverse
Thus, the Ionian and heptatonic chromatic scales are the only heptatonic scales in 12-TET with single-note rotations that cycle through the entire chromatic scale. I’ve only found five other heptatonic scales in 12-TET with single-note rotations of any sort, and they’re complicated. In fact, they have an asterisk: all five require moving notes from the start of the scale to the end, or vice versa, which means that in some respects, they aren’t even single-note transformations. Two of these scales respectively have the following interval spacing:
Alternating heptamode: semitone, whole tone, semitone, whole tone, semitone, whole tone, minor third
Jhankāradhvani ♯5: whole tone, semitone, whole tone, semitone, whole tone, semitone, minor third
Now, if we try to use every letter of the scale, it might not even be obvious that this even works as a single-note transformation. (I haven’t given these proper names yet; sorry.)
| Alternating heptamode & alternating heptamode inverse | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♭ | E♭ | F♭ | G♭ | A𝄫 | B♭ | ½ | 1 | ½ | 1 | ½ | 1½ | 1 | ||
| C♮ | D♭ | E♭ | F♭ | G♮ | A♮ | B♭ | ½ | 1 | ½ | 1½ | 1 | ½ | 1 | ||
| C♮ | D♭ | E♮ | F♯ | G♮ | A♮ | B♭ | ½ | 1½ | 1 | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♭ | B𝄫 | 1 | ½ | 1 | ½ | 1 | ½ | 1½ | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♮ | B♮ | 1 | ½ | 1 | ½ | 1½ | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♯ | G♯ | A♮ | B♮ | 1 | ½ | 1½ | 1 | ½ | 1 | ½ | ||
| C♮ | D♯ | E♯ | F♯ | G♯ | A♮ | B♮ | 1½ | 1 | ½ | 1 | ½ | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♭ | G♭ | A𝄫 | B𝄫 | ½ | 1 | ½ | 1 | ½ | 1 | 1½ | ||
| C♮ | D♭ | E♭ | F♭ | G♭ | A♮ | B♭ | ½ | 1 | ½ | 1 | 1½ | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♯ | G♮ | A♮ | B♭ | ½ | 1 | 1½ | ½ | 1 | ½ | 1 | ||
| C♮ | D♯ | E♮ | F♯ | G♮ | A♮ | B♭ | 1½ | ½ | 1 | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♭ | B♮ | 1 | ½ | 1 | ½ | 1 | 1½ | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♯ | A♮ | B♮ | 1 | ½ | 1 | 1½ | ½ | 1 | ½ | ||
| C♮ | D♮ | E♯ | F♯ | G♯ | A♮ | B♮ | 1 | 1½ | ½ | 1 | ½ | 1 | ½ | ||
Let’s try just using sharps instead of trying to use every letter of the scale – which is still confusing, since the notes don’t stay in the same positions between scales. For instance, D♯ appears in all scales except 1321212 (the third scale) and 2312121 (the last), but in 3212121 (the seventh scale) and 3121212 (the fourth from the last), it’s the second note of the scale, and in all others it appears in, it’s the third note. Likewise, F♯ appears in all scales except 1213212 (the second scale) and 2123121 (the second from the last), and it switches between positions four and five.
| Alternating heptamode & alternating heptamode inverse (revisited) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | C♯ | D♯ | E♮ | F♯ | G♮ | A♯ | ½ | 1 | ½ | 1 | ½ | 1½ | 1 | ||
| C♮ | C♯ | D♯ | E♮ | G♮ | A♮ | A♯ | ½ | 1 | ½ | 1½ | 1 | ½ | 1 | ||
| C♮ | C♯ | E♮ | F♯ | G♮ | A♮ | A♯ | ½ | 1½ | 1 | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | D♯ | F♮ | F♯ | G♯ | A♮ | 1 | ½ | 1 | ½ | 1 | ½ | 1½ | ||
| C♮ | D♮ | D♯ | F♮ | F♯ | A♮ | B♮ | 1 | ½ | 1 | ½ | 1½ | 1 | ½ | ||
| C♮ | D♮ | D♯ | F♯ | G♯ | A♮ | B♮ | 1 | ½ | 1½ | 1 | ½ | 1 | ½ | ||
| C♮ | D♯ | F♮ | F♯ | G♯ | A♮ | B♮ | 1½ | 1 | ½ | 1 | ½ | 1 | ½ | ||
| C♮ | C♯ | D♯ | E♮ | F♯ | G♮ | A♮ | ½ | 1 | ½ | 1 | ½ | 1 | 1½ | ||
| C♮ | C♯ | D♯ | E♮ | F♯ | A♮ | A♯ | ½ | 1 | ½ | 1 | 1½ | ½ | 1 | ||
| C♮ | C♯ | D♯ | F♯ | G♮ | A♮ | A♯ | ½ | 1 | 1½ | ½ | 1 | ½ | 1 | ||
| C♮ | D♯ | E♮ | F♯ | G♮ | A♮ | A♯ | 1½ | ½ | 1 | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | D♯ | F♮ | F♯ | G♯ | B♮ | 1 | ½ | 1 | ½ | 1 | 1½ | ½ | ||
| C♮ | D♮ | D♯ | F♮ | G♯ | A♮ | B♮ | 1 | ½ | 1 | 1½ | ½ | 1 | ½ | ||
| C♮ | D♮ | F♮ | F♯ | G♯ | A♮ | B♮ | 1 | 1½ | ½ | 1 | ½ | 1 | ½ | ||
In case it isn’t obvious why these don’t cycle through the entire chromatic scale: in both cases, the single-note rotation moves a note by a minor third. Four minor thirds make up an octave, so we will only cycle through four notes of the chromatic scale.
That said, we’re somewhat examining this scale through the wrong lens. The issue is that the Ionian scale’s root only changes once in each of its transformation cycles: between Locrian and Lydian. As I hinted at, that’s not actually true of this scale. If we examine the pattern of notes being moved, we notice something interesting:
- 1212132 to 1213212: F♯ is raised to A
- 1213212 to 1321212: D♯ is raised to F♯
- 1321212 to 2121213: C is raised to D♯
…and this is where things get confusing.
With Ionian’s scale transformations, we only have to mess with the root once: between Locrian and Lydian. Alternating heptamode isn’t so simple: we have to transpose the root on multiple occasions. In fact, only eight notes can ever be part of this set of transformations, and going forward, I’ll represent them all with flats except F♯, since doing so will make E the only repeated letter. Thus, starting on C with 1212132 gives us:
| Alternating heptamode: A closer examination | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | C¹ | D♭¹ | E♭¹ | E¹ | F♯¹ | G¹ | A¹ | B♭¹ | C² | D♭² | E♭² | E² | F♯² | G² | A² |
| 121213 | C | D♭ | E♭ | E | F♯ | G | B♭ | ||||||||
| 121213 | C | D♭ | E♭ | E | G | A | B♭ | ||||||||
| 121213 | C | D♭ | E | F♯ | G | A | B♭ | ||||||||
| 121213 | D♭ | E♭ | E | F♯ | G | A | B♭ | ||||||||
| 121213 | D♭ | E♭ | E | F♯ | G | B♭ | C | ||||||||
| 121213 | D♭ | E♭ | E | G | A | B♭ | C | ||||||||
| 121213 | D♭ | E | F♯ | G | A | B♭ | C | ||||||||
| 121213 | E♭ | E | F♯ | G | A | B♭ | D♭ | ||||||||
| 121213 | E♭ | E | F♯ | G | B♭ | C | D♭ | ||||||||
| 121213 | E♭ | E | G | A | B♭ | C | D♭ | ||||||||
| 121213 | E | F♯ | G | A | B♭ | C | D♭ | ||||||||
| 121213 | E | F♯ | G | A | B♭ | D♭ | E♭ | ||||||||
| 121213 | E | F♯ | G | B♭ | C | D♭ | E♭ | ||||||||
| 121213 | E | G | A | B♭ | C | D♭ | E♭ | ||||||||
| 121213 | F♯ | G | A | B♭ | C | D♭ | E | ||||||||
| 121213 | F♯ | G | A | B♭ | D♭ | E♭ | E | ||||||||
| 121213 | F♯ | G | B♭ | C | D♭ | E♭ | E | ||||||||
| 121213 | G | A | B♭ | C | D♭ | E♭ | E | ||||||||
| 121213 | G | A | B♭ | C | D♭ | E | F♯ | ||||||||
| 121213 | G | A | B♭ | D♭ | E♭ | E | F♯ | ||||||||
| 121213 | G | B♭ | C | D♭ | E♭ | E | F♯ | ||||||||
| 121213 | A | B♭ | C | D♭ | E♭ | E | G | ||||||||
| 121213 | A | B♭ | C | D♭ | E | F♯ | G | ||||||||
| 121213 | A | B♭ | D♭ | E♭ | E | F♯ | G | ||||||||
| 121213 | B♭ | C | D♭ | E♭ | E | F♯ | G | ||||||||
| 121213 | B♭ | C | D♭ | E♭ | E | G | A | ||||||||
| 121213 | B♭ | C | D♭ | E | F♯ | G | A | ||||||||
| 121213 | B♭ | D♭ | E♭ | E | F♯ | G | A | ||||||||
In each row, the orange note will move in the next scale, and the blue note has just moved. Note that these are always separated by a tritone within the same scale, and the note always moves up by a minor third from one scale to the next (except from 3212121 to 1212132, when it moves down by a major sixth). The note that moves is also always either a minor third below or a major sixth above the note that moved in the previous scale. The four notes that move by minor thirds and major sixths form C, E♭, F♯, and A diminished seventh chords; the four notes that only ever move by octaves form D♭, E, G, and A♯ diminished seventh chords.
I’m going to confess that I don’t perfectly understand what’s going on here either, but I think what’s happening is that whenever we move the root, or whenever we move a note across the root, we have to move a note from the end of the scale to the start, or vice versa. But note that both this scale and its inverse contain three pairs of whole tones and semitones, and one additional minor third. A whole tone and a semitone, of course, add up to a minor third. Thus, although this is in some ways the least regular interval distribution we’ve examined, the fact that it can be divided into four three-semitone regions (three with two notes, one with one) necessitates a rotation that lines up exactly with the parent tonality. In short, while a scale needs to be nearly regular to be possible to rotate all its notes across the entire chromatic scale, the intervals themselves must be impossible to subdivide into regions of exactly equal size.
These scales’ complements experience exactly the same issue:
Major pentatonic ♭234: semitone, whole tone, minor third, minor third, minor third
Major pentatonic ♭23: whole tone, semitone, minor third, minor third, minor third
I will leave discerning why as an exercise for the reader.
Melodic Phrygian (or Neapolitan “major”, as it’s misleadingly known)
Another scale with a similar transformation is just as confusing to track. It’s most frequently known by the misleading name “Neapolitan major”. This is a terrible name, because it’s objectively not a major scale – it doesn’t have a major third above the root! The reasoning behind the name is that a Neapolitan chord starts on a scale’s flat second degree, and it has a major sixth above the root, while its counterpart the Neapolitan minor has a minor sixth above the root. This is an awful justification: I can’t think of any scales for which “major” and “minor” mean the sixth scale degree rather than the third. A name that requires an explanation to make sense isn’t very helpful! I instead call them, respectively, melodic Phrygian and harmonic Phrygian; anyone who knows Phrygian, melodic minor, and harmonic minor can reasonably infer what those mean.
Melodic Phrygian transforms as follows:
| Analyzing the melodic Phrygian scale | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♯ | D♯ | E♯ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | ½ | ½ | 1 | 1 | 1 | ||
| C♯ | D♯ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | ½ | 1 | 1 | 1 | 1 | ||
| C♯ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | ½ | ½ | 1 | 1 | 1 | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♮ | ½ | 1 | 1 | 1 | 1 | 1 | ½ | ||
| C♭ | D♭ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | 1 | 1 | 1 | ½ | ½ | ||
| C♭ | D♭ | E♭ | F♮ | G♮ | A♭ | B𝄫 | 1 | 1 | 1 | 1 | ½ | ½ | 1 | ||
| C♭ | D♭ | E♭ | F♮ | G♭ | A𝄫 | B𝄫 | 1 | 1 | 1 | ½ | ½ | 1 | 1 | ||
(Lydian dominant ♭6 is also, equally inaccurately, called Lydian minor. I won’t dignify this with further discussion.)
Once again, trying to use all seven note names paints a misleading picture. It looks like more than one note is moving per scale transformation until we rename the notes. In the following table, I’ve renamed C♭ to B♮, E♯ to F♮, E♭ to D♯, G♭ to F♯, A𝄫 to G♮, A♭ to G♯, B𝄫 to A♮, and B♭ to A♯. I’ve also reshuffled the modes so that melodic Phrygian is at the bottom, intensified the border each time we move the root note, and highlighted the note we move in each scale transformation. Since we move the note by a whole step each time, we skip half the possible transpositions. This set gives us melodic Phrygian on A♯, G♯, F♯, E, D, and C; and its other modes on B, A, G, F, D♯, and C♯.
| Transforming the melodic Phrygian scale | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| B♮ | C♯ | D♯ | F♮ | G♮ | A♮ | A♯ | 1 | 1 | 1 | 1 | 1 | ½ | ½ | ||
| B♮ | C♯ | D♯ | F♮ | G♮ | G♯ | A♮ | 1 | 1 | 1 | 1 | ½ | ½ | 1 | ||
| B♮ | C♯ | D♯ | F♮ | F♯ | G♮ | A♮ | 1 | 1 | 1 | ½ | ½ | 1 | 1 | ||
| B♮ | C♯ | D♯ | E♮ | F♮ | G♮ | A♮ | 1 | 1 | ½ | ½ | 1 | 1 | 1 | ||
| B♮ | C♯ | D♮ | D♯ | F♮ | G♮ | A♮ | 1 | ½ | ½ | 1 | 1 | 1 | 1 | ||
| B♮ | C♮ | C♯ | D♯ | F♮ | G♮ | A♮ | ½ | ½ | 1 | 1 | 1 | 1 | 1 | ||
| A♯ | B♮ | C♯ | D♯ | F♮ | G♮ | A♮ | ½ | 1 | 1 | 1 | 1 | 1 | ½ | ||
| A♮ | B♮ | C♯ | D♯ | F♮ | G♮ | G♯ | 1 | 1 | 1 | 1 | 1 | ½ | ½ | ||
| A♮ | B♮ | C♯ | D♯ | F♮ | F♯ | G♮ | 1 | 1 | 1 | 1 | ½ | ½ | 1 | ||
| A♮ | B♮ | C♯ | D♯ | E♮ | F♮ | G♮ | 1 | 1 | 1 | ½ | ½ | 1 | 1 | ||
| A♮ | B♮ | C♯ | D♮ | D♯ | F♮ | G♮ | 1 | 1 | ½ | ½ | 1 | 1 | 1 | ||
| A♮ | B♮ | C♮ | C♯ | D♯ | F♮ | G♮ | 1 | ½ | ½ | 1 | 1 | 1 | 1 | ||
| A♮ | A♯ | B♮ | C♯ | D♯ | F♮ | G♮ | ½ | ½ | 1 | 1 | 1 | 1 | 1 | ||
| G♯ | A♮ | B♮ | C♯ | D♯ | F♮ | G♮ | ½ | 1 | 1 | 1 | 1 | 1 | ½ | ||
| G♮ | A♮ | B♮ | C♯ | D♯ | F♮ | F♯ | 1 | 1 | 1 | 1 | 1 | ½ | ½ | ||
| G♮ | A♮ | B♮ | C♯ | D♯ | E♮ | F♮ | 1 | 1 | 1 | 1 | ½ | ½ | 1 | ||
| G♮ | A♮ | B♮ | C♯ | D♮ | D♯ | F♮ | 1 | 1 | 1 | ½ | ½ | 1 | 1 | ||
| G♮ | A♮ | B♮ | C♮ | C♯ | D♯ | F♮ | 1 | 1 | ½ | ½ | 1 | 1 | 1 | ||
| G♮ | A♮ | A♯ | B♮ | C♯ | D♯ | F♮ | 1 | ½ | ½ | 1 | 1 | 1 | 1 | ||
| G♮ | G♯ | A♮ | B♮ | C♯ | D♯ | F♮ | ½ | ½ | 1 | 1 | 1 | 1 | 1 | ||
| F♯ | G♮ | A♮ | B♮ | C♯ | D♯ | F♮ | ½ | 1 | 1 | 1 | 1 | 1 | ½ | ||
| F♮ | G♮ | A♮ | B♮ | C♯ | D♯ | E♮ | 1 | 1 | 1 | 1 | 1 | ½ | ½ | ||
| F♮ | G♮ | A♮ | B♮ | C♯ | D♮ | D♯ | 1 | 1 | 1 | 1 | ½ | ½ | 1 | ||
| F♮ | G♮ | A♮ | B♮ | C♮ | C♯ | D♯ | 1 | 1 | 1 | ½ | ½ | 1 | 1 | ||
| F♮ | G♮ | A♮ | A♯ | B♮ | C♯ | D♯ | 1 | 1 | ½ | ½ | 1 | 1 | 1 | ||
| F♮ | G♮ | G♯ | A♮ | B♮ | C♯ | D♯ | 1 | ½ | ½ | 1 | 1 | 1 | 1 | ||
| F♮ | F♯ | G♮ | A♮ | B♮ | C♯ | D♯ | ½ | ½ | 1 | 1 | 1 | 1 | 1 | ||
| E♮ | F♮ | G♮ | A♮ | B♮ | C♯ | D♯ | ½ | 1 | 1 | 1 | 1 | 1 | ½ | ||
| D♯ | F♮ | G♮ | A♮ | B♮ | C♯ | D♮ | 1 | 1 | 1 | 1 | 1 | ½ | ½ | ||
| D♯ | F♮ | G♮ | A♮ | B♮ | C♮ | C♯ | 1 | 1 | 1 | 1 | ½ | ½ | 1 | ||
| D♯ | F♮ | G♮ | A♮ | A♯ | B♮ | C♯ | 1 | 1 | 1 | ½ | ½ | 1 | 1 | ||
| D♯ | F♮ | G♮ | G♯ | A♮ | B♮ | C♯ | 1 | 1 | ½ | ½ | 1 | 1 | 1 | ||
| D♯ | F♮ | F♯ | G♮ | A♮ | B♮ | C♯ | 1 | ½ | ½ | 1 | 1 | 1 | 1 | ||
| D♯ | E♮ | F♮ | G♮ | A♮ | B♮ | C♯ | ½ | ½ | 1 | 1 | 1 | 1 | 1 | ||
| D♮ | D♯ | F♮ | G♮ | A♮ | B♮ | C♯ | ½ | 1 | 1 | 1 | 1 | 1 | ½ | ||
| C♯ | D♯ | F♮ | G♮ | A♮ | B♮ | C♮ | 1 | 1 | 1 | 1 | 1 | ½ | ½ | ||
| C♯ | D♯ | F♮ | G♮ | A♮ | A♯ | B♮ | 1 | 1 | 1 | 1 | ½ | ½ | 1 | ||
| C♯ | D♯ | F♮ | G♮ | G♯ | A♮ | B♮ | 1 | 1 | 1 | ½ | ½ | 1 | 1 | ||
| C♯ | D♯ | F♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | ½ | ½ | 1 | 1 | 1 | ||
| C♯ | D♯ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | ½ | 1 | 1 | 1 | 1 | ||
| C♯ | D♮ | D♯ | F♮ | G♮ | A♮ | B♮ | ½ | ½ | 1 | 1 | 1 | 1 | 1 | ||
| C♮ | C♯ | D♯ | F♮ | G♮ | A♮ | B♮ | ½ | 1 | 1 | 1 | 1 | 1 | ½ | ||
Apathetic minor & Pacific
I’ve found one other pair of scales that exhibit this behavior. Unsurprisingly, they’re each other’s mirror images. I’ll call them “apathetic minor” and “Pacific”, but again, I don’t have good names for most of their modes yet. (For the record, “apathetic minor” is the mode labelled “1131141”, and Pacific is the mode labelled “1311141”. Again, I apologize for not baving better names for these.) Rooted on C, they look like this:
| Apathetic minor and Pacific on C | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♯ | E♮ | F♮ | G𝄪 | A♯ | B♮ | 1½ | ½ | ½ | 2 | ½ | ½ | ½ | ||
| C♮ | D𝄪 | E♯ | F♯ | G♮ | A♯ | B♮ | 2 | ½ | ½ | ½ | 1½ | ½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♭ | A♯ | B♮ | ½ | 1½ | ½ | ½ | 2 | ½ | ½ | ||
| C♮ | D♭ | E♯ | F♯ | G♮ | A♭ | B♮ | ½ | 2 | ½ | ½ | ½ | 1½ | ½ | ||
| C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B♮ | ½ | ½ | 1½ | ½ | ½ | 2 | ½ | ||
| C♮ | D♭ | E𝄫 | F♯ | G♮ | A♭ | B𝄫 | ½ | ½ | 2 | ½ | ½ | ½ | 1½ | ||
| C♮ | D♭ | E𝄫 | F𝄫 | G♭ | A𝄫 | B𝄫♭ | ½ | ½ | ½ | 1½ | ½ | ½ | 2 | ||
| C♮ | D𝄪 | E♯ | F♯ | G𝄪 | A♯ | B♮ | 2 | ½ | ½ | 1½ | ½ | ½ | ½ | ||
| C♮ | D♯ | E♮ | F♮ | G♭ | A♯ | B♮ | 1½ | ½ | ½ | ½ | 2 | ½ | ½ | ||
| C♮ | D♭ | E♯ | F♯ | G♮ | A♯ | B♮ | ½ | 2 | ½ | ½ | 1½ | ½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♭ | A𝄫 | B♮ | ½ | 1½ | ½ | ½ | ½ | 2 | ½ | ||
| C♮ | D♭ | E𝄫 | F♯ | G♮ | A♭ | B♮ | ½ | ½ | 2 | ½ | ½ | 1½ | ½ | ||
| C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B𝄫♭ | ½ | ½ | 1½ | ½ | ½ | ½ | 2 | ||
| C♮ | D♭ | E𝄫 | F𝄫 | G♮ | A♭ | B𝄫 | ½ | ½ | ½ | 2 | ½ | ½ | 1½ | ||
Which, again, is extremely difficult to parse. Let’s simplify.
| Apathetic minor and Pacific on C (simplified) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♯ | E♮ | F♮ | A♮ | A♯ | B♮ | 1½ | ½ | ½ | 2 | ½ | ½ | ½ | ||
| C♮ | E♮ | F♮ | F♯ | G♮ | A♯ | B♮ | 2 | ½ | ½ | ½ | 1½ | ½ | ½ | ||
| C♮ | C♯ | E♮ | F♮ | F♯ | A♯ | B♮ | ½ | 1½ | ½ | ½ | 2 | ½ | ½ | ||
| C♮ | C♯ | F♮ | F♯ | G♮ | G♯ | B♮ | ½ | 2 | ½ | ½ | ½ | 1½ | ½ | ||
| C♮ | C♯ | D♮ | F♮ | F♯ | G♮ | B♮ | ½ | ½ | 1½ | ½ | ½ | 2 | ½ | ||
| C♮ | C♯ | D♮ | F♯ | G♮ | G♯ | A♮ | ½ | ½ | 2 | ½ | ½ | ½ | 1½ | ||
| C♮ | C♯ | D♮ | D♯ | F♯ | G♮ | G♯ | ½ | ½ | ½ | 1½ | ½ | ½ | 2 | ||
| C♮ | E♮ | F♮ | F♯ | A♮ | A♯ | B♮ | 2 | ½ | ½ | 1½ | ½ | ½ | ½ | ||
| C♮ | D♯ | E♮ | F♮ | F♯ | A♯ | B♮ | 1½ | ½ | ½ | ½ | 2 | ½ | ½ | ||
| C♮ | C♯ | F♮ | F♯ | G♮ | A♯ | B♮ | ½ | 2 | ½ | ½ | 1½ | ½ | ½ | ||
| C♮ | C♯ | E♮ | F♮ | F♯ | G♮ | B♮ | ½ | 1½ | ½ | ½ | ½ | 2 | ½ | ||
| C♮ | C♯ | D♮ | F♯ | G♮ | G♯ | B♮ | ½ | ½ | 2 | ½ | ½ | 1½ | ½ | ||
| C♮ | C♯ | D♮ | F♮ | F♯ | G♮ | G♯ | ½ | ½ | 1½ | ½ | ½ | ½ | 2 | ||
| C♮ | C♯ | D♮ | D♯ | G♮ | G♯ | A♮ | ½ | ½ | ½ | 2 | ½ | ½ | 1½ | ||
Of course, the transformations again require moving the root more often than that. And since we’re moving notes by six semitones, we again skip ⅚ of the chromatic scale for each mode.
| Single-note transformations of apathetic minor | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♯ | E♮ | F♮ | A♮ | A♯ | B♮ | 1½ | ½ | ½ | 2 | ½ | ½ | ½ | ||
| B♮ | D♯ | E♮ | F♮ | F♯ | A♮ | A♯ | 2 | ½ | ½ | ½ | 1½ | ½ | ½ | ||
| B♮ | C♮ | D♯ | E♮ | F♮ | A♮ | A♯ | ½ | 1½ | ½ | ½ | 2 | ½ | ½ | ||
| A♯ | B♮ | D♯ | E♮ | F♮ | F♯ | A♮ | ½ | 2 | ½ | ½ | ½ | 1½ | ½ | ||
| A♯ | B♮ | C♮ | D♯ | E♮ | F♮ | A♮ | ½ | ½ | 1½ | ½ | ½ | 2 | ½ | ||
| A♮ | A♯ | B♮ | D♯ | E♮ | F♮ | F♯ | ½ | ½ | 2 | ½ | ½ | ½ | 1½ | ||
| A♮ | A♯ | B♮ | C♮ | D♯ | E♮ | F♮ | ½ | ½ | ½ | 1½ | ½ | ½ | 2 | ||
| F♯ | A♮ | A♯ | B♮ | D♯ | E♮ | F♮ | 1½ | ½ | ½ | 2 | ½ | ½ | ½ | ||
| F♮ | A♮ | A♯ | B♮ | C♮ | D♯ | E♮ | 2 | ½ | ½ | ½ | 1½ | ½ | ½ | ||
| F♮ | F♯ | A♮ | A♯ | B♮ | D♯ | E♮ | ½ | 1½ | ½ | ½ | 2 | ½ | ½ | ||
| E♮ | F♮ | A♮ | A♯ | B♮ | C♮ | D♯ | ½ | 2 | ½ | ½ | ½ | 1½ | ½ | ||
| E♮ | F♮ | F♯ | A♮ | A♯ | B♮ | D♯ | ½ | ½ | 1½ | ½ | ½ | 2 | ½ | ||
| D♯ | E♮ | F♮ | A♮ | A♯ | B♮ | C♮ | ½ | ½ | 2 | ½ | ½ | ½ | 1½ | ||
| D♯ | E♮ | F♮ | F♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | 1½ | ½ | ½ | 2 | ||
| Single-note transformations of Pacific | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| D♮ | D♯ | E♮ | F♮ | A♮ | A♯ | B♮ | ½ | ½ | ½ | 2 | ½ | ½ | 1½ | ||
| B♮ | D♯ | E♮ | F♮ | G♯ | A♮ | A♯ | 2 | ½ | ½ | 1½ | ½ | ½ | ½ | ||
| B♮ | D♮ | D♯ | E♮ | F♮ | A♮ | A♯ | 1½ | ½ | ½ | ½ | 2 | ½ | ½ | ||
| A♯ | B♮ | D♯ | E♮ | F♮ | G♯ | A♮ | ½ | 2 | ½ | ½ | 1½ | ½ | ½ | ||
| A♯ | B♮ | D♮ | D♯ | E♮ | F♮ | A♮ | ½ | 1½ | ½ | ½ | ½ | 2 | ½ | ||
| A♮ | A♯ | B♮ | D♯ | E♮ | F♮ | G♯ | ½ | ½ | 2 | ½ | ½ | 1½ | ½ | ||
| A♮ | A♯ | B♮ | D♮ | D♯ | E♮ | F♮ | ½ | ½ | 1½ | ½ | ½ | ½ | 2 | ||
| G♯ | A♮ | A♯ | B♮ | D♯ | E♮ | F♮ | ½ | ½ | ½ | 2 | ½ | ½ | 1½ | ||
| F♮ | A♮ | A♯ | B♮ | D♮ | D♯ | E♮ | 2 | ½ | ½ | 1½ | ½ | ½ | ½ | ||
| F♮ | G♯ | A♮ | A♯ | B♮ | D♯ | E♮ | 1½ | ½ | ½ | ½ | 2 | ½ | ½ | ||
| E♮ | F♮ | A♮ | A♯ | B♮ | D♮ | D♯ | ½ | 2 | ½ | ½ | 1½ | ½ | ½ | ||
| E♮ | F♮ | G♯ | A♮ | A♯ | B♮ | D♯ | ½ | 1½ | ½ | ½ | ½ | 2 | ½ | ||
| D♯ | E♮ | F♮ | A♮ | A♯ | B♮ | D♮ | ½ | ½ | 2 | ½ | ½ | 1½ | ½ | ||
| D♯ | E♮ | F♮ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | 1½ | ½ | ½ | ½ | 2 | ||
A brief analysis of note distributions
It may also be instructive to compare how many times each note appears on each mode of each scale on C.
| Note distributions across parallel modes | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Scale | C | C♯ | D | D♯ | E | F | F♯ | G | G♯ | A | A♯ | B |
| Ionian | 7 | 2 | 5 | 4 | 3 | 6 | 2 | 6 | 3 | 4 | 5 | 2 |
| Major pentatonic | 5 | 0 | 3 | 2 | 1 | 4 | 0 | 4 | 1 | 2 | 3 | 0 |
| Chromatic heptatonic | 7 | 6 | 5 | 4 | 3 | 2 | 2 | 2 | 3 | 4 | 5 | 6 |
| Chromatic pentatonic | 5 | 4 | 3 | 2 | 1 | 0 | 0 | 0 | 1 | 2 | 3 | 4 |
| Alternating heptamode | 7 | 3 | 3 | 6 | 3 | 3 | 6 | 3 | 3 | 6 | 3 | 3 |
| Major pentatonic ♭234 | 5 | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 1 |
| Alternating heptamode inverse | 7 | 3 | 3 | 6 | 3 | 3 | 6 | 3 | 3 | 6 | 3 | 3 |
| Major pentatonic ♭23 | 5 | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 1 |
| Melodic Phrygian | 7 | 2 | 6 | 2 | 6 | 2 | 6 | 2 | 6 | 2 | 6 | 2 |
| Augmented ninth | 5 | 0 | 4 | 0 | 4 | 0 | 4 | 0 | 4 | 0 | 4 | 0 |
| Apathetic minor | 7 | 5 | 3 | 2 | 3 | 5 | 6 | 5 | 3 | 2 | 3 | 5 |
| Rāga Saugandhini | 5 | 3 | 1 | 0 | 1 | 3 | 4 | 3 | 1 | 0 | 1 | 3 |
| Pacific | 7 | 5 | 3 | 2 | 3 | 5 | 6 | 5 | 3 | 2 | 3 | 5 |
| Rāga Saugandhini inverse | 5 | 3 | 1 | 0 | 1 | 3 | 4 | 3 | 1 | 0 | 1 | 3 |
- The reflective symmetry about F♯ seen above holds uniformly for every scale.
- As a consequence of #1, a scale and its mirror image always have identical note distributions. This explains the identical note distributions seen above.
-
After each heptatonic scale, I list its pentatonic complement. Each pentatonic scale uses every note of the chromatic scale in two fewer modes than its heptatonic complement does. This principle appears to hold uniformly for all scale complements, even counting the null set as the chromatic scale’s complement.
- Hexatonic scales and their complements always have identical note distributions.
- Heptatonic distributions equal their pentatonic complements’ distributions plus two.
- Octatonic distributions equal their tetratonic complements’ distributions plus four.
- Enneatonic distributions equal their three-note complements’ distributions plus six.
- Decatonic distributions equal their two-note complements’ distributions plus eight.
- The hendecatonic scale’s distribution equals the octave’s distribution plus ten.
- The dodecatonic scale’s distribution equals the null set’s distribution plus twelve.
This has a few further implications. At the start of this section, we noted that the hendecatonic scale must use each note of the chromatic scale except the root in all but one of its modes. For similar reasons, every note of the chromatic scale except the root appears in:
- At least eight modes of every decatonic scale.
- At least six modes of every enneatonic scale.
- At least four modes of every octatonic scale.
- At least two modes of every heptatonic scale.
Meanwhile, all scales must use the root in every mode – this is a requirement of being a scale.
- Decrementing alternating heptamode and melodic Phrygian’s root counts by 1 would cause them to repeat the same pattern four and six times, respectively. This makes them very nearly rotationally as well as reflectively symmetrical. Explaining the full implications of this would require covering some principles we haven’t addressed yet, so we’ll return to these scales when we discuss modes of limited transposition (which, to be clear, these scales aren’t; however, they’re single-note transformations thereof).
Why Ionian’s scale mutation requires a coprime scale length with 12
For the sake of Science, let’s try using a scale generator with four notes separated by perfect fifths. Starting on F, we’ll get a scale of C-D-F-G. What would that set of transformations look like?
| Tetratonic truncation of Ionian | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| RID | MID | SID | IID | 1 | 2 | 3 | 4 | Intervals | |||
| C♮ | D♯ | F♮ | A♯ | 1½ | 1 | 2½ | 1 | ||||
| C♮ | F♮ | G♮ | A♯ | 2½ | 1 | 1½ | 1 | ||||
| C♮ | D♮ | F♮ | G♮ | 1 | 1½ | 1 | 2½ | ||||
| C♮ | D♮ | G♮ | A♮ | 1 | 2½ | 1 | 1½ | ||||
| E♮ | G♮ | A♮ | D♮ | 1½ | 1 | 2½ | 1 | ||||
(I’ll explain these names and/or replace them with better ones soon™.)
Each note moves by a major third. When we move the root, we also must move the scale’s second note to the end. That’s fair enough – we’ve seen it before – but because we’re moving notes by four semitones, we won’t cycle through the entire chromatic scale. Our next root will be G♯, and then we’ll be back on C, having skipped three quarters of the chromatic scale. (Egad. …Sorry.)
OK, so what about C-D-E-F-G-A?
| Hexatonic truncation of Ionian (semitone transposition) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| RID | MID | SID | IID | 1 | 2 | 3 | 4 | 5 | 6 | Intervals | |||||
| C♮ | D♮ | E♮ | G♮ | A♮ | B♮ | 1 | 1 | 1½ | 1 | 1 | ½ | ||||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | 1 | 1 | ½ | 1 | 1 | 1½ | ||||
| C♮ | D♮ | F♮ | G♮ | A♮ | A♯ | 1 | 1½ | 1 | 1 | ½ | 1 | ||||
| C♮ | D♮ | D♯ | F♮ | G♮ | A♯ | 1 | ½ | 1 | 1 | 1½ | 1 | ||||
| C♮ | D♯ | F♮ | G♮ | G♯ | A♯ | 1½ | 1 | 1 | ½ | 1 | 1 | ||||
| C♮ | C♯ | D♯ | F♮ | G♯ | A♯ | ½ | 1 | 1 | 1½ | 1 | 1 | ||||
| B♮ | C♯ | D♯ | F♯ | G♯ | A♯ | 1 | 1 | 1½ | 1 | 1 | ½ | ||||
| B♮ | C♯ | D♯ | E♮ | F♯ | G♯ | 1 | 1 | ½ | 1 | 1 | 1½ | ||||
This will cycle us through the entire circle of fifths, but…
- Between F71.β and F71.ε, B moves down by a tritone to F.
- Between F71.ε and F71.γ, E moves up by a tritone to A♯.
- Between F71.γ and F71.ζ, A moves down by a tritone to D♯.
- Between F71.ζ and F71.δ, D moves up by a tritone to G♯.
- Between F71.δ and F71.α, G moves down by a tritone to C♯.
But then the pattern breaks. The transformation from F71.α on C to F71.β on B requires moving two notes, not one: F down to B, and C up to F♯ – both by a tritone, but we’re no longer doing single-note transformations.
Another giveaway is that the interval pattern breaks – indeed, as we can see by examining F71.ε on B, we’d need even more transformations to continue it.
We can continue the pattern by transposing the scale by a tritone rather than a semitone:
| Hexatonic truncation of Ionian (tritone transposition) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| RID | MID | SID | IID | 1 | 2 | 3 | 4 | 5 | 6 | Intervals | |||||
| C♮ | D♮ | E♮ | G♮ | A♮ | B♮ | 1 | 1 | 1½ | 1 | 1 | ½ | ||||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | 1 | 1 | ½ | 1 | 1 | 1½ | ||||
| C♮ | D♮ | F♮ | G♮ | A♮ | A♯ | 1 | 1½ | 1 | 1 | ½ | 1 | ||||
| C♮ | D♮ | D♯ | F♮ | G♮ | A♯ | 1 | ½ | 1 | 1 | 1½ | 1 | ||||
| C♮ | D♯ | F♮ | G♮ | G♯ | A♯ | 1½ | 1 | 1 | ½ | 1 | 1 | ||||
| C♮ | C♯ | D♯ | F♮ | G♯ | A♯ | ½ | 1 | 1 | 1½ | 1 | 1 | ||||
| F♯ | G♯ | A♯ | C♯ | D♯ | F♮ | 1 | 1 | 1½ | 1 | 1 | ½ | ||||
| F♯ | G♯ | A♯ | B♮ | C♯ | D♯ | 1 | 1 | ½ | 1 | 1 | 1½ | ||||
But of course, we skip ⅚ of the chromatic scale by doing this.
Well, what about octatonic, then? As far as I can work out, there is a single-note set of transformations, but again, it will not cycle us through the entire chromatic scale.
| Octatonic expansions of Ionian | |||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| RID | MID | SID | IID | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Intervals | |||||||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | A♯ | B♮ | 1 | 1 | ½ | 1 | 1 | ½ | ½ | ½ | ||||
| C♮ | D♮ | E♮ | F♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | ½ | ½ | ½ | 1 | 1 | ½ | ||||
| C♮ | C♯ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | ½ | ½ | 1 | 1 | ½ | 1 | 1 | ½ | ||||
| B♮ | C♯ | D♮ | E♮ | F♯ | G♮ | G♯ | A♮ | 1 | ½ | 1 | 1 | ½ | ½ | ½ | 1 | ||||
| B♮ | C♯ | D♮ | D♯ | E♮ | F♯ | G♯ | A♮ | 1 | ½ | ½ | ½ | 1 | 1 | ½ | 1 | ||||
| A♯ | B♮ | C♯ | D♯ | E♮ | F♯ | G♯ | A♮ | ½ | 1 | 1 | ½ | 1 | 1 | ½ | ½ | ||||
| A♯ | B♮ | C♯ | D♯ | E♮ | F♮ | F♯ | G♯ | ½ | 1 | 1 | ½ | ½ | ½ | 1 | 1 | ||||
| A♯ | B♮ | C♮ | C♯ | D♯ | F♮ | F♯ | G♯ | ½ | ½ | ½ | 1 | 1 | ½ | 1 | 1 | ||||
| G♯ | A♯ | C♮ | C♯ | D♯ | F♮ | F♯ | G♮ | 1 | 1 | ½ | 1 | 1 | ½ | ½ | ½ | ||||
This set of mutations is difficult to parse, but as far as I can make out, this is what’s happening:
- H28.δ to H28.η: A♯ moves down four semitones to F♯.
- H28.η to H28.β: One semitone below F♯, F moves down four semitones to C♯.
- H28.β to H28.ε: One semitone below C♯, C moves down four semitones to G♯. Because we moved the root below B, it now moves to the front of the scale.
- H28.ε to H28.θ: One semitone below G♯, G moves down four semitones to D♯.
- H28.θ to H28.γ: One semitone below D♯, D moves down four semitones to A♯. A♯ becomes the new root, since it is below our previous root of B.
- H28.γ to H28.ζ: One semitone below A♯, A moves down four semitones to F.
- H28.ζ to H28.α: One semitone below F, E moves down four semitones to C.
- H28.α to H28.δ: One semitone below C, B moves down four semitones to G. Because we moved the root below G♯, it now moves to the front of the scale.
As we can see, this sequence of transformations will move the entire scale by major thirds each cycle, so this set of transformations will only transpose each mode to one third of the chromatic scale. This is a direct result of the scale length sharing a factor with 12: it’s surely no coincidence that their mutual factor is exactly equal to the number of semitones by which this set of transformations moves the scale each cycle. Effectively, for this form of mutation to work, the generator must use an interval size that is coprime with the parent temperament, and it must use a scale length that is also coprime with the parent temperament. So, the intervals that will work are one semitone, five semitones, seven semitones, and eleven semitones, and the scale lengths that will work are one note, five notes, seven notes, and eleven notes.
Transformations of the Ionian scale
Harmonic minor
Since learning harmonic and melodic minor’s modes is absolutely essential for anyone who wants to play jazz, I’ve created several sets of tables to help people visualize them better.
| Harmonic minor vs. modes from Aeolian (rooted on C, linear order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | 1 | ½ | 1 | 1 | ½ | 1½ | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♮ | B♭ | ½ | 1 | 1 | ½ | 1½ | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♯ | A♮ | B♮ | 1 | 1 | ½ | 1½ | ½ | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♭ | 1 | ½ | 1½ | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♭ | ½ | 1½ | ½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♯ | E♮ | F♯ | G♮ | A♮ | B♮ | 1½ | ½ | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♭ | G♭ | A♭ | B𝄫 | ½ | 1 | ½ | 1 | 1 | ½ | 1½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
Thus, as the first set of mode names suggests, harmonic minor’s modes respectively raise by a half-step:
- Aeolian’s seventh degree,
- Locrian’s sixth degree,
- Ionian’s fifth degree,
- Dorian’s fourth degree,
- Phrygian’s third degree,
- Lydian’s second degree,
- C-C-C-C-COMBO BREAKER!
Since we’re transposing every mode to C, we can’t raise Mixolydian’s first degree, because it’s the first degree! Instead, we must lower every other degree by a half-step. Say wha?
Somehow, it’s actually both even weirder than that, and not weird at all: what we do in the above table is the equivalent of raising the first degree. Since we’re constraing ourselves to a root of C, raising the first note of a scale by a half-step requires us to lower every note of that scale by a half-step. This results in the first note being the only scale degree we don’t lower: ½ − ½ = 0.
In practice, though, it’s usually already raised for us: it’s harmonic minor’s seventh degree! Let’s see what happens when we root these modes on the corresponding notes of their respective parent C minor scales:
| Harmonic minor vs. modes from Aeolian (rooted on scale, linear order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | 1 | ½ | 1 | 1 | ½ | 1½ | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | C♮ | ½ | 1 | 1 | ½ | 1½ | ½ | 1 | ||
| D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| E♭ | F♮ | G♮ | A♭ | B♮ | C♮ | D♮ | 1 | 1 | ½ | 1½ | ½ | 1 | ½ | ||
| E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | D♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| F♮ | G♮ | A♭ | B♮ | C♮ | D♮ | E♭ | 1 | ½ | 1½ | ½ | 1 | ½ | 1 | ||
| F♮ | G♮ | A♭ | B♭ | C♮ | D♮ | E♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| G♮ | A♭ | B♮ | C♮ | D♮ | E♭ | F♮ | ½ | 1½ | ½ | 1 | ½ | 1 | 1 | ||
| G♮ | A♭ | B♭ | C♮ | D♮ | E♭ | F♮ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| A♭ | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | 1½ | ½ | 1 | ½ | 1 | 1 | ½ | ||
| A♭ | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | ½ | 1 | ½ | 1 | 1 | ½ | 1½ | ||
| B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
Thus, B♭ Mixolydian is to C Aeolian as B Mixolydian ♯1 is to C harmonic minor: each starts from its parent scale’s seventh note. There’s actually nothing odd going on here at all; it’s exactly how modes are supposed to behave. C Mixolydian ♯1 equates to lowering every note of C Mixolydian except C by a half-step – and to raising only the B in B Mixolydian by a half-step.
Observant readers may have noticed that the “rooted on C” table above actually appears to contain several shifts. The missing puzzle piece is that it lists the modes in ascending order rather than “circle of fifths” order, which I did to make the scales’ intervals easier to relate to each other. So let’s return to “circle of fiths” order.
| Harmonic minor vs. modes from Aeolian (rooted on C, in “circle of fifths” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♯ | E♮ | F♯ | G♮ | A♮ | B♮ | 1½ | ½ | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♯ | A♮ | B♮ | 1 | 1 | ½ | 1½ | ½ | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♭ | G♭ | A♭ | B𝄫 | ½ | 1 | ½ | 1 | 1 | ½ | 1½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♭ | 1 | ½ | 1½ | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | 1 | ½ | 1 | 1 | ½ | 1½ | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♭ | ½ | 1½ | ½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♮ | B♭ | ½ | 1 | 1 | ½ | 1½ | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
We now see that, broadly speaking, in all except one case, each scale has one fewer sharp or one more flat than its predecessor two entries above. The clear outlier is Mixolydian ♯1, and this table may further clarify why the mode corresponding to Mixolydian is the one thus affected. Mixolydian corresponds to Ionian’s fifth scale degree, Dorian’s fourth scale degree, Phrygian’s third scale degree… and that’s the degree that harmonic minor’s modes raise. For completeness, here are the modes in “circle of fifths” order, rooted to their respective notes within their parent C minor scales:
| Harmonic minor vs. modes from Aeolian (rooted on scale, in “circle of fifths” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| A♭ | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | 1½ | ½ | 1 | ½ | 1 | 1 | ½ | ||
| A♭ | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| E♭ | F♮ | G♮ | A♭ | B♮ | C♮ | D♮ | 1 | 1 | ½ | 1½ | ½ | 1 | ½ | ||
| E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | D♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | ½ | 1 | ½ | 1 | 1 | ½ | 1½ | ||
| B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| F♮ | G♮ | A♭ | B♮ | C♮ | D♮ | E♭ | 1 | ½ | 1½ | ½ | 1 | ½ | 1 | ||
| F♮ | G♮ | A♭ | B♭ | C♮ | D♮ | E♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | 1 | ½ | 1 | 1 | ½ | 1½ | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| G♮ | A♭ | B♮ | C♮ | D♮ | E♭ | F♮ | ½ | 1½ | ½ | 1 | ½ | 1 | 1 | ||
| G♮ | A♭ | B♭ | C♮ | D♮ | E♭ | F♮ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | C♮ | ½ | 1 | 1 | ½ | 1½ | ½ | 1 | ||
| D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
Analysis of chord tonality by scale position:
| Chord tonalities by scale position & mode (harmonic minor, linear order) | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
| 1 | min | dim | AUG | min | Maj | Maj | dim | ||
| 2 | dim | AUG | min | Maj | Maj | dim | min | ||
| 3 | AUG | min | Maj | Maj | dim | min | dim | ||
| 4 | min | Maj | Maj | dim | min | dim | AUG | ||
| 5 | Maj | Maj | dim | min | dim | AUG | min | ||
| 6 | Maj | dim | min | dim | AUG | min | Maj | ||
| 7 | dim | min | dim | AUG | min | Maj | Maj | ||
| Chord tonalities by scale position & mode (harmonic minor, “circle of fifths” order) | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
| 4 | min | Maj | Maj | dim | min | dim | AUG | ||
| 1 | min | dim | AUG | min | Maj | Maj | dim | ||
| 5 | Maj | Maj | dim | min | dim | AUG | min | ||
| 2 | dim | AUG | min | Maj | Maj | dim | min | ||
| 6 | Maj | dim | min | dim | AUG | min | Maj | ||
| 3 | AUG | min | Maj | Maj | dim | min | dim | ||
| 7 | dim | min | dim | AUG | min | Maj | Maj | ||
The circle of fifths table here may clarify one reason harmonic minor requires so many more changes to rotate than Ionian does. Note how only one chord stays the same between any two successive rows of the harmonic minor “circle of fifths” table, while four stayed the same between any two successive rows of Ionian’s table (e.g., Ionian and Mixolydian both have I, ii, IV, and vi chords). For each mode rooted on C, we get the following chords:
| Chords for C harmonic minor’s parallel modes | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
| 1 | C min | D dim | E♭ aug | F min | G maj | A♭ maj | B dim | ||
| 2 | C dim | D♭ aug | E♭ min | F maj | G♭ maj | A dim | B♭ min | ||
| 3 | C aug | D min | E maj | F maj | G♯ dim | A min | B dim | ||
| 4 | C min | D maj | E♭ maj | F♯ dim | G min | A dim | B♭ aug | ||
| 5 | C maj | D♭ maj | E dim | F min | G dim | A♭ aug | B♭ min | ||
| 6 | C maj | D♯ dim | E min | F♯ dim | G aug | A min | B maj | ||
| 7 | C dim | D♭ min | E♭ dim | F♭ aug | G min | A♭ maj | B𝄫 maj | ||
Meanwhile, the chords for C harmonic minor’s relative modes are as follows:
| Chords for C harmonic minor’s relative modes | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
| 1 | C min | D dim | E♭ aug | F min | G maj | A♭ maj | B dim | ||
| 2 | D dim | E♭ aug | F min | G maj | A♭ maj | B dim | C min | ||
| 3 | E♭ aug | F min | G maj | A♭ maj | B dim | C min | D dim | ||
| 4 | F min | G maj | A♭ maj | B dim | C min | D dim | E♭ aug | ||
| 5 | G maj | A♭ maj | B dim | C min | D dim | E♭ aug | F min | ||
| 6 | A♭ maj | B dim | C min | D dim | E♭ aug | F min | G maj | ||
| 7 | B dim | C min | D dim | E♭ aug | F min | G maj | A♭ maj | ||
Melodic minor
Melodic minor is perhaps better related to the modes starting with Ionian. I haven’t drawn borders this time, because… well, it’s easier to show the table, then explain.
| Melodic minor vs. modes from Ionian (rooted on C, linear order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♯ | G♯ | A♮ | B♮ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♭ | B♭ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♭ | G♭ | A♭ | B♭ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
The dual mnemonics for each mode of melodic minor in this table effectively show how we can derive melodic minor and each of its modes in two different ways from two different modes of Ionian:
- Melodic minor is Ionian with a flat third or Dorian with a sharp seventh.
- Jazz minor inverse is Dorian with a flat second or Phrygian with a sharp sixth.
- Lydian augmented is Phrygian with a flat first* or Lydian with a sharp fifth.
- Lydian dominant is Lydian with a flat seventh or Mixolydian with a sharp fourth.
- Aeolian dominant is Mixolydian with a flat sixth or Aeolian with a sharp third.
- Half-diminished is Aeolian with a flat fifth or Locrian with a sharp second.
- Super-Locrian is Locrian with a flat fourth or Ionian with a sharp first*.
Asterisks are necessary for the first scale degree when transposing every scale degree to C. When improvising on an existing scale, the same principles apply as with harmonic minor’s Mixolydian ♯1 – the mode’s root will already be transposed within the scale you’re playing, so you just have to bear that in mind when thinking of what notes to play above it. This may be clearer in the following table, which shows how the above modes relate to C melodic minor and C Ionian:
| Melodic minor vs. modes from Ionian (rooted on scale, linear order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | ||
| D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | C♮ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | C♮ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | ||
| E♮ | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| E♭ | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | ||
| F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | E♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 | ||
| G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | F♮ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| G♮ | A♮ | B♮ | C♮ | D♮ | E♭ | F♮ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||
| A♮ | B♮ | C♮ | D♮ | E♮ | F♮ | G♮ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| A♮ | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | ||
| B♮ | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
In short, E♭ Phrygian ♭1’s root note is already flat in its parent scale – you don’t have to flat it again!
“Circle of fifths” order makes it clear that the big note shift from Phrygian to Lydian augmented occurs in the “rooted on C” chart for the same reason the note shift between Mixolydian and Mixolydian ♯1 occurs with the harmonic minor scale: rooting everything to C means we can’t lower the first note and must instead raise the other notes by however much we’d have lowered the first note.
| Melodic minor vs. modes from Ionian (rooted on C, in “circle of fifths” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♭ | B♭ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♯ | G♯ | A♮ | B♮ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♭ | G♭ | A♭ | B♭ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
Phrygian mode corresponds to Ionian’s third degree; that’s the note melodic minor lowers from Ionian. Thus, Phrygian is the mode that undergoes the note shift in the above table. Moreover, C Lydian augmented raises every note of C Phrygian except its root because its parent scale lowers its corresponding note.
For completeness, here’s “circle of fifths” order without transposition.
| Melodic minor vs. modes from Ionian (rooted on scale, in “circle of fifths” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | E♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | ||
| G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | F♮ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| G♮ | A♮ | B♮ | C♮ | D♮ | E♭ | F♮ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||
| D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | C♮ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | C♮ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | ||
| A♮ | B♮ | C♮ | D♮ | E♮ | F♮ | G♮ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| A♮ | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | ||
| E♮ | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| E♭ | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | ||
| B♮ | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
Of course, as the red names in the “rooted on C” table suggest, we can also get melodic minor by raising a different set of notes on a different series of modes. I’ve changed the first set of scale names accordingly, and since this interpretation of melodic minor raises pitches from its parent modes instead of lowering them, I’ve printed it first in this table. Note also Ionian’s different root key here (B♭ major instead of C major).
| Melodic minor vs. modes from Dorian (rooted on scale, linear order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | C♮ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | ||
| D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | C♮ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| E♭ | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | ||
| E♭ | F♮ | G♮ | A♮ | B♭ | C♮ | D♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | E♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 | ||
| F♮ | G♮ | A♮ | B♭ | C♮ | D♮ | E♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| G♮ | A♮ | B♮ | C♮ | D♮ | E♭ | F♮ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||
| G♮ | A♮ | B♭ | C♮ | D♮ | E♭ | F♮ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| A♮ | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | ||
| A♮ | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
| B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
Here’s a comparison of these transformations in “circle of fifths” order, rooted to C:
| Melodic minor vs. modes from Dorian (rooted on C, in “circle of fifths” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♮ | E♮ | F♯ | G♯ | A♮ | B♮ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♭ | G♭ | A♭ | B♭ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♭ | B♭ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
And in “circle of fifths” order rooted on their parent scales:
| Melodic minor vs. modes from Dorian (rooted on scale, in “circle of fifths” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| E♭ | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | ||
| E♭ | F♮ | G♮ | A♮ | B♭ | C♮ | D♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
| B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | E♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 | ||
| F♮ | G♮ | A♮ | B♭ | C♮ | D♮ | E♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| G♮ | A♮ | B♮ | C♮ | D♮ | E♭ | F♮ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||
| G♮ | A♮ | B♭ | C♮ | D♮ | E♭ | F♮ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | C♮ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | ||
| D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | C♮ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| A♮ | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | ||
| A♮ | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
Analysis of chord tonality by scale position. (For reasons that will become clearer in the next section, I’m using Super-Locrian as the top scale of melodic minor’s “circle of fifths” order.)
| Chord tonalities by scale position & mode (melodic minor, linear order) | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
| 1 | min | min | AUG | Maj | Maj | dim | dim | ||
| 2 | min | AUG | Maj | Maj | dim | dim | min | ||
| 3 | AUG | Maj | Maj | dim | dim | min | min | ||
| 4 | Maj | Maj | dim | dim | min | min | AUG | ||
| 5 | Maj | dim | dim | min | dim | AUG | Maj | ||
| 6 | dim | dim | min | min | AUG | Maj | Maj | ||
| 7 | dim | min | min | AUG | Maj | Maj | min | ||
| Chord tonalities by scale position & mode (melodic minor, “circle of fifths” order) | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
| 7 | dim | min | min | AUG | Maj | Maj | min | ||
| 4 | Maj | Maj | dim | dim | min | min | AUG | ||
| 1 | min | min | AUG | Maj | Maj | dim | dim | ||
| 5 | Maj | dim | dim | min | min | AUG | Maj | ||
| 2 | min | AUG | Maj | Maj | dim | dim | min | ||
| 6 | dim | dim | min | min | AUG | Maj | Maj | ||
| 3 | AUG | Maj | Maj | dim | dim | min | min | ||
Oddly, melodic minor’s sucessive modes in circle of fifths order have no chords in common.
| Chords for C melodic minor’s parallel modes | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
| 1 | C min | D min | E♭ aug | F maj | G maj | A dim | B dim | ||
| 2 | C min | D♭ aug | E♭ maj | F maj | G dim | A dim | B♭ min | ||
| 3 | C aug | D maj | E maj | F♯ dim | G♯ dim | A min | B min | ||
| 4 | C maj | D maj | E dim | F♯ dim | G min | A min | B♭ aug | ||
| 5 | C maj | D dim | E dim | F min | G min | A♭ aug | B♭ maj | ||
| 6 | C dim | D dim | E♭ min | F min | G♭ aug | A♭ maj | B♭ maj | ||
| 7 | C dim | D♭ min | E♭ min | F♭ aug | G♭ maj | A♭ maj | B♭ min | ||
| Chords for C melodic minor’s relative modes | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
| 1 | C min | D min | E♭ aug | F maj | G maj | A dim | B dim | ||
| 2 | D min | E♭ aug | F maj | G maj | A dim | B dim | C min | ||
| 3 | E♭ aug | F maj | G maj | A dim | B dim | C min | D min | ||
| 4 | F maj | G maj | A dim | B dim | C min | D min | E♭ aug | ||
| 5 | G maj | A dim | B dim | C min | D min | E♭ aug | F maj | ||
| 6 | A dim | B dim | C min | D min | E♭ aug | F maj | G maj | ||
| 7 | B dim | C min | D min | E♭ aug | F maj | G maj | A min | ||
The Ionian scale’s stability
Let’s use a slightly more flexible root to compare melodic minor’s modes to Ionian’s in both directions – Ionian’s modes, melodic minor’s modes, and Dorian’s modes. Note especially how much stabler the Ionian scale’s root is.
| Melodic minor vs. Ionian & Dorian (rooted on C±½, “circle of fifths” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♯ | D♮ | E♮ | F♯ | G♯ | A♮ | B♮ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♯ | G♯ | A♮ | B♮ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| C♯ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| C♯ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♭ | B♭ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| C♭ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | ||
| C♭ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♭ | G♭ | A♭ | B♭ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
| C♭ | D♭ | E♭ | F♭ | G♭ | A♭ | B♭ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
None of these representations are perfect, but together, they may help illuminate how these scales’ modes are related. As you can see, it’s quite messy – we have to move our root up or down a half-step at points to preserve relationships to the Ionian scale and the circle of fifths. Whether we read melodic minor as or as , we must move its root three times in a row to keep the circle of fifths progression stable:
- Half-diminished to Lydian augmented: lower it a half-step
- Lydian augmented to Super-Locrian: raise it a half-step
- Super-Locrian to Lydian dominant: lower it a half-step
If the above table continued, its next three modes would be B Lydian, B Lydian dominant, and B Mixolydian – and a case could be made for rewriting its last six rows as B♯ Phrygian, B Lydian augmented, B Lydian, B♯ Locrian, B♯ Super-Locrian, and B Ionian.
One further set of comparisons involves melodic minor and Mixolydian. Oddly, this lines up better in several respects: in particular, it lines up each scale’s symmetrical modes (Aeolian dominant and Dorian) and balances accidentals across the comparison (i.e., C♯ Super-Locrian and C Lydian each have one sharp; C Ionian has no accidentals, while C Lydian dominant has a sharp and a flat).
| Melodic minor vs. modes from Mixolydian (rooted on C±½, linear order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♭ | B♭ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| C♯ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| C♭ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | ||
| Melodic minor vs. modes from Mixolydian (rooted on C±½, “circle of fifths” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| C♯ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♭ | B♭ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| C♭ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | ||
Although this comparison swaps two interval pairs from the Ionian modes we’re comparing them to, it also only swaps two of their notes (which may include the root). There are two ways to read this set of transformations:
- Melodic minor swaps Mixolydian’s second interval with its third and its sixth with its seventh.
- Melodic minor swaps Mixolydian’s second and third intervals with its sixth and seventh.
The latter is probably the more helpful way to read it. Since both these interval pairs collectively add up to three semitones, only the notes within each interval pair move.
To preseve the pattern, harmonic minor’s modes must shift their roots in similar ways to melodic minor’s, except more unpredictably spaced (which feels inevitable, since its intervals are also less evenly spaced):
- Ionian augmented (#2) to Super-Locrian ♭7 (#3)
- Super-Locrian ♭7 (#3) to Lydian diminished (#4)
- Maqam Tarznauyn (#7) to Aeolian harmonic (#1)
| Harmonic minor’s modes revisited, or, please throw some snacks down this rabbit hole | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♯ | E♮ | F♯ | G♮ | A♮ | B♮ | 1½ | ½ | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♯ | A♮ | B♮ | 1 | 1 | ½ | 1½ | ½ | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| C♯ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | ½ | 1 | 1 | ½ | 1½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♭ | 1 | ½ | 1½ | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | 1 | ½ | 1 | 1 | ½ | 1½ | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♭ | ½ | 1½ | ½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♮ | B♭ | ½ | 1 | 1 | ½ | 1½ | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
The Ionian scale’s descent is comparatively stable: each transformation lowers only one note of the scale, and it lowers that note only by a semitone. As far as I can ascertain, it is the only heptatonic scale for which this is true. This occurs in part because it comes as close as any heptatonic scale in twelve-tone equal temperament can come to having its notes evenly spaced, without being precisely even. Two whole tones, a semitone, three whole tones, and a semitone. The fact that these notes, in turn, traverse the circle of fifths from F to B is the other part of the puzzle.
Descending through Ionian’s modes in circle of fifths order lowers one note every transformation by a semitone. Few other scale transformations are so simple. Harmonic and melodic minor’s transformations each lower two notes by a semitone and raise a third note by a semitone:
| Harmonic minor & melodic minor’s “circle of fifths” progressions | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Shift from Previous Note | |||||||
|
|
C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | −½ | 0 | 0 | 0 | 0 | 0 | 0 | |
|
|
C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 0 | 0 | 0 | −½ | 0 | 0 | 0 | |
|
|
C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 0 | 0 | 0 | 0 | 0 | 0 | −½ | |
|
|
C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 0 | 0 | −½ | 0 | 0 | 0 | 0 | |
|
|
C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 0 | 0 | 0 | 0 | 0 | −½ | 0 | |
|
|
C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | 0 | −½ | 0 | 0 | 0 | 0 | 0 | |
|
|
C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | 0 | 0 | 0 | 0 | −½ | 0 | 0 | |
|
|
C♭ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | −½ | 0 | 0 | 0 | 0 | 0 | 0 | |
|
|
C♮ | D♯ | E♮ | F♯ | G♮ | A♮ | B♮ | −½ | +½ | 0 | 0 | 0 | −½ | 0 | |
|
|
C♮ | D♮ | E♮ | F♮ | G♯ | A♮ | B♮ | 0 | −½ | 0 | −½ | +½ | 0 | 0 | |
|
|
C♯ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | +½ | 0 | 0 | 0 | −½ | 0 | −½ | |
|
|
C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♭ | −½ | 0 | −½ | +½ | 0 | 0 | 0 | |
|
|
C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | 0 | 0 | 0 | −½ | 0 | −½ | +½ | |
|
|
C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♭ | 0 | −½ | +½ | 0 | 0 | 0 | −½ | |
|
|
C♮ | D♭ | E♭ | F♮ | G♭ | A♮ | B♭ | 0 | 0 | −½ | 0 | −½ | +½ | 0 | |
|
|
C♭ | D♮ | E♭ | F♮ | G♭ | A♭ | B♭ | −½ | +½ | 0 | 0 | 0 | −½ | 0 | |
|
|
C♭ | D♭ | E♭ | F♭ | G♮ | A♭ | B♭ | 0 | −½ | 0 | −½ | +½ | 0 | 0 | |
|
|
C♮ | D♭♮ | E♭♮ | F♭♮ | G♭♮ | A♭♮ | B𝄫 | +½ | 0 | 0 | 0 | −½ | 0 | −½ | |
|
|
C♭ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B𝄫 | −½ | 0 | −½ | +½ | 0 | 0 | 0 | |
| ♮ | C♮ | D♮ | E♮ | F♯ | G♯ | A♮ | B♮ | −½ | −½ | 0 | 0 | +½ | 0 | 0 | |
| C♯ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | +½ | 0 | 0 | −½ | −½ | 0 | 0 | ||
| ♮ | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♭ | −½ | 0 | 0 | +½ | 0 | 0 | −½ | |
| ♮ | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 0 | 0 | −½ | −½ | 0 | 0 | +½ | |
| ♮ | C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♭ | 0 | 0 | +½ | 0 | 0 | −½ | −½ | |
| ♮ | C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♭ | 0 | −½ | −½ | 0 | 0 | +½ | 0 | |
| ♮ | C♮ | D♮ | E♭ | F♮ | G♭ | A♭ | B♭ | 0 | +½ | 0 | 0 | −½ | −½ | 0 | |
| C♭ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | −½ | −½ | 0 | 0 | +½ | 0 | 0 | ||
| ♮ | C♮ | D♭ | E♭ | F♭ | G♭ | A♭ | B♭ | +½ | 0 | 0 | −½ | −½ | 0 | 0 | |
Note a few additional patterns here:
- Harmonic minor:
-
The fourth scale degree changes three times over four consecutive transformations:
- C Ionian ♯5 shifts F♯ to F♮. (This is the normal circle of fifths transformation.)
- C Dorian ♯4 shifts F back to F♯.
- C Aeolian ♯7 changes F♯ once more back to F♮.
-
The seventh scale degree also changes three times over four consecutive transformations:
- C♯ Mixolydian ♯1 shifts B to B♭. (This is the normal circle of fifths transformation.)
- C Dorian ♯4 shifts B♭ back to B♮.
- C Aeolian ♯7 changes B once more back to F♭.
-
Tritonality:
- These notes are a tritone apart from each other in the base scale.
- The Ionian scale contains only one tritone: F and B.
- Harmonic minor contains two: D is also a tritone away from A♭.
- Transforming Ionian to harmonic minor thus requires adding a second tritone.
-
- Melodic minor:
-
The fourth scale degree changes three times over three consecutive transformations:
- C♯ Ionian ♯1 (C♯ Locrian ♭4) shifts F♯ to F♮. (This is the normal circle of fifths transformation.)
- C Mixolydian ♯4 (C Lydian ♭7) shifts F back to F♯.
- C Dorian ♯7 (C Ionian ♭3) changes F♯ once more back to F♮.
-
The seventh scale degree also changes three times over three consecutive transformations:
- C♯ Mixolydian ♯4 (C♯ Lydian ♭7) shifts B to B♭. (This is the normal circle of fifths transformation.)
- C Dorian ♯7 (C Ionian ♭3) shifts B♭ back to F♮.
- C Aeolian ♯3 (C Mixolydian ♭6) changes F♯ once more back to F♮.
-
Tritonality:
- These notes are also a tritone apart from each other in the base scale.
- Like harmonic minor, melodic minor has two tritones: E♭ to A.
- Transforming Ionian to melodic minor thus requires adding a second tritone.
-
-
Melodic minor’s second set of tritones is exactly a half-step removed from harmonic minor’s. This is a direct consequence of their mathematical relationships in 12-TET:
- C Aeolian is tonally identical to C Dorian ♭6 (conversely: C Dorian is tonally identical to C Aeolian ♯6).
- C melodic minor is tonally identical to C Dorian ♯7
- C harmonic minor is tonally identical to C Aeolian ♯7.
As a direct consequence of the above, the following statements are logically equivalent:
- Changing C Dorian (C Aeolian ♯6) to C Aeolian (C Dorian ♭6) requires only changing A to A♭.
- Changing C melodic minor to C harmonic minor requires only changing A to A♭.
- Changing C Dorian ♯7 to C Aeolian ♯7 requires only changing A to A♭.
- The Ionian scale is one of 12-TET’s only two heptatonic scales with only a single tritone. It also has the most uniform note spacing of any heptatonic scale in 12-TET, as I’ll prove mathematically below.
-
As a mathematical consequence of the following premises:
- C Aeolian follows C Dorian in the Ionian scale’s circle of fifths order.
- C melodic minor is C Dorian ♯7.
- C harmonic minor is C Aeolian ♯7.
- Shifting harmonic minor’s transformed mode up a row (Aeolian to Dorian) gets us melodic minor.
We can express melodic minor’s transformations of Ionian’s modes using:
- The mode from one row of harmonic minor’s transformations of Ionian’s modes
- The key and scale degree shift from the next
We can thus consider melodic minor a sort of midpoint between two transformations of harmonic minor:
One weird trick to transform harmonic minor to melodic minor Harmonic minor − ½ Melodic minor Harmonic minor + ½ C Lydian ♯2 C Lydian ♯5 C Ionian ♯5 C Ionian ♯5 C♯ Ionian ♯1 C♯ Mixolydian ♯1 C♯ Mixolydian ♯1 C Mixolydian ♯4 C Dorian ♯4 C Dorian ♯4 C Dorian ♯7 C Aeolian ♯7 C Aeolian ♯7 C Aeolian ♯3 C Phrygian ♯3 C Phrygian ♯3 C Phrygian ♯6 C Locrian ♯6 C Locrian ♯6 C Locrian ♯2 C♭ Lydian ♯2 C♭ Lydian ♯2 C♭ Lydian ♯5 C♭ Ionian ♯5 - If you’re starting to suspect that absolutely none of this is coincidental, you’re right.
(At some point, I plan to make equivalents of §3’s charts for at least melodic minor and harmonic minor, and perhaps for some of the Greek scales I discuss below as well… but not until I’ve written programs to automate their generation, which could take anywhere from a few days to months.)
Mathematical proof of even spacing
Ionian has the most even interval distribution any seven-note scale can have in 12-TET, and I’ll prove it.
-
In n-tone equal temperament, for an s-note scale, divide n by s to determine the mean interval size, expressed in units of 1/n octave (i.e., the smallest possible interval; in 12-TET, a semitone).
12/7 is 1.714285714….
-
If there is a remainder, discard it to get the scale’s lower interval size, and add one to the result to get its larger interval size. If there is no remainder, skip the rest of this proof; you’re done (but, as Oliver Messiaen would’ve already told you, you won’t have any other modes).
That’s 1. So all intervals in the scale should be at least a semitone (and, in fact, must be).
-
Take the modulus of the temperament size and the scale size to determine how many intervals will get an extra 1/n octave added to them.
12 modulo 7 is 5. Thus, we have five leftover half-steps to add to five of the intervals.
-
Subtract the number of leftovers from the number of notes in the scale to determine how many intervals don’t get an extra 1/n octave.
7 - 5 is 2. Thus, five intervals have added semitones, two don’t. Five whole steps, two half steps.
-
To determine the most even interval distribution, first divide the number of occurrences of the more frequent interval by the number of occurrences of the less frequent one.
- If the divisor is 0, you need to work on your reading comprehension: step 2 already told you to stop.
- If the less frequent interval occurs multiple times and the quotient is an integer, the scale will repeat the same pattern multiple times per octave and is therefore a mode of limited transposition.
- If the less frequent interval occurs nearly as often as the more frequent one, you may need to repeat some of these steps, replacing n with the count of the more frequent interval and s with the count of the less frequent one.
5 / 2 = 2.5. There should be a median of 2.5 occurrences of the more frequent interval in a row.
-
The interval distribution should take roughly this form: small group of more frequent interval, less frequent interval, large group of more frequent interval, less frequent interval. If there are more than two groups of the less frequent interval, determining the optimal distribution may require repeating some of these steps, replacing with n the more frequent interval’s count and s with the less frequent one’s count.
Since we can’t have a whole step exactly 2.5 times in a row, we’ll have to have two in one group and three in another. That gives us two whole tones, a semitone, three whole tones, and a semitone. That’s Ionian. I literally just described the Ionian scale. Median number of whole tones in a row: 2.5. Therefore, its semitones are as evenly spread out as they possibly can be between its whole tones.
Surely that also applies to its complement, right? Let’s look at the pentatonic scale.
-
In n-tone equal temperament, for an s-note scale, divide n by s to determine the mean interval size, expressed in units of 1/n octave (i.e., the smallest possible interval; in 12-TET, a semitone).
12/5 is 2.4.
-
If there is a remainder, discard it to get the scale’s lower interval size, and add one to the result to get its larger interval size. If there is no remainder, skip the rest of this proof; you’re done (but, as Oliver Messiaen would’ve already told you, you won’t have any other modes).
That’s 2. So all intervals in the scale should be at least a whole tone.
-
Take the modulus of the temperament size and the scale size to determine how many intervals will get an extra 1/n octave added to them.
12 modulo 5 is 2. So we have two leftover half-steps to add to two of the intervals.
-
Subtract the number of leftovers from the number of notes in the scale to determine how many intervals don’t get an extra 1/n octave.
5 - 2 is 3. Thus, two intervals have extra semitones, three won’t. Three whole steps, two minor thirds.
-
To determine the most even interval distribution, first divide the number of occurrences of the more frequent interval by the number of occurrences of the less frequent one.
- If the divisor is 0, you need to work on your reading comprehension: step 2 already told you to stop.
- If the less frequent interval occurs multiple times and the quotient is an integer, the scale will repeat the same pattern multiple times per octave and is therefore a mode of limited transposition.
- If the less frequent interval occurs nearly as often as the more frequent one, you may need to repeat some of these steps, replacing n with the count of the more frequent interval and s with the count of the less frequent one.
3 / 2 = 1.5. There should be a median of 1.5 occurrences of the more frequent interval in a row.
-
The interval distribution should take roughly this form: small group of more frequent interval, less frequent interval, large group of more frequent interval, less frequent interval. If there are more than two groups of the less frequent interval, determining the optimal distribution may require repeating some of these steps, replacing with n the more frequent interval’s count and s with the less frequent one’s count.
Since we can’t have exactly 1.5 whole steps in a row, we’ll have to have two in one group and one by itself. That gives us two whole tones, a minor third, a whole tone, and a minor third. Which, again, is the pentatonic scale.
Of course, we already knew this. If the Ionian scale has the most even interval distribution a heptatonic scale can have in 12-TET, its complement must also have the most even interval distribution a pentatonic scale can have in 12-TET, by definition. Nonetheless, it’s nice to prove it mathematically.
Other single-note transformations of Ionian
So far, we’ve almost exclusively explored single-note transformations of Ionian and its modes:
- Ionian ♭3 is melodic minor.
- Ionian ♯1 is Super-Locrian, melodic minor’s sixth mode (counting melodic minor itself as the first).
- Ionian ♯5 is Ionian augmented, harmonic minor’s fifth mode.
Do other single-note transformations exist? Yes, but fewer than you might expect:
-
As we already know, two transformations create other modes of Ionian:
- Ionian ♯4 is Lydian.
- Ionian ♭7 is Mixolydian.
-
Four transformations would create hexatonic scales due to note duplication:
- Ionian ♭1: C♭ is enharmonically equivalent to B.
- Ionian ♯3: E♯ is enharmonically equivalent to F.
- Ionian ♭4: F♭ is enharmonically equivalent to E.
- Ionian ♯7: B♯ is enharmonically equivalent to C.
Thus, eight single-note transformations create heptatonic scales that aren’t other modes of Ionian. (I’ve printed the five we haven’t yet explored in bold, blue text.)
| Threshold of Transformation | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♯ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♮ | B♮ | ½ | 1½ | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♯ | E♮ | F♮ | G♮ | A♮ | B♮ | 1½ | ½ | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♭ | A♮ | B♮ | 1 | 1 | ½ | ½ | 1½ | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♯ | A♮ | B♮ | 1 | 1 | ½ | 1½ | ½ | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♮ | 1 | 1 | ½ | 1 | ½ | 1½ | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♯ | B♮ | 1 | 1 | ½ | 1 | 1½ | ½ | ½ | ||
Expanding those gives us:
| 4. Expand, expand, expand. Clear forest, make land, fresh blood on hands | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♮ | B♮ | ½ | 1½ | ½ | 1 | 1 | 1 | ½ | ||
| C♭ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1½ | ½ | 1 | 1 | 1 | ½ | ½ | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B𝄫 | ½ | 1 | 1 | 1 | ½ | ½ | 1½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♭ | B♮ | 1 | 1 | 1 | ½ | ½ | 1½ | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♭ | A♮ | B♭ | 1 | 1 | ½ | ½ | 1½ | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B♭ | 1 | ½ | ½ | 1½ | ½ | 1 | 1 | ||
| C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B♭ | ½ | ½ | 1½ | ½ | 1 | 1 | 1 | ||
| C♮ | D♯ | E♮ | F♮ | G♮ | A♮ | B♮ | 1½ | ½ | ½ | 1 | 1 | 1 | ½ | ||
| C♯ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | ½ | 1 | 1 | 1 | ½ | 1½ | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♮ | ½ | 1 | 1 | 1 | ½ | 1½ | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♯ | B♮ | 1 | 1 | 1 | ½ | 1½ | ½ | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♯ | A♮ | B♭ | 1 | 1 | ½ | 1½ | ½ | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♭ | 1 | ½ | 1½ | ½ | ½ | 1 | 1 | ||
| C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♭ | ½ | 1½ | ½ | ½ | 1 | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♭ | A♮ | B♮ | 1 | 1 | ½ | ½ | 1½ | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♭ | G♮ | A♮ | B♭ | 1 | ½ | ½ | 1½ | 1 | ½ | 1 | ||
| C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B♭ | ½ | ½ | 1½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♮ | F♯ | G♮ | A♮ | B♮ | ½ | 1½ | 1 | ½ | 1 | 1 | ½ | ||
| C♭ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1½ | 1 | ½ | 1 | 1 | ½ | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B𝄫 | 1 | ½ | 1 | 1 | ½ | ½ | 1½ | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A𝄫 | B♭ | ½ | 1 | 1 | ½ | ½ | 1½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♯ | B♮ | 1 | 1 | ½ | 1 | 1½ | ½ | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♯ | A♮ | B♭ | 1 | ½ | 1 | 1½ | ½ | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♭ | ½ | 1 | 1½ | ½ | ½ | 1 | 1 | ||
| C♮ | D♮ | E♯ | F♯ | G♮ | A♮ | B♮ | 1 | 1½ | ½ | ½ | 1 | 1 | ½ | ||
| C♮ | D♯ | E♮ | F♮ | G♮ | A♮ | B♭ | 1½ | ½ | ½ | 1 | 1 | ½ | 1 | ||
| C♯ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | ½ | 1 | 1 | ½ | 1 | 1½ | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♮ | ½ | 1 | 1 | ½ | 1 | 1½ | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♮ | 1 | 1 | ½ | 1 | ½ | 1½ | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♮ | B♭ | 1 | ½ | 1 | ½ | 1½ | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B♭ | ½ | 1 | ½ | 1½ | ½ | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♮ | 1 | ½ | 1½ | ½ | 1 | 1 | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♮ | B♭ | ½ | 1½ | ½ | 1 | 1 | ½ | 1 | ||
| C♭ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1½ | ½ | 1 | 1 | ½ | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B𝄫 | ½ | 1 | 1 | ½ | 1 | ½ | 1½ | ||
Or, in circle of fifths order:
| 5. Why just shells? Why limit yourself? She sells seashells; sell oil as well | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♭ | B♮ | 1 | 1 | 1 | ½ | ½ | 1½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♮ | B♮ | ½ | 1½ | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♭ | A♮ | B♭ | 1 | 1 | ½ | ½ | 1½ | ½ | 1 | ||
| C♭ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1½ | ½ | 1 | 1 | 1 | ½ | ½ | ||
| C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B♭ | 1 | ½ | ½ | 1½ | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B𝄫 | ½ | 1 | 1 | 1 | ½ | ½ | 1½ | ||
| C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B♭ | ½ | ½ | 1½ | ½ | 1 | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♯ | B♮ | 1 | 1 | 1 | ½ | 1½ | ½ | ½ | ||
| C♮ | D♯ | E♮ | F♮ | G♮ | A♮ | B♮ | 1½ | ½ | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♯ | A♮ | B♭ | 1 | 1 | ½ | 1½ | ½ | ½ | 1 | ||
| C♯ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | ½ | 1 | 1 | 1 | ½ | 1½ | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♭ | 1 | ½ | 1½ | ½ | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♮ | ½ | 1 | 1 | 1 | ½ | 1½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♭ | ½ | 1½ | ½ | ½ | 1 | 1 | 1 | ||
| C♮ | D♭ | E♮ | F♯ | G♮ | A♮ | B♮ | ½ | 1½ | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♭ | A♮ | B♮ | 1 | 1 | ½ | ½ | 1½ | 1 | ½ | ||
| C♭ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1½ | 1 | ½ | 1 | 1 | ½ | ½ | ||
| C♮ | D♮ | E♭ | F♭ | G♮ | A♮ | B♭ | 1 | ½ | ½ | 1½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B𝄫 | 1 | ½ | 1 | 1 | ½ | ½ | 1½ | ||
| C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B♭ | ½ | ½ | 1½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A𝄫 | B♭ | ½ | 1 | 1 | ½ | ½ | 1½ | 1 | ||
| C♮ | D♮ | E♯ | F♯ | G♮ | A♮ | B♮ | 1 | 1½ | ½ | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♯ | B♮ | 1 | 1 | ½ | 1 | 1½ | ½ | ½ | ||
| C♮ | D♯ | E♮ | F♮ | G♮ | A♮ | B♭ | 1½ | ½ | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♯ | A♮ | B♭ | 1 | ½ | 1 | 1½ | ½ | ½ | 1 | ||
| C♯ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | ½ | 1 | 1 | ½ | 1 | 1½ | ||
| C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♭ | ½ | 1 | 1½ | ½ | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♮ | ½ | 1 | 1 | ½ | 1 | 1½ | ½ | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♮ | 1 | ½ | 1½ | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♮ | 1 | 1 | ½ | 1 | ½ | 1½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♮ | B♭ | ½ | 1½ | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♮ | B♭ | 1 | ½ | 1 | ½ | 1½ | ½ | 1 | ||
| C♭ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1½ | ½ | 1 | 1 | ½ | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B♭ | ½ | 1 | ½ | 1½ | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B𝄫 | ½ | 1 | 1 | ½ | 1 | ½ | 1½ | ||
Note that Dorian ♭1 is the mathematical inverse of Dorian ♯1, as is Dorian ♯5 of Dorian ♭4. Dorian ♭5 is likewise the mathematical inverse of Dorian ♯4, harmonic minor’s fourth mode. To clarify:
| 6. Guns, sell stocks, sell diamonds, sell rocks, sell water to a fish, sell the time to a clock | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♭ | B♮ | 1 | 1 | 1 | ½ | ½ | 1½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♭ | ½ | 1½ | ½ | ½ | 1 | 1 | 1 | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♮ | B♮ | ½ | 1½ | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♮ | ½ | 1 | 1 | 1 | ½ | 1½ | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♭ | A♮ | B♭ | 1 | 1 | ½ | ½ | 1½ | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♭ | 1 | ½ | 1½ | ½ | ½ | 1 | 1 | ||
| C♭ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1½ | ½ | 1 | 1 | 1 | ½ | ½ | ||
| C♯ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | ½ | 1 | 1 | 1 | ½ | 1½ | ||
| C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B♭ | 1 | ½ | ½ | 1½ | ½ | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♯ | A♮ | B♭ | 1 | 1 | ½ | 1½ | ½ | ½ | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B𝄫 | ½ | 1 | 1 | 1 | ½ | ½ | 1½ | ||
| C♮ | D♯ | E♮ | F♮ | G♮ | A♮ | B♮ | 1½ | ½ | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B♭ | ½ | ½ | 1½ | ½ | 1 | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♯ | B♮ | 1 | 1 | 1 | ½ | 1½ | ½ | ½ | ||
| C♮ | D♭ | E♮ | F♯ | G♮ | A♮ | B♮ | ½ | 1½ | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♮ | ½ | 1 | 1 | ½ | 1 | 1½ | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♭ | A♮ | B♮ | 1 | 1 | ½ | ½ | 1½ | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♭ | ½ | 1 | 1½ | ½ | ½ | 1 | 1 | ||
| C♭ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1½ | 1 | ½ | 1 | 1 | ½ | ½ | ||
| C♯ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | ½ | 1 | 1 | ½ | 1 | 1½ | ||
| C♮ | D♮ | E♭ | F♭ | G♮ | A♮ | B♭ | 1 | ½ | ½ | 1½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♯ | A♮ | B♭ | 1 | ½ | 1 | 1½ | ½ | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B𝄫 | 1 | ½ | 1 | 1 | ½ | ½ | 1½ | ||
| C♮ | D♯ | E♮ | F♮ | G♮ | A♮ | B♭ | 1½ | ½ | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B♭ | ½ | ½ | 1½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♯ | B♮ | 1 | 1 | ½ | 1 | 1½ | ½ | ½ | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A𝄫 | B♭ | ½ | 1 | 1 | ½ | ½ | 1½ | 1 | ||
| C♮ | D♮ | E♯ | F♯ | G♮ | A♮ | B♮ | 1 | 1½ | ½ | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♮ | 1 | ½ | 1½ | ½ | 1 | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♮ | B♭ | ½ | 1 | 1 | ½ | 1½ | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♮ | 1 | 1 | ½ | 1 | ½ | 1½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♭ | ½ | 1½ | ½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♮ | B♭ | ½ | 1½ | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | 1 | ½ | 1 | 1 | ½ | 1½ | ½ | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♮ | B♭ | 1 | ½ | 1 | ½ | 1½ | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♭ | 1 | ½ | 1½ | ½ | 1 | ½ | 1 | ||
| C♭ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1½ | ½ | 1 | 1 | ½ | 1 | ½ | ||
| C♯ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | ½ | 1 | 1 | ½ | 1½ | ||
| C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B♭ | ½ | 1 | ½ | 1½ | ½ | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♯ | A♮ | B♮ | 1 | 1 | ½ | 1½ | ½ | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B𝄫 | ½ | 1 | 1 | ½ | 1 | ½ | 1½ | ||
| C♮ | D♯ | E♮ | F♯ | G♮ | A♮ | B♮ | 1½ | ½ | 1 | ½ | 1 | 1 | ½ | ||
Scale transformations and symmetry
The above data aren’t very easy to parse. Clearly, some of these transformations produce more symmetrical and (for lack of a better term) stabler scales than others. The question is, why? I’m still piecing together the answer, but a piece of the puzzle has to do with note distributions.
The Ionian scale is internally symmetrical: Dorian mode has the same interval order forwards and backwards. Ionian is tied among heptatonic scales in 12-TET for the smallest number of semitone intervals (only two), and it has fewer tritones than any other heptatonic scale in 12-TET. Also – and this is probably the most important fact here – Ionian has the most uniform note distribution among heptatonic scales in 12-TET, but – and this part is equally important – it isn’t completely uniform. As we’ll see later when we discuss modes of limited transposition, completely uniform scales don’t even have modes.
Transforming a single note can thus completely destroy the scale symmetry. As it happens, of all the single-note transformations that produce symmetrical scales, both sets produce versions of the melodic minor scale, which we’ve already examined. Not coincidentally, this also is the closest heptatonic scale within 12-TET to Ionian’s stability respective to the circle of fifths: while one must transform each note three times for its equivalent of Ionian’s descent through its modes, at least it’s the same note three times in a row.
Another important note is that each transformation of Ionian that raises a note has an equal and opposite transformation that lowers a note and produces the first transformation’s reflection. For twelve of the fourteen transformations that produce modes of melodic minor, that reflection is another mode of melodic minor; for the remaining two, that reflection is itself, but is applied to a different mode of Ionian:
- Aeolian ♯3 produces the same (symmetrical) result as Mixolydian ♭6.
- Aeolian ♭5 produces Locrian ♯2’s mirror image. (Or, Mixolydian ♯4 produces Lydian ♭7’s mirror image.)
While writing Ionian’s modes this way is pretentious, we can say the same of its own internal transformations:
- Mixolydian ♭3 produces the same (symmetrical) result as Aeolian ♯6, namely, Dorian mode.
- Aeolian ♭2 produces Mixolydian ♯7’s mirror image. (Or, Locrian ♯5 produces Lydian ♭4’s mirror image.)
All single-note transformations of the Ionian scale that don’t create hexatonic scales or modes of melodic minor or Ionian result in scales with enantiomorphs (Attic Greek: ἐναντίος, enantíos, opposite, + μορφή, morphḗ, form), which all appear in different sets of single-note transformations of the Ionian scale. Only scales that cannot be transformed into their inversions by rotation have enantiomorphs. Thus:
- If any mode of a scale has palindromic intervals, that scale does not have enantiomorphs. Aeolian ♯3 has palindromic intervals and therefore no enantiomorph.
- If a scale’s mirror image is one of its own modes, it does not have an an enantiomorph. Ionian is the mirror image of Phrygian. Ionian has no enantiomorph.
In the following table, I’ve taken the liberty of rotating Lydian ♯5 to the end of the first set of scale comparisons, and Locrian ♭4 to the start of the second. I had several reasons for this:
- This places the symmetrical mode, Aeolian ♯3 / Mixolydian ♭6, in the center of the comparisons.
- The table’s other comparisons are between two discrete sets of scale transformations, but here, we compare a set of scale transformations to itself. This places our comparison in sync with itself.
-
These are the table’s only comparisons of single-note transformations that can be derived from two discrete parent modes. Oddly, shifting the scales like this actually approximates our usual circle of fifths order:
- Both sets now open with Ionian ♯1 / Locrian ♭4, between which is Lydian.
- Both sets now close with Lydian ♯5 / Phrygian ♭1, between which is Locrian.
- Remember in the pentatonic scale analysis how I said symmetrical modes should be circle-order comparisons’ central rows? Now it is.
-
This results in a few additional quirks:
- The 7×7 interval inset has 180° rotational symmetry.
- Both halves also have identical interval distributions, with a pattern that spans across them.
Other one-note transformations don’t produce symmetrical scales; therefore, they have reflections.
- Dorian ♯5 is Dorian ♭4’s reflection.
- Dorian ♯4 is Dorian ♭5’s reflection.
- Dorian ♯1 is Dorian ♭1’s reflection (if we disregard transposition of the root).
I specifically used Dorian mode for these examples because it’s symmetrical in the base scale, but we can still make similar comparisons for the other six modes, since they each have reflections within the Ionian scale:
- Ionian’s reflection is Phrygian.
- Locrian’s reflection is Lydian.
- Aeolian’s reflection is Mixolydian.
Thus, the reflection of a transformation of a non-palindromic mode applies to the parent mode’s reflection:
- Aeolian ♭4 is Mixolydian ♯5’s reflection.
- Aeolian ♯4 is Mixolydian ♭5’s reflection.
- Ionian ♯5 is Phrygian ♭4’s reflection.
- Phrygian ♯4 is Ionian ♭5’s reflection.
- Ionian ♭3 is Phrygian ♯6’s reflection.
- Phrygian ♯3 is Ionian ♭6’s reflection.
We can observe all this in the table below.
| 7. Press on the gas, take your foot off the brakes; then run to be the president of the United States | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♯ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♭ | B♭ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | ||
| C♭ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | ||
| C♯ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | ½ | 1 | ½ | 1 | 1 | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♭ | B♭ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | ||
| C♭ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | ||
| C♮ | D♭ | E♮ | F♯ | G♮ | A♮ | B♮ | ½ | 1½ | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♭ | A♮ | B♮ | 1 | 1 | ½ | ½ | 1½ | 1 | ½ | ||
| C♭ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1½ | 1 | ½ | 1 | 1 | ½ | ½ | ||
| C♮ | D♮ | E♭ | F♭ | G♮ | A♮ | B♭ | 1 | ½ | ½ | 1½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B𝄫 | 1 | ½ | 1 | 1 | ½ | ½ | 1½ | ||
| C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B♭ | ½ | ½ | 1½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A𝄫 | B♭ | ½ | 1 | 1 | ½ | ½ | 1½ | 1 | ||
| C♮ | D♮ | E♯ | F♯ | G♮ | A♮ | B♮ | 1 | 1½ | ½ | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♯ | B♮ | 1 | 1 | ½ | 1 | 1½ | ½ | ½ | ||
| C♮ | D♯ | E♮ | F♮ | G♮ | A♮ | B♭ | 1½ | ½ | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♯ | A♮ | B♭ | 1 | ½ | 1 | 1½ | ½ | ½ | 1 | ||
| C♯ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | ½ | 1 | 1 | ½ | 1 | 1½ | ||
| C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♭ | ½ | 1 | 1½ | ½ | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♮ | ½ | 1 | 1 | ½ | 1 | 1½ | ½ | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♮ | 1 | ½ | 1½ | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♮ | 1 | 1 | ½ | 1 | ½ | 1½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♮ | B♭ | ½ | 1½ | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♭ | A♮ | B♭ | 1 | ½ | 1 | ½ | 1½ | ½ | 1 | ||
| C♭ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1½ | ½ | 1 | 1 | ½ | 1 | ½ | ||
| C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B♭ | ½ | 1 | ½ | 1½ | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B𝄫 | ½ | 1 | 1 | ½ | 1 | ½ | 1½ | ||
| C♮ | D♯ | E♮ | F♯ | G♮ | A♮ | B♮ | 1½ | ½ | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♯ | A♮ | B♮ | 1 | 1 | ½ | 1½ | ½ | 1 | ½ | ||
| C♯ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | ½ | 1 | 1 | ½ | 1½ | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♭ | 1 | ½ | 1½ | ½ | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | 1 | ½ | 1 | 1 | ½ | 1½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♭ | ½ | 1½ | ½ | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♮ | B♭ | ½ | 1 | 1 | ½ | 1½ | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♭ | B♮ | 1 | 1 | 1 | ½ | ½ | 1½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♮ | B♮ | ½ | 1½ | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♭ | A♮ | B♭ | 1 | 1 | ½ | ½ | 1½ | ½ | 1 | ||
| C♭ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1½ | ½ | 1 | 1 | 1 | ½ | ½ | ||
| C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B♭ | 1 | ½ | ½ | 1½ | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B𝄫 | ½ | 1 | 1 | 1 | ½ | ½ | 1½ | ||
| C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B♭ | ½ | ½ | 1½ | ½ | 1 | 1 | 1 | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♯ | B♮ | 1 | 1 | 1 | ½ | 1½ | ½ | ½ | ||
| C♮ | D♯ | E♮ | F♮ | G♮ | A♮ | B♮ | 1½ | ½ | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♯ | A♮ | B♭ | 1 | 1 | ½ | 1½ | ½ | ½ | 1 | ||
| C♯ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | ½ | 1 | 1 | 1 | ½ | 1½ | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♭ | 1 | ½ | 1½ | ½ | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♮ | ½ | 1 | 1 | 1 | ½ | 1½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♭ | ½ | 1½ | ½ | ½ | 1 | 1 | 1 | ||
One obvious symmetrical scale can’t be produced with a single-note transformation to the Ionian scale (though we can produce it by swapping two intervals; it’s also equivalent to the whole-tone scale with a note added). Its interval distribution is quite far from uniform, and it’s also all but impossible to relate to any sort of circle of fifths order. We’ve already studied it at length, but I haven’t shown it with every mode rooted on C, so here it is.
| 8. Big smile, mate; big wave, that’s great; now the truth is overrated; tell lies out the gate | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♮ | E♮ | F♮ | G♭ | A♭ | B♭ | 1 | 1 | ½ | ½ | 1 | 1 | 1 | ||
| C♮ | D♮ | E♭ | F♭ | G♭ | A♭ | B♭ | 1 | ½ | ½ | 1 | 1 | 1 | 1 | ||
| C♮ | D♭ | E𝄫 | F♭ | G♭ | A♭ | B♭ | ½ | ½ | 1 | 1 | 1 | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♮ | ½ | 1 | 1 | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♯ | A♯ | B♮ | 1 | 1 | 1 | 1 | 1 | ½ | ½ | ||
| C♮ | D♮ | E♮ | F♯ | G♯ | A♮ | B♭ | 1 | 1 | 1 | 1 | ½ | ½ | 1 | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♭ | B♭ | 1 | 1 | 1 | ½ | ½ | 1 | 1 | ||
I’m also bizarrely partial to the Major Phyrgian scale, which has its own fearful symmetry, to coin a phrase. It’s more closely related to the Ancient Greek chromatic genus, which I cover below in the section on Ancient Greek harmony, than it is to the Ionian scale. We’ll therefore revisit it later.
| 9. Polarise the people; controversy is the game; it don’t matter if they hate you if they all say your name | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♭ | ½ | 1½ | ½ | ½ | 1½ | ½ | 1 | ||
| C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♮ | 1½ | ½ | ½ | 1½ | ½ | 1 | ½ | ||
| C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B𝄫 | ½ | ½ | 1½ | ½ | 1 | ½ | 1½ | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | ||
| C♮ | D♯ | E♮ | F♯ | G♮ | A♯ | B♮ | 1½ | ½ | 1 | ½ | 1½ | ½ | ½ | ||
| C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B𝄫 | ½ | 1 | ½ | 1½ | ½ | ½ | 1½ | ||
| C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♮ | 1 | ½ | 1½ | ½ | ½ | 1½ | ½ | ||
“What immortal hand or eye
Could frame thy fearful symmetry?”
Rotational symmetry: Modes of limited transposition
Definition
The scales we’ve examined in detail thus far have as many modes as they have notes. A seven-note scale multiplied by twelve notes in the chromatic scale gives us eighty-four possible permutations of modes and root notes; pentatonic scales likewise have sixty possible permutations. However, this is not true of every scale (although, as we will eventually see, it is true of every pentatonic and heptatonic scale in 12-TET specifically).
French composer Oliver Messiaen (1908-1992) deemed a scale scales with fewer modes than notes a mode of limited transposition (acronym: MoLT). Such scales can be “simplified” into repetitions smaller than an octave, so they have rotational symmetry: rotating them by their internal repetition produces the same mode. Thus:
- A mode of limited transposition can be transformed into itself by a parallel rotation no larger than a tritone and no smaller than a semitone. (This rotation must not change its interval order in any way.)
- Modes of limited transposition reuse the same sets of notes across multiple transpositions.
- Modes of limited transposition have fewer modes than notes.
As an example, consider the C augmented chord, C-E-G♯, a stack of four-semitone intervals.
- A four-semitone parallel rotation transforms it into itself.
- When transposed to E, it becomes E-G♯-C, the same set of notes it has on C.
- Since its intervals are all the same, it has only one mode.
- Further explanation: Counting E-G♯-C as the C augmented chord’s second relative mode would be double-counting modes since, by definition, E-G♯-C must be its first relative mode on E. Likewise, G♯-C-E must be its first relative mode on G♯. Therefore, it has only a single mode.
For all three reasons, it is therefore a mode of limited transposition. By contrast, let’s consider Ionian.
- Ionian cannot transform into itself by sub-octave rotations.
- Ionian transposes into unique pitch sets for each note of the chromatic scale. (I’ve used circle of fifths order so that the note composition only changes one note per line, and I’ve highlighted the root in green.)
Ionian’s pitch sets across the chromatic scale Root C C♯ D E♭ E F F♯ G G♯ A B♭ B D♭ C♮ D♭ E♭ F♮ G♭ A♭ B♭ A♭ C♮ D♭ E♭ F♮ G♮ A♭ B♭ E♭ C♮ D♮ E♭ F♮ G♮ A♭ B♭ B♭ C♮ D♮ E♭ F♮ G♮ A♮ B♭ F♮ C♮ D♮ E♮ F♮ G♮ A♮ B♭ C♮ C♮ D♮ E♮ F♮ G♮ A♮ B♮ G♮ C♮ D♮ E♮ F♯ G♮ A♮ B♮ D♮ C♯ D♮ E♮ F♯ G♮ A♮ B♮ A♮ C♯ D♮ E♮ F♯ G♯ A♮ B♮ E♮ C♯ D♯ E♮ F♯ G♯ A♮ B♮ B♮ C♯ D♯ E♮ F♯ G♯ A♯ B♮ F♯ C♯ D♯ E♯ F♯ G♯ A♯ B♮ - Ionian, of course, has seven notes and seven modes.
Messiaen identified seven possible patterns (beyond the chromatic scale in its entirety); eight “truncations” also remove notes in ways that conform to the patterns, and the chromatic scale itself meets Messiaen’s definition of a mode of limited transposition, making for a total of sixteen.
Mode 1: The whole-tone scale
Since the whole-tone scale repeats a single interval six times (W-W-W-W-W-W), it has only one mode (i.e., itself) that may only be made from two sets of notes:
| Whole-tone note sets | ||||||
|---|---|---|---|---|---|---|
| # | 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | C♮ | D♮ | E♮ | F♯ | G♯ | A♯ |
| 2 | C♯ | D♯ | F♮ | G♮ | A♮ | B♮ |
Multiplying one mode by six repetitions by two note sets gives us a total of twelve transpositions:
| Transpositions of the whole-tone scale | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| T | P | 1 | 2 | 3 | 4 | 5 | 6 | P | 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | 1 | C♮ | D♮ | E♮ | F♯ | G♯ | A♯ | 2 | C♯ | D♯ | F♮ | G♮ | A♮ | B♮ |
| 2 | 1 | D♮ | E♮ | F♯ | G♯ | A♯ | C♮ | 2 | D♯ | F♮ | G♮ | A♮ | B♮ | C♯ |
| 3 | 1 | E♮ | F♯ | G♯ | A♯ | C♮ | D♮ | 2 | F♮ | G♮ | A♮ | B♮ | C♯ | D♯ |
| 4 | 1 | F♯ | G♯ | A♯ | C♮ | D♮ | E♮ | 2 | G♮ | A♮ | B♮ | C♯ | D♯ | F♮ |
| 5 | 1 | G♯ | A♯ | C♮ | D♮ | E♮ | F♯ | 2 | A♮ | B♮ | C♯ | D♯ | F♮ | G♮ |
| 6 | 1 | A♯ | C♮ | D♮ | E♮ | F♯ | G♯ | 2 | B♮ | C♯ | D♯ | F♮ | G♮ | A♮ |
The whole-tone scale is the first mode of limited transposition, and the only one that has no other modes. (A few truncations of the modes of limited transposition also have no other modes, as we shall see below.)
Mode 2: The octatonic scale
The octatonic scale (H-W-H-W-H-W-H-W), Messiaen’s second mode of limited transposition, repeats a two-note pattern every three half-steps. Since its pattern has only two notes, it has only two modes:
| The second mode’s modes | ||||||||
|---|---|---|---|---|---|---|---|---|
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 1 | ½ | 1 | ½ | 1 | ½ | 1 | ½ | 1 |
| 2 | 1 | ½ | 1 | ½ | 1 | ½ | 1 | ½ |
And since its pattern spans three half-steps, there are only three possible note sets:
| The second mode’s notes | ||||||||
|---|---|---|---|---|---|---|---|---|
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 1 | C | C♯ | D♯ | E | F♯ | G | A | A♯ |
| 2 | C♯ | D | E | F | G | G♯ | A♯ | B |
| 3 | D | D♯ | F | F♯ | G♯ | A | B | C |
Does it make sense why we have to stop counting here? C octatonic’s third mode would start on D♯, but it would contain exactly the same notes as D♯ octatonic’s first mode, in exactly the same order! We can’t count them both, so the octatonic scale has six total permutations of modes and note sets.
But if we have to stop counting modes at the end of each cluster, how do we calculate the number of discrete transpositions of the scale and its modes? As far as I can work out, the calculation is quite simple:
- (
8 / 4 = 2)Divide the scale’s note count by its repetitions per octave to count its modes. - (
2 × 12 = 24)Multiply by the number of transpositions (which is always 12 in 12-TET).
As we see below, the octatonic scale indeed has twenty-four total transpositions:
| Transposing the second mode of limited transposition | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M | T | P | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | M | T | P | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 1 | 1 | C♮ | C♯ | D♯ | E♮ | F♯ | G♮ | A♮ | A♯ | 1 | 1 | C♯ | D♯ | E♮ | F♯ | G♮ | A♮ | A♯ | C♮ | ||
| 2 | C♯ | D♮ | E♮ | F♮ | G♮ | G♯ | A♯ | B♮ | 2 | D♮ | E♮ | F♮ | G♮ | G♯ | A♯ | B♮ | C♯ | ||||
| 3 | D♮ | D♯ | F♮ | F♯ | G♯ | A♮ | B♮ | C♮ | 3 | D♯ | F♮ | F♯ | G♯ | A♮ | B♮ | C♮ | D♮ | ||||
| 2 | 1 | D♯ | E♮ | F♯ | G♮ | A♮ | A♯ | C♮ | C♯ | 2 | 1 | E♮ | F♯ | G♮ | A♮ | A♯ | C♮ | C♯ | D♯ | ||
| 2 | E♮ | F♮ | G♮ | G♯ | A♯ | B♮ | C♯ | D♮ | 2 | F♮ | G♮ | G♯ | A♯ | B♮ | C♯ | D♮ | E♮ | ||||
| 3 | F♮ | F♯ | G♯ | A♮ | B♮ | C♮ | D♮ | D♯ | 3 | F♯ | G♯ | A♮ | B♮ | C♮ | D♮ | D♯ | F♮ | ||||
| 3 | 1 | F♯ | G♮ | A♮ | A♯ | C♮ | C♯ | D♯ | E♮ | 3 | 1 | G♮ | A♮ | A♯ | C♮ | C♯ | D♯ | E♮ | F♯ | ||
| 2 | G♮ | G♯ | A♯ | B♮ | C♯ | D♮ | E♮ | F♮ | 2 | G♯ | A♯ | B♮ | C♯ | D♮ | E♮ | F♮ | G♮ | ||||
| 3 | G♯ | A♮ | B♮ | C♮ | D♮ | D♯ | F♮ | F♯ | 3 | A♮ | B♮ | C♮ | D♮ | D♯ | F♮ | F♯ | G♯ | ||||
| 4 | 1 | A♮ | A♯ | C♮ | C♯ | D♯ | E♮ | F♯ | G♮ | 4 | 1 | A♯ | C♮ | C♯ | D♯ | E♮ | F♯ | G♮ | A♮ | ||
| 2 | A♯ | B♮ | C♯ | D♮ | E♮ | F♮ | G♮ | G♯ | 2 | B♮ | C♯ | D♮ | E♮ | F♮ | G♮ | G♯ | A♯ | ||||
| 3 | B♮ | C♮ | D♮ | D♯ | F♮ | F♯ | G♯ | A♮ | 3 | C♮ | D♮ | D♯ | F♮ | F♯ | G♯ | A♮ | B♮ | ||||
Mode 3: The triple chromatic scale
The third mode of limited transposition repeats a three-interval pattern across four half-steps; thus, it has three unique modes that may be constructed from four possible sets of notes, for twelve total permutations of notes per four-half-step cluster:
| The third mode of limited transposition | |||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M | S | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Intervals | ||||||||
| 1 | C♮ | D♮ | D♯ | E♮ | F♯ | G♮ | G♯ | A♯ | B♮ | 1 | ½ | ½ | 1 | ½ | ½ | 1 | ½ | ½ | |
| 2 | C♯ | D♯ | E♮ | F♮ | G♮ | G♯ | A♮ | B♮ | C♮ | ||||||||||
| 3 | D♮ | E♮ | F♮ | F♯ | G♯ | A♮ | A♯ | C♮ | C♯ | ||||||||||
| 4 | D♯ | F♮ | F♯ | G♮ | A♮ | A♯ | B♮ | C♯ | D♮ | ||||||||||
| 1 | D♮ | D♯ | E♮ | F♯ | G♮ | G♯ | A♯ | B♮ | C♮ | ½ | ½ | 1 | ½ | ½ | 1 | ½ | ½ | 1 | |
| 2 | D♯ | E♮ | F♮ | G♮ | G♯ | A♮ | B♮ | C♮ | C♯ | ||||||||||
| 3 | E♮ | F♮ | F♯ | G♯ | A♮ | A♯ | C♮ | C♯ | D♮ | ||||||||||
| 4 | F♮ | F♯ | G♮ | A♮ | A♯ | B♮ | C♯ | D♮ | D♯ | ||||||||||
| 1 | D♯ | E♮ | F♯ | G♮ | G♯ | A♯ | B♮ | C♮ | D♮ | ½ | 1 | ½ | ½ | 1 | ½ | ½ | 1 | ½ | |
| 2 | E♮ | F♮ | G♮ | G♯ | A♮ | B♮ | C♮ | C♯ | D♯ | ||||||||||
| 3 | F♮ | F♯ | G♯ | A♮ | A♯ | C♮ | C♯ | D♮ | E♮ | ||||||||||
| 4 | F♯ | G♮ | A♮ | A♯ | B♮ | C♯ | D♮ | D♯ | F♮ | ||||||||||
I leave filling in the rest of the table as an exercise for the reader. A quick hint: You should wind up with three sets of twelve scales that each walk up the chromatic scale by half-steps, for a total of thirty-six.
Mode 4: The double chromatic scale
Modes 4 through 7 all repeat patterns of various lengths twice an octave. Since the fourth mode of limited transposition has eight total notes, it has forty-eight possible transpositions.
| The fourth mode of limited transposition | |||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M | S | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Intervals | |||||||
| 1 | C♮ | C♯ | D♮ | F♮ | F♯ | G♮ | G♯ | B♮ | ½ | ½ | 1½ | ½ | ½ | ½ | 1½ | ½ | |
| 2 | C♯ | D♮ | D♯ | F♯ | G♮ | G♯ | A♮ | C♮ | |||||||||
| 3 | D♮ | D♯ | E♮ | G♮ | G♯ | A♮ | A♯ | C♯ | |||||||||
| 4 | D♯ | E♮ | F♮ | G♯ | A♮ | A♯ | B♮ | D♮ | |||||||||
| 5 | E♮ | F♮ | F♯ | A♮ | A♯ | B♮ | C♮ | D♯ | |||||||||
| 6 | F♮ | F♯ | G♮ | A♯ | B♮ | C♮ | C♯ | E♮ | |||||||||
| 1 | C♮ | C♯ | E♮ | F♮ | F♯ | G♮ | A♯ | B♮ | ½ | 1½ | ½ | ½ | ½ | 1½ | ½ | ½ | |
| 2 | C♯ | D♮ | F♮ | F♯ | G♮ | G♯ | B♮ | C♮ | |||||||||
| 3 | D♮ | D♯ | F♯ | G♮ | G♯ | A♮ | C♮ | C♯ | |||||||||
| 4 | D♯ | E♮ | G♮ | G♯ | A♮ | A♯ | C♯ | D♮ | |||||||||
| 5 | E♮ | F♮ | G♯ | A♮ | A♯ | B♮ | D♮ | D♯ | |||||||||
| 6 | F♮ | F♯ | A♮ | A♯ | B♮ | C♮ | D♯ | E♮ | |||||||||
| 1 | C♮ | D♯ | E♮ | F♮ | F♯ | A♮ | A♯ | B♮ | 1½ | ½ | ½ | ½ | 1½ | ½ | ½ | ½ | |
| 2 | C♯ | E♮ | F♮ | F♯ | G♮ | A♯ | B♮ | C♮ | |||||||||
| 3 | D♮ | F♮ | F♯ | G♮ | G♯ | B♮ | C♮ | C♯ | |||||||||
| 4 | D♯ | F♯ | G♮ | G♯ | A♮ | C♮ | C♯ | D♮ | |||||||||
| 5 | E♮ | G♮ | G♯ | A♮ | A♯ | C♯ | D♮ | D♯ | |||||||||
| 6 | F♮ | G♯ | A♮ | A♯ | B♮ | D♮ | D♯ | E♮ | |||||||||
| 1 | C♮ | C♯ | D♮ | D♯ | F♯ | G♮ | G♯ | A♮ | ½ | ½ | ½ | 1½ | ½ | ½ | ½ | 1½ | |
| 2 | C♯ | D♮ | D♯ | E♮ | G♮ | G♯ | A♮ | A♯ | |||||||||
| 3 | D♮ | D♯ | E♮ | F♮ | G♯ | A♮ | A♯ | B♮ | |||||||||
| 4 | D♯ | E♮ | F♮ | F♯ | A♮ | A♯ | B♮ | C♮ | |||||||||
| 5 | E♮ | F♮ | F♯ | G♮ | A♯ | B♮ | C♮ | C♯ | |||||||||
| 6 | F♮ | F♯ | G♮ | G♯ | B♮ | C♮ | C♯ | D♮ | |||||||||
Mode 5: The tritone chromatic scale
The fifth mode has six notes and repeats twice an octave; therefore, it has thirty-six possible transpositions.
| The fifth mode of limited transposition | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M | S | 1 | 2 | 3 | 4 | 5 | 6 | Intervals | |||||
| 1 | C♮ | C♯ | F♮ | F♯ | G♮ | B♮ | ½ | 2 | ½ | ½ | 2 | ½ | |
| 2 | C♯ | D♮ | F♯ | G♮ | G♯ | C♮ | |||||||
| 3 | D♮ | D♯ | G♮ | G♯ | A♮ | C♯ | |||||||
| 4 | D♯ | E♮ | G♯ | A♮ | A♯ | D♮ | |||||||
| 5 | E♮ | F♮ | A♮ | A♯ | B♮ | D♯ | |||||||
| 6 | F♮ | F♯ | A♯ | B♮ | C♮ | E♮ | |||||||
| 1 | C♮ | E♮ | F♮ | F♯ | A♯ | B♮ | 2 | ½ | ½ | 2 | ½ | ½ | |
| 2 | C♯ | F♮ | F♯ | G♮ | B♮ | C♮ | |||||||
| 3 | D♮ | F♯ | G♮ | G♯ | C♮ | C♯ | |||||||
| 4 | D♯ | G♮ | G♯ | A♮ | C♯ | D♮ | |||||||
| 5 | E♮ | G♯ | A♮ | A♯ | D♮ | D♯ | |||||||
| 6 | F♮ | A♮ | A♯ | B♮ | D♯ | E♮ | |||||||
| 1 | C♮ | C♯ | D♮ | F♯ | G♮ | G♯ | ½ | ½ | 2 | ½ | ½ | 2 | |
| 2 | C♯ | D♮ | D♯ | G♮ | G♯ | A♮ | |||||||
| 3 | D♮ | D♯ | E♮ | G♯ | A♮ | A♯ | |||||||
| 4 | D♯ | E♮ | F♮ | A♮ | A♯ | B♮ | |||||||
| 5 | E♮ | F♮ | F♯ | A♯ | B♮ | C♮ | |||||||
| 6 | F♮ | F♯ | G♮ | B♮ | C♮ | C♯ | |||||||
Mode 6: The whole-tone chromatic scale
Messiaen’s sixth mode of limited transposition repeats a four-note, six-half-step pattern. Four modes, six note combinations per mode, twenty-four note permutations per cluster, two clusters per octave, forty-eight permutations of roots and modes across the entire chromatic scale.
| The sixth mode of limited transposition | |||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M | S | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Intervals | |||||||
| 1 | C♮ | D♮ | E♮ | F♮ | F♯ | G♯ | A♯ | B♮ | 1 | 1 | ½ | ½ | 1 | 1 | ½ | ½ | |
| 2 | C♯ | D♯ | F♮ | F♯ | G♮ | A♮ | B♮ | C♮ | |||||||||
| 3 | D♮ | E♮ | F♯ | G♮ | G♯ | A♯ | C♮ | C♯ | |||||||||
| 4 | D♯ | F♮ | G♮ | G♯ | A♮ | B♮ | C♯ | D♮ | |||||||||
| 5 | E♮ | F♯ | G♯ | A♮ | A♯ | C♮ | D♮ | D♯ | |||||||||
| 6 | F♮ | G♮ | A♮ | A♯ | B♮ | C♯ | D♯ | E♮ | |||||||||
| 1 | C♮ | D♮ | D♯ | E♮ | F♯ | G♯ | A♮ | A♯ | 1 | ½ | ½ | 1 | 1 | ½ | ½ | 1 | |
| 2 | C♯ | D♯ | E♮ | F♮ | G♮ | A♮ | A♯ | B♮ | |||||||||
| 3 | D♮ | E♮ | F♮ | F♯ | G♯ | A♯ | B♮ | C♮ | |||||||||
| 4 | D♯ | F♮ | F♯ | G♮ | A♮ | B♮ | C♮ | C♯ | |||||||||
| 5 | E♮ | F♯ | G♮ | G♯ | A♯ | C♮ | C♯ | D♮ | |||||||||
| 6 | F♮ | G♮ | G♯ | A♮ | B♮ | C♯ | D♮ | D♯ | |||||||||
| 1 | C♮ | C♯ | D♮ | E♮ | F♯ | G♮ | G♯ | A♯ | ½ | ½ | 1 | 1 | ½ | ½ | 1 | 1 | |
| 2 | C♯ | D♮ | D♯ | F♮ | G♮ | G♯ | A♮ | B♮ | |||||||||
| 3 | D♮ | D♯ | E♮ | F♯ | G♯ | A♮ | A♯ | C♮ | |||||||||
| 4 | D♯ | E♮ | F♮ | G♮ | A♮ | A♯ | B♮ | C♯ | |||||||||
| 5 | E♮ | F♮ | F♯ | G♯ | A♯ | B♮ | C♮ | D♮ | |||||||||
| 6 | F♮ | F♯ | G♮ | A♮ | B♮ | C♮ | C♯ | D♯ | |||||||||
| 1 | C♮ | C♯ | D♯ | F♮ | F♯ | G♮ | A♮ | B♮ | ½ | 1 | 1 | ½ | ½ | 1 | 1 | ½ | |
| 2 | C♯ | D♮ | E♮ | F♯ | G♮ | G♯ | A♯ | C♮ | |||||||||
| 3 | D♮ | D♯ | F♮ | G♮ | G♯ | A♮ | B♮ | C♯ | |||||||||
| 4 | D♯ | E♮ | F♯ | G♯ | A♮ | A♯ | C♮ | D♮ | |||||||||
| 5 | E♮ | F♮ | G♮ | A♮ | A♯ | B♮ | C♯ | D♯ | |||||||||
| 6 | F♮ | F♯ | G♯ | A♯ | B♮ | C♮ | D♮ | E♮ | |||||||||
Mode 7: Duplex genus secundum inverse
This decatonic scale repeats the same pattern twice an octave; thus, it has sixty possible transpositions.
| The seventh mode of limited transposition | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M | S | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Intervals | |||||||||
| 1 | C♮ | C♯ | D♮ | D♯ | F♮ | F♯ | G♮ | G♯ | A♮ | B♮ | ½ | ½ | ½ | 1 | ½ | ½ | ½ | ½ | 1 | ½ | |
| 2 | C♯ | D♮ | D♯ | E♮ | F♯ | G♮ | G♯ | A♮ | A♯ | C♮ | |||||||||||
| 3 | D♮ | D♯ | E♮ | F♮ | G♮ | G♯ | A♮ | A♯ | B♮ | C♯ | |||||||||||
| 4 | D♯ | E♮ | F♮ | F♯ | G♯ | A♮ | A♯ | B♮ | C♮ | D♮ | |||||||||||
| 5 | E♮ | F♮ | F♯ | G♮ | A♮ | A♯ | B♮ | C♮ | C♯ | D♯ | |||||||||||
| 6 | F♮ | F♯ | G♮ | G♯ | A♯ | B♮ | C♮ | C♯ | D♮ | E♮ | |||||||||||
| 1 | C♮ | C♯ | D♮ | E♮ | F♮ | F♯ | G♮ | G♯ | A♯ | B♮ | ½ | ½ | 1 | ½ | ½ | ½ | ½ | 1 | ½ | ½ | |
| 2 | C♯ | D♮ | D♯ | F♮ | F♯ | G♮ | G♯ | A♮ | B♮ | C♮ | |||||||||||
| 3 | D♮ | D♯ | E♮ | F♯ | G♮ | G♯ | A♮ | A♯ | C♮ | C♯ | |||||||||||
| 4 | D♯ | E♮ | F♮ | G♮ | G♯ | A♮ | A♯ | B♮ | C♯ | D♮ | |||||||||||
| 5 | E♮ | F♮ | F♯ | G♯ | A♮ | A♯ | B♮ | C♮ | D♮ | D♯ | |||||||||||
| 6 | F♮ | F♯ | G♮ | A♮ | A♯ | B♮ | C♮ | C♯ | D♯ | E♮ | |||||||||||
| 1 | C♮ | C♯ | D♯ | E♮ | F♮ | F♯ | G♮ | A♮ | A♯ | B♮ | ½ | 1 | ½ | ½ | ½ | ½ | 1 | ½ | ½ | ½ | |
| 2 | C♯ | D♮ | E♮ | F♮ | F♯ | G♮ | G♯ | A♯ | B♮ | C♮ | |||||||||||
| 3 | D♮ | D♯ | F♮ | F♯ | G♮ | G♯ | A♮ | B♮ | C♮ | C♯ | |||||||||||
| 4 | D♯ | E♮ | F♯ | G♮ | G♯ | A♮ | A♯ | C♮ | C♯ | D♮ | |||||||||||
| 5 | E♮ | F♮ | G♮ | G♯ | A♮ | A♯ | B♮ | C♯ | D♮ | D♯ | |||||||||||
| 6 | F♮ | F♯ | G♯ | A♮ | A♯ | B♮ | C♮ | D♮ | D♯ | E♮ | |||||||||||
| 1 | C♮ | D♮ | D♯ | E♮ | F♮ | F♯ | G♯ | A♮ | A♯ | B♮ | 1 | ½ | ½ | ½ | ½ | 1 | ½ | ½ | ½ | ½ | |
| 2 | C♯ | D♯ | E♮ | F♮ | F♯ | G♮ | A♮ | A♯ | B♮ | C♮ | |||||||||||
| 3 | D♮ | E♮ | F♮ | F♯ | G♮ | G♯ | A♯ | B♮ | C♮ | C♯ | |||||||||||
| 4 | D♯ | F♮ | F♯ | G♮ | G♯ | A♮ | B♮ | C♮ | C♯ | D♮ | |||||||||||
| 5 | E♮ | F♯ | G♮ | G♯ | A♮ | A♯ | C♮ | C♯ | D♮ | D♯ | |||||||||||
| 6 | F♮ | G♮ | G♯ | A♮ | A♯ | B♮ | C♯ | D♮ | D♯ | E♮ | |||||||||||
| 1 | C♮ | C♯ | D♮ | D♯ | E♮ | F♯ | G♮ | G♯ | A♮ | A♯ | ½ | ½ | ½ | ½ | 1 | ½ | ½ | ½ | ½ | 1 | |
| 2 | C♯ | D♮ | D♯ | E♮ | F♮ | G♮ | G♯ | A♮ | A♯ | B♮ | |||||||||||
| 3 | D♮ | D♯ | E♮ | F♮ | F♯ | G♯ | A♮ | A♯ | B♮ | C♮ | |||||||||||
| 4 | D♯ | E♮ | F♮ | F♯ | G♮ | A♮ | A♯ | B♮ | C♮ | C♯ | |||||||||||
| 5 | E♮ | F♮ | F♯ | G♮ | G♯ | A♯ | B♮ | C♮ | C♯ | D♮ | |||||||||||
| 6 | F♮ | F♯ | G♮ | G♯ | A♮ | B♮ | C♮ | C♯ | D♮ | D♯ | |||||||||||
Truncations & implications
To illustrate MoLT truncation, let’s strip every other note of the first two MoLTs. The results may look familiar:
-
Whole-tone: An augmented chord.
Transposing the augmented chord P/T 1 2 3 P/T 1 2 3 P/T 1 2 3 P/T 1 2 3 A/1 C♮ E♮ G♯ A/2 C♯ F♮ A♮ A/3 D♮ F♯ A♯ A/4 D♯ G♮ B♮ B/1 E♮ G♯ C♮ B/2 F♮ A♮ C♯ B/3 F♯ A♯ D♮ B/4 G♮ B♮ D♯ C/1 G♯ C♮ E♮ C/2 A♮ C♯ F♮ C/3 A♯ D♮ F♯ C/4 B♮ D♯ G♮ -
Octatonic: A diminished seventh chord.
Transposing the diminished seventh chord P/T 1 2 3 4 P/T 1 2 3 4 P/T 1 2 3 4 P/T 1 2 3 4 A/1 C♮ D♯ F♯ A♮ B/1 D♯ F♯ A♮ C♮ C/1 F♯ A♮ C♮ D♯ D/1 A♮ C♮ D♯ F♯ A/2 C♯ E♮ G♮ A♯ B/2 E♮ G♮ A♯ C♯ C/2 G♮ A♯ C♯ E♮ D/2 A♯ C♯ E♮ G♮ A/3 D♮ F♮ G♯ B♮ B/3 F♮ G♯ B♮ D♮ C/3 G♯ B♮ D♮ F♮ D/3 B♮ D♮ F♮ G♯
(Note: Each arrangement of the three diminished seventh chords appears on the same line as its rotations; by contrast, the augmented chord’s three-note sets are 3×3 squares. For each chord, A/2 precedes B/1.)
Some additional notes:
- Since each chord truncates its parent mode in ways that maintain its symmetry, each has only twelve transpositions. Put another way, each inversion of a diminished seventh chord or an augmented chord contains the same notes and interval sequence as its parent; it just starts on a different note. By contrast, major and minor chords each have first and second inversions and thus thirty-six discrete transpositions in 12-TET.
-
In most cases, scale truncation refers merely to removing at least one note from a scale. However, references to truncated modes of limited transposition usually refer specifically to truncations that maintain at least some of their parent scales’ rotational symmetry. Thus, if a parent scale has tritone rotational symmetry, such truncations may only remove notes in pairs from the following scale degrees:
- 1 and 7
- 2 and 8
- 3 and 9
- 4 and 10
- 5 and 11
- 6 and 12
- 1 and 7
- 2 and 8
- 3 and 9
- 4 and 10
- 5 and 11
- 6 and 12
We may truncate a scale with rotational symmetry across minor thirds in ways that preserve only rotational symmetry across tritones, or the whole-tone scale in ways that preserve rotational symmetry only across major thirds or only across tritones, but we must preserve at least some of its rotational symmetry. (By contrast, scales with rotational symmetry only across major thirds or only across tritones must be truncated in ways that preserve that symmetry, since it cannot be reduced to less frequent symmetry.)
-
Messaien omitted several scales on the grounds that they just truncated scales he’d already listed, but he arguably applied this principle rather arbitrarily:
- Every scale in his list except mode 3 truncates mode 7.
- Mode 1 also truncates modes 3 and 6.
- Mode 5 also truncates modes 4 and 6.
Modes 1 & 2 repeat more than their parents, so there’s a reasonable case for treating them differently, but modes 4–6 don’t. As we’ll see below, sixteen total scales in 12-TET fit the criteria of a MoLT.
- Although Messiaen didn’t classify the chromatic scale as a MoLT, it fits all the criteria for one. Every other MoLT is thus by definition a truncation of the chromatic scale.
- The Ionian scale, W-W-H-W-W-W-H, is a nice contrast with MoLTs: the extra W between its W-W-H patterns makes it impossible to simplify into a smaller pattern. It therefore has seven modes, which each have twelve variants, for a total of eighty-four permutations.
- An important consequence of MoLTs in music theory is that they have ambiguous tonalities, since multiple keys could be the root. In many cases, these notes are a tritone apart, which can provide interesting opportunities for tritone substitutions, but it also provides challenges that are absent when writing for less ambiguous tonalities such as the Ionian scale.
- MoLTs must have at least twelve permutations in twelve-tone equal temperament: even scales with only one mode can still start on all twelve notes of the chromatic scale.
-
This is a complete list of every set of notes in 12-TET with only a single mode:
- The chromatic scale
- The whole-tone scale
- The diminished seventh chord
- The augmented chord
- The tritone
- The octave
It’s no coincidence that every last one of these intervals, expressed in semitones, divides evenly into 12. (Note that, because the octave has both one mode and one note, it is not a mode of limited transposition.)
Mode 0: The chromatic scale
The chromatic scale itself merits further discussion. It repeats one interval twelve times (H-H-H-H-H-H-H-H-H-H-H-H). Thus, it has only one mode, which in turn may only be made from one permutation of notes. The chromatic scale can therefore be transposed in twelve different ways across the entire, um, chromatic scale:
| Permuatations of the chromatic scale | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| 1 | C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ |
| 2 | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | C♮ |
| 3 | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | C♮ | C♯ |
| 4 | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | C♮ | C♯ | D♮ |
| 5 | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | C♮ | C♯ | D♮ | D♯ |
| 6 | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | C♮ | C♯ | D♮ | D♯ | E♮ |
| 7 | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | C♮ | C♯ | D♮ | D♯ | E♮ | F♮ |
| 8 | G♮ | G♯ | A♮ | A♯ | B♮ | C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ |
| 9 | G♯ | A♮ | A♯ | B♮ | C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ |
| 10 | A♮ | A♯ | B♮ | C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ |
| 11 | A♯ | B♮ | C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ |
| 12 | B♮ | C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ |
By extension, we might consider the chromatic scale the Zeroth Mode of Limited Transposition: all other scales in twelve-tone equal temperament are truncations of it.
All modes of limited transposition in 12-TET
In 12-TET, sixteen scales meet all necessary criteria for modes of limited transposition, with thirty-eight modes between them (in all, they’re missing sixty-four modes). I listed intervals in semitones and only listed one mode per scale. §10 (Scale counts by size) lists all modes of each.
| MoLTs at a glance | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Notes | Intervals (semitones) | Modes | Notes | Intervals (semitones) | Modes | Notes | Intervals (semitones) | Modes | Notes | Intervals (semitones) | Modes |
| 12 | 1 | 10 | 5 | 9 | 3 | 8 | 2 | ||||
| 8 | 4 | 8 | 4 | 6 | 1 | 6 | 2 | ||||
| 6 | 3 | 6 | 3 | 6 | 3 | 4 | 1 | ||||
| 4 | 2 | 4 | 2 | 3 | 1 | 2 | 1 | ||||
These scales have the following note distributions across their parallel modes:
| Note distributions of parallel modes of limited transposition | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Scale | C | C♯ | D | D♯ | E | F | F♯ | G | G♯ | A | A♯ | B |
| Mode 0 (111111111111) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| Mode 7 (1112111121) | 5 | 4 | 4 | 4 | 4 | 4 | 5 | 4 | 4 | 4 | 4 | 4 |
| Mode 3 (211211211) | 3 | 2 | 2 | 2 | 3 | 2 | 2 | 2 | 3 | 2 | 2 | 2 |
| Mode 4 (11311131) | 4 | 3 | 2 | 2 | 2 | 3 | 4 | 3 | 2 | 2 | 2 | 3 |
| Mode 6 (22112211) | 4 | 2 | 3 | 2 | 3 | 2 | 4 | 2 | 3 | 2 | 3 | 2 |
| Mode 2 (12121212) | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 |
| Mode 5 (141141) | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 2 | 1 | 0 | 1 | 2 |
| Mode 2 Truncation 1 (321321) | 3 | 1 | 1 | 2 | 1 | 1 | 3 | 1 | 1 | 2 | 1 | 1 |
| Mode 3 Truncation (131313) | 2 | 1 | 0 | 1 | 2 | 1 | 0 | 1 | 2 | 1 | 0 | 1 |
| Mode 2 Truncation 2 (231231) | 3 | 1 | 1 | 2 | 1 | 1 | 3 | 1 | 1 | 2 | 1 | 1 |
| Mode 1 (222222) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| Mode 5 Truncation 1 (1515) | 2 | 1 | 0 | 0 | 0 | 1 | 2 | 1 | 0 | 0 | 0 | 1 |
| Mode 6 Truncation (2424) | 2 | 0 | 1 | 0 | 1 | 0 | 2 | 0 | 1 | 0 | 1 | 0 |
| Diminished Seventh (3333) | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |
| Augmented Chord (444) | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| Tritone (66) | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
Adjusting these numbers to account for the missing modes gives us the following distributions:
| Note distributions of parallel modes of limited transposition (adjusted for inflation) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Scale | C | C♯ | D | D♯ | E | F | F♯ | G | G♯ | A | A♯ | B |
| Mode 0 (111111111111) | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| Mode 7 (1112111121) | 10 | 8 | 8 | 8 | 8 | 8 | 10 | 8 | 8 | 8 | 8 | 8 |
| Mode 3 (211211211) | 9 | 6 | 6 | 6 | 9 | 6 | 6 | 6 | 9 | 6 | 6 | 6 |
| Mode 4 (11311131) | 8 | 6 | 4 | 4 | 4 | 6 | 8 | 6 | 4 | 4 | 4 | 6 |
| Mode 6 (22112211) | 8 | 4 | 6 | 4 | 6 | 4 | 8 | 4 | 6 | 4 | 6 | 4 |
| Mode 2 (12121212) | 8 | 4 | 4 | 8 | 4 | 4 | 8 | 4 | 4 | 8 | 4 | 4 |
| Mode 5 (141141) | 6 | 4 | 2 | 0 | 2 | 4 | 6 | 4 | 2 | 0 | 2 | 4 |
| Mode 2 Truncation 1 (321321) | 6 | 2 | 2 | 4 | 2 | 2 | 6 | 2 | 2 | 4 | 2 | 2 |
| Mode 3 Truncation (131313) | 6 | 3 | 0 | 3 | 6 | 3 | 0 | 3 | 6 | 3 | 0 | 3 |
| Mode 2 Truncation 2 (231231) | 6 | 2 | 2 | 4 | 2 | 2 | 6 | 2 | 2 | 4 | 2 | 2 |
| Mode 1 (222222) | 6 | 0 | 6 | 0 | 6 | 0 | 6 | 0 | 6 | 0 | 6 | 0 |
| Mode 5 Truncation 1 (1515) | 4 | 2 | 0 | 0 | 0 | 2 | 4 | 2 | 0 | 0 | 0 | 2 |
| Mode 6 Truncation (2424) | 4 | 0 | 2 | 0 | 4 | 0 | 2 | 0 | 4 | 0 | 2 | 0 |
| Diminished Seventh (3333) | 4 | 0 | 0 | 4 | 0 | 0 | 4 | 0 | 0 | 4 | 0 | 0 |
| Augmented Chord (444) | 3 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| Tritone (66) | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
Single-note transformations of modes of limited transposition
I mentioned that I’d return to melodic Phrygian, alternating heptamode, alternating heptamode inverse, apathetic minor, and Pacific after we’d covered modes of limited transposition. 12-TET cannot contain heptatonic modes of limited transposition, but melodic Phrygian, alternating heptamode, and alternating heptamode inverse are single-note transformations of modes of limited transposition. For instance:
- C melodic Phrygian is C♯ whole-tone with an added C.
- C melodic Phrygian is also C whole-tone chromatic’s fourth mode with a deleted F♯.
- C alternating heptamode is C octatonic with a deleted B♭.
- C alternating heptamode inverse is C octatonic inverse with a deleted D.
- C apathetic minor is C double chromatic with a deleted G♯.
- C Pacific is C double chromatic’s second mode with a deleted A♯.
The scale transformations we apply to each of these scales to produce their modes consist solely of moving the added or deleted note by the interval at which their parent modes of limited transposition repeat, which is why they don’t take us through the entire chromatic scale. Here’s a note distribution comparison, adjusted for inflation limited transposition (i.e., the whole-tone scale is short 5 modes and the octatonic scale is short 3, so to facilitate 1:1 comparisons, I’ve multiplied the whole-tone scale’s values by 6 and the octatonic scale’s values by 4).
| Note distributions across parallel modes (…again‽ But that trick never works!) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Scale | C | C♯ | D | D♯ | E | F | F♯ | G | G♯ | A | A♯ | B |
| Ionian | 7 | 2 | 5 | 4 | 3 | 6 | 2 | 6 | 3 | 4 | 5 | 2 |
| Chromatic heptatonic | 7 | 6 | 5 | 4 | 3 | 2 | 2 | 2 | 3 | 4 | 5 | 6 |
| Melodic Phrygian | 7 | 2 | 6 | 2 | 6 | 2 | 6 | 2 | 6 | 2 | 6 | 2 |
| Whole-tone | 6 | 0 | 6 | 0 | 6 | 0 | 6 | 0 | 6 | 0 | 6 | 0 |
| (Difference) | 1 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 |
| Whole-tone chromatic | 8 | 4 | 6 | 4 | 6 | 4 | 8 | 4 | 6 | 4 | 6 | 4 |
| Melodic Phrygian | 7 | 2 | 6 | 2 | 6 | 2 | 6 | 2 | 6 | 2 | 6 | 2 |
| (Difference) | 1 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 0 | 2 | 0 | 2 |
| Octatonic | 8 | 4 | 4 | 8 | 4 | 4 | 8 | 4 | 4 | 8 | 4 | 4 |
| Alternating heptamode | 7 | 3 | 3 | 6 | 3 | 3 | 6 | 3 | 3 | 6 | 3 | 3 |
| (Difference) | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 |
| Double chromatic | 8 | 6 | 4 | 4 | 4 | 6 | 8 | 6 | 4 | 4 | 4 | 6 |
| Apathetic minor | 7 | 5 | 3 | 2 | 3 | 5 | 6 | 5 | 3 | 2 | 3 | 5 |
| (Difference) | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 |
Note also that the whole-tone scale is rotationally symmetrical by two semitones, the octatonic scale is rotationally symmetrical by three semitones, and the double chromatic and whole-tone chromatic scales are rotationally symmetrical by six semitones. That is, the whole-tone scale’s distribution repeats every two semitones, the octatonic scale’s every three semitones, and the double chromatic and whole-tone chromatic scales’ every six semitones. This is axiomatically equivalent to their being modes of limited transposition.
Now, note how melodic Phrygian, alternating heptamode, and apathetic minor come to duplicating those distributions: in fact, disregarding the root note, alternating heptamode has exactly (octatonic’s adjusted distributions × ¾), and melodic Phrygian has exactly ((whole-tone’s adjusted distributions × ⅔) + 2). The root note is the outlier here because, by definition, it must always occur in every mode of any scale.
It should be fairly obvious why melodic Phrygian and its modes are 12-TET’s only heptatonic single-note transformations of the whole-tone scale: we may only add notes into six gaps, which are all whole steps; thus, all such transformations must change one whole step into two half-steps.
12-TET contains three different octatonic modes of limited transposition, namely Messiaen’s modes 2, 4, and 6. Can we derive other similar heptatonic scales in similar ways? That is, can deleting a single note of the octatonic, double chromatic, or whole-tone chromatic scales create other scales whose modes can in turn all be derived through single-note transformations to themselves? I believe I’ve exhausted all the possibilities, but I haven’t mathematically proved that yet.
Microtonal corollaries
-
In any
t-TET witht > 12 and t⟨en.wiktionary. org /wiki /modulo⟩ 12 = 0, i.e.:t = 12 × n- n is an integer
n > 1
The 12-tone chromatic scale is a MoLT. Since it never has more than one mode, it always has exactly
tpossible permutations of modes and root notes. Thus, in 24-TET, it has 24 possible permutations of modes and root notes; in 36-TET, it has 36; in 48-TET, it has 48; and so on. Four particular corollaries to MoLTs within 12-TET itself apply:The 12-tone chromatic scale as a mode of limited microtonal transposition In x-TET:the 12-TCS has x/ 12 modes:the same number yhas in 12-TET:24 2 the whole-tone scale 36 3 the diminished seventh chord 48 4 the augmented chord 72 6 the tritone In other words, y consists of evenly spaced intervals whose ratio to 12 equals 12’s ratio to x.
- The 12-tone chromatic scale is also a MoLT in 60-TET, 84-TET, 96-TET, 108-TET, 120-TET, and so on; the analogy just works less perfectly in those tonalities because 12 doesn’t have an integer quotient with their ratios to 12 (i.e., 12 / (60 / 12) = 12² / 60 = 2.4, i.e., not an integer).
-
Any scale’s number of repetitions per octave in any n-TET must be an integer factor of n. Thus:
- All scales in 12-TET must repeat exactly 12, 6, 4, 3, 2, or 1 times per octave.
- For any prime number n, the only possible MoLT in n-TET (e.g., 13-TET, 17-TET, 19-TET) is the n-tone chromatic scale.
- For any composite number n, its n-TET (e.g., 15-TET, 22-TET, 24-TET) must have MoLTs other than its own chromatic scale.
Wikipedia’s article asserts both b and c, without fully explaining them, but the most parsimonious explanation is prime factorization:
- Let t-TET exist, where t is composite.
- Let f be an integer factor of t.
- Therefore, t / f must yield an integer n.
-
As a result, a scale must exist in t-TET with:
- intervals uniformly spaced n notes apart
- f repetitions per octave
- n discrete transpositions
-
A MoLT’s note count need not be an integer factor of its parent temperament, since its scale complement must also be a MoLT: that is, removing a repeating sequence of notes from the temperament’s chromatic scale results in a repeating note sequence (usually a different one, unless it equates to every other note of the chromatic scale). Thus:
- 12-TET contains not just the expected 2, 3, 4, and 6-note MoLTs, but also 8, 9, and 10-note MoLTs.
- 15-TET contains 3-note, 5-note, 10-note, and 12-note MoLTs.
- 22-TET contains 2-note, 11-note, and 20-note MoLTs.
- 24-TET contains 2-note, 3-note, 4-note, 6-note, 8-note, 9-note, 10-note, 12-note, 14-note, 15-note, 16-note, 18-note, 20-note, 21-note, and 22-note MoLTs.
Many of these equate to the parent temperament minus one of its integer factors, but things can get complicated in temperaments with large numbers of factors. Let’s look again at 24-TET.
- 16, 18, 20, 21, and 22 are relatively straightforward: they equal 24 minus various integer factors of itself.
- However, 10 is not an integer factor of 24, nor is it 24 minus one of its own integer factors. But both 12 and 2 are factors of 24, and 12 − 2 = 10.
- The recursion doesn’t even stop there, since subtracting a 10-note MoLT from 24-TET’s chromatic scale produces a 14-note MoLT.
-
And it goes yet further:
- Subtracting the 2-note MoLT (tritone) from any other 24-TET MoLT that repeats twice an octave produces a MoLT.
- Subtracting the 3-note MoLT (augmented chord) from any other 24-TET MoLT that repeats three times an octave produces a MoLT.
Thus, all multiples of 2 and 3 have MoLTs in 24-TET.
The only way I currently know to calculate the total number of MoLTs within any composite equal temperament is to brute-force the calculation. I’m sure better methods exist, but I expect them to be complicated. I haven’t even figured out a reliable formula for all possible MoLT sizes within a composite equal temperament at a glance.
Achiral scales: Reflective & translational symmetry
Reflective symmetry
Rotational symmetry is not the only kind of symmetry a scale may possess. As it turns out, counting the octave and the chromatic scale, sixty-four modes of fifty-one scales in twelve-tone equal temperament have interval distributions with reflective symmetry. They are listed below.
(The scale names are based on interval ordering: the opening letter refers to the number of notes within the scale. The three-digit number orders every mode of every scale with that number of notes by “alphabetizing” the intervals. The two-digit number orders the scales based on their lowest-numbered modes. The Greek letters order the modes, where the most left-aligned is α; subsequent Greek letters each rotate the scale one interval to the left. “SC” means “Scale Complement”. I haven’t yet listed it for every scale, but I’ll get to it soon™.
I may replace this numbering with the now-ubiquitous pitch class sets theorist Allen Forte (1926-2014) gave them in his book The Structure of Atonal Music (1973), since it’s especially convenient for analysis that non-hexatonic scales have the same Forte numbers as their complements, but to be honest, I don’t fully understand Forte’s numbering system yet.)
| One-note reflective symmetry: the octave | ||||
|---|---|---|---|---|
| Mode | 1 | Intervals | SC | |
| C | 6 | K01 | ||
| Two-note reflective symmetry: the tritone | ||||||
|---|---|---|---|---|---|---|
| Mode | 1 | 2 | Intervals | SC | ||
| C♮ | F♯ | 3 | 3 | J06 | ||
| Three-note reflective symmetry (five scales, five modes) | ||||||||
|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | Intervals | SC | |||
| C♮ | C♯ | B♮ | ½ | 5 | ½ | I01 | ||
| C♮ | D♮ | A♯ | 1 | 4 | 1 | I05 | ||
| C♮ | D♯ | A♮ | 1½ | 3 | 1½ | I13 | ||
| C♮ | E♮ | G♯ | 2 | 2 | 2 | I19 | ||
| C♮ | F♮ | G♮ | 2½ | 1 | 2½ | I16 | ||
| Tetratonic reflective symmetry (three scales, five modes) | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | Intervals | SC | ||||
| C♮ | C♯ | F♯ | B♮ | ½ | 2½ | 2½ | ½ | H21 | ||
| C♮ | D♮ | F♯ | A♯ | 1 | 2 | 2 | 1 | H36 | ||
| C♮ | D♯ | F♯ | A♮ | 1½ | 1½ | 1½ | 1½ | H43 | ||
| C♮ | E♮ | F♯ | G♯ | 2 | 1 | 1 | 2 | H36 | ||
| C♮ | F♮ | F♯ | G♮ | 2½ | ½ | ½ | 2½ | H21 | ||
| Pentatonic reflective symmetry (ten scales, ten modes) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | Intervals | SC | |||||
| C♮ | C♯ | D♮ | A♯ | B♮ | ½ | ½ | 4 | ½ | ½ | G01 | ||
| C♮ | C♯ | D♯ | A♮ | B♮ | ½ | 1 | 3 | 1 | ½ | G09 | ||
| C♮ | D♮ | D♯ | A♮ | A♯ | 1 | ½ | 3 | ½ | 1 | G11 | ||
| C♮ | C♯ | E♮ | G♯ | B♮ | ½ | 1½ | 2 | 1½ | ½ | G44 | ||
| C♮ | D♯ | E♮ | G♯ | A♮ | 1½ | ½ | 2 | ½ | 1½ | G55 | ||
| C♮ | C♯ | F♮ | G♮ | B♮ | ½ | 2 | 1 | 2 | ½ | G45 | ||
| C♮ | E♮ | F♮ | G♮ | G♯ | 2 | ½ | 1 | ½ | 2 | G37 | ||
| C♮ | D♮ | E♮ | G♯ | A♯ | 1 | 1 | 2 | 1 | 1 | G49 | ||
| C♮ | D♮ | F♮ | G♮ | A♯ | 1 | 1½ | 1 | 1½ | 1 | G66 | ||
| C♮ | D♯ | F♮ | G♮ | A♮ | 1½ | 1 | 1 | 1 | 1½ | G64 | ||
| Hexatonic reflective symmetry (six scales, ten modes) | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | Intervals | SC | ||||||
| C♮ | C♯ | D♮ | F♯ | A♯ | B♮ | ½ | ½ | 2 | 2 | ½ | ½ | F22 | ||
| C♮ | C♯ | F♮ | F♯ | G♮ | B♮ | ½ | 2 | ½ | ½ | 2 | ½ | F47 | ||
| C♮ | E♮ | F♮ | F♯ | G♮ | G♯ | 2 | ½ | ½ | ½ | ½ | 2 | F22 | ||
| C♮ | D♮ | D♯ | F♯ | A♮ | A♯ | 1 | ½ | 1½ | 1½ | ½ | 1 | F60 | ||
| C♮ | D♮ | F♮ | F♯ | G♮ | A♯ | 1 | 1½ | ½ | ½ | 1½ | 1 | F67 | ||
| C♮ | D♯ | F♮ | F♯ | G♮ | A♮ | 1½ | 1 | ½ | ½ | 1 | 1½ | F58 | ||
| C♮ | C♯ | D♯ | F♯ | A♮ | B♮ | ½ | 1 | 1½ | 1½ | 1 | ½ | F58 | ||
| C♮ | C♯ | E♮ | F♯ | G♯ | B♮ | ½ | 1½ | 1 | 1 | 1½ | ½ | F67 | ||
| C♮ | D♯ | E♮ | F♯ | G♯ | A♮ | 1½ | ½ | 1 | 1 | ½ | 1½ | F60 | ||
| C♮ | D♮ | E♮ | F♯ | G♯ | A♯ | 1 | 1 | 1 | 1 | 1 | 1 | F80 | ||
| Heptatonic reflective symmetry (ten scales, ten modes) | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | SC | |||||||
| C♮ | D♭ | E𝄫 | F𝄫 | G𝄪 | A♯ | B♮ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | E01 | ||
| C♮ | D♭ | E𝄫 | F♭ | G♯ | A♯ | B♮ | ½ | ½ | 1 | 2 | 1 | ½ | ½ | E13 | ||
| C♮ | D♭ | E♭ | F♭ | G♯ | A♮ | B♮ | ½ | 1 | ½ | 2 | ½ | 1 | ½ | E17 | ||
| C♮ | D♮ | E♭ | F♭ | G♯ | A♮ | B♭ | 1 | ½ | ½ | 2 | ½ | ½ | 1 | E20 | ||
| C♮ | D♭ | E𝄫 | F♮ | G♮ | A♯ | B♮ | ½ | ½ | 1½ | 1 | 1½ | ½ | ½ | E34 | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | E55 | ||
| C♮ | D♯ | E♮ | F♮ | G♮ | A♭ | B𝄫 | 1½ | ½ | ½ | 1 | ½ | ½ | 1½ | E31 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | E65 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | E66 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♮ | ½ | 1 | 1 | 1 | 1 | 1 | ½ | E64 | ||
| Octatonic reflective symmetry (five scales, ten modes) | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Intervals | SC | ||||||||
| C♮ | C♯ | D♮ | D♯ | F♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | 1½ | 1½ | ½ | ½ | ½ | D09 | ||
| C♮ | C♯ | D♮ | F♮ | F♯ | G♮ | A♯ | B♮ | ½ | ½ | 1½ | ½ | ½ | 1½ | ½ | ½ | D22 | ||
| C♮ | C♯ | E♮ | F♮ | F♯ | G♮ | G♯ | B♮ | ½ | 1½ | ½ | ½ | ½ | ½ | 1½ | ½ | D22 | ||
| C♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | 1½ | ½ | ½ | ½ | ½ | ½ | ½ | 1½ | D09 | ||
| C♮ | D♮ | D♯ | E♮ | F♯ | G♯ | A♮ | A♯ | 1 | ½ | ½ | 1 | 1 | ½ | ½ | 1 | D41 | ||
| C♮ | D♮ | D♯ | F♮ | F♯ | G♮ | A♮ | A♯ | 1 | ½ | 1 | ½ | ½ | 1 | ½ | 1 | D40 | ||
| C♮ | D♮ | E♮ | F♮ | F♯ | G♮ | G♯ | A♯ | 1 | 1 | ½ | ½ | ½ | ½ | 1 | 1 | D34 | ||
| C♮ | C♯ | D♯ | F♮ | F♯ | G♮ | A♮ | B♮ | ½ | 1 | 1 | ½ | ½ | 1 | 1 | ½ | D41 | ||
| C♮ | C♯ | D♯ | E♮ | F♯ | G♯ | A♮ | B♮ | ½ | 1 | ½ | 1 | 1 | ½ | 1 | ½ | D40 | ||
| C♮ | C♯ | D♮ | E♮ | F♯ | G♯ | A♯ | B♮ | ½ | ½ | 1 | 1 | 1 | 1 | ½ | ½ | D34 | ||
| Enneatonic reflective symmetry (five scales, five modes) | ||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Intervals | SC | |||||||||
| C♮ | C♯ | D♮ | D♯ | E♮ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | ½ | 2 | ½ | ½ | ½ | ½ | C01 | ||
| C♮ | C♯ | D♮ | D♯ | F♮ | G♮ | A♮ | A♯ | B♮ | ½ | ½ | ½ | 1 | 1 | 1 | ½ | ½ | ½ | C10 | ||
| C♮ | C♯ | D♮ | E♮ | F♮ | G♮ | G♯ | A♯ | B♮ | ½ | ½ | 1 | ½ | 1 | ½ | 1 | ½ | ½ | C16 | ||
| C♮ | C♯ | D♯ | E♮ | F♮ | G♮ | G♯ | A♮ | B♮ | ½ | 1 | ½ | ½ | 1 | ½ | ½ | 1 | ½ | C19 | ||
| C♮ | D♮ | D♯ | E♮ | F♮ | G♮ | G♯ | A♮ | A♯ | 1 | ½ | ½ | ½ | 1 | ½ | ½ | ½ | 1 | C13 | ||
| Decatonic reflective symmetry (three scales, five modes) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | Intervals | ||||||||||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♯ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | ½ | 1 | 1 | ½ | ½ | ½ | ½ | ||
| C♮ | C♯ | D♮ | D♯ | F♮ | F♯ | G♮ | A♮ | A♯ | B♮ | ½ | ½ | ½ | 1 | ½ | ½ | 1 | ½ | ½ | ½ | ||
| C♮ | C♯ | D♮ | E♮ | F♮ | F♯ | G♮ | G♯ | A♯ | B♮ | ½ | ½ | 1 | ½ | ½ | ½ | ½ | 1 | ½ | ½ | ||
| C♮ | C♯ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | B♮ | ½ | 1 | ½ | ½ | ½ | ½ | ½ | ½ | 1 | ½ | ||
| C♮ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | 1 | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | 1 | ||
| Hendecatonic reflective symmetry (one scale, one mode) | |||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | Intervals | |||||||||||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | G♮ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | ½ | ½ | 1 | ½ | ½ | ½ | ½ | ½ | ||
| Dodecatonic reflective symmetry (the chromatic scale) | |||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | Intervals | ||||||||||||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | ||
There are 351 discrete scales in twelve-tone equal temperament. Of these, 256 are chiral scales that cannot be transformed into their inverses by rotation. As a result, another scale serves as their enantiomorph (Greek: ἐναντίος, enantíos, opposite, and μορφή, morphḗ, form). The most familiar achiral scale is surely harmonic minor. Fittingly, its enantiomorph, Mixolydian ♭2, is harmonic major’s fifth mode; in turn, of course, harmonic major’s enantiomorph, Phrygian dominant, is harmonic minor’s fifth mode.
By contrast, achiral scales can be transformed into their inverses by rotation and thus have no enantiomorphs: every reflection of any mode of an achiral scale is either itself (if the mode itself is symmetrical) or another of its modes. For instance, Ionian’s reflection is Phrygian; Dorian’s reflection is itself. The classic metaphor is that achiral scales are socks that fit either foot, while chiral scales are mittens that only fit one hand.
Every symmetrical heptatonic scale in twelve-tone equal temperament is a complement of a symmetrical pentatonic scale. The following table shows the interval distributions of symmetrical pentatonic scales and their complements side-by-side, first using my order for the heptatonics, then my order for the pentatonics. You may have noticed that I ordered the scales above so that their interval distributions would form aesthetic patterns. An interesting consequence of this is that their complements’ symmetrical modes form their own aesthetic patterns, which are quite dissimilar from the originals, but quite similar to each other.
(I should clarify that, while these tables list these scales’ symmetrical modes side-by-side, scale complements apply on a scale-wide basis, not a modal basis, simply because the process of forming a scale complement requires rotating the scale, and not necessarily by a constant amount. Moreover, in 12-TET, only hexatonic scales have complements of the same size; thus, 1:1 modal relationships between complements cannot always exist.)
| Symmetrical scale complements (5 & 7) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| heptatonic order | |||||||||||||||
| Pentatonic | Intervals | Heptatonic | Intervals | ||||||||||||
| ½ | ½ | 4 | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||||
| ½ | 1 | 3 | 1 | ½ | ½ | ½ | 1 | 2 | 1 | ½ | ½ | ||||
| ½ | 1½ | 2 | 1½ | ½ | ½ | 1 | ½ | 2 | ½ | 1 | ½ | ||||
| ½ | 2 | 1 | 2 | ½ | 1 | ½ | ½ | 2 | ½ | ½ | 1 | ||||
| 1 | ½ | 3 | ½ | 1 | ½ | ½ | 1½ | 1 | 1½ | ½ | ½ | ||||
| 1½ | ½ | 2 | ½ | 1½ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | ||||
| 2 | ½ | 1 | ½ | 2 | 1½ | ½ | ½ | 1 | ½ | ½ | 1½ | ||||
| 1½ | 1 | 1 | 1 | 1½ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||||
| 1 | 1½ | 1 | 1½ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||||
| 1 | 1 | 2 | 1 | 1 | ½ | 1 | 1 | 1 | 1 | 1 | ½ | ||||
| pentatonic order | |||||||||||||||
| Pentatonic | Intervals | Heptatonic | Intervals | ||||||||||||
| ½ | ½ | 4 | ½ | ½ | ½ | ½ | ½ | 3 | ½ | ½ | ½ | ||||
| ½ | 1 | 3 | 1 | ½ | ½ | ½ | 1 | 2 | 1 | ½ | ½ | ||||
| 1 | ½ | 3 | ½ | 1 | ½ | ½ | 1½ | 1 | 1½ | ½ | ½ | ||||
| ½ | 1½ | 2 | 1½ | ½ | ½ | 1 | ½ | 2 | ½ | 1 | ½ | ||||
| 1½ | ½ | 2 | ½ | 1½ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | ||||
| ½ | 2 | 1 | 2 | ½ | 1 | ½ | ½ | 2 | ½ | ½ | 1 | ||||
| 2 | ½ | 1 | ½ | 2 | 1½ | ½ | ½ | 1 | ½ | ½ | 1½ | ||||
| 1 | 1 | 2 | 1 | 1 | ½ | 1 | 1 | 1 | 1 | 1 | ½ | ||||
| 1 | 1½ | 1 | 1½ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||||
| 1½ | 1 | 1 | 1 | 1½ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | ||||
A similar pattern occurs with three-note scales and the enneatonics:
| Symmetrical scale complements (3 & 9) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Three-note | Intervals | Enneatonic | Intervals | ||||||||||||
| ½ | 5 | ½ | ½ | ½ | ½ | ½ | 2 | ½ | ½ | ½ | ½ | ||||
| 1 | 4 | 1 | ½ | ½ | ½ | 1 | 1 | 1 | ½ | ½ | ½ | ||||
| 1½ | 3 | 1½ | ½ | ½ | 1 | ½ | 1 | ½ | 1 | ½ | ½ | ||||
| 2 | 2 | 2 | ½ | 1 | ½ | ½ | 1 | ½ | ½ | 1 | ½ | ||||
| 2½ | 1 | 2½ | 1 | ½ | ½ | ½ | 1 | ½ | ½ | ½ | 1 | ||||
Translational symmetry
I’ve mentioned that 12-TET contains 351 scales, that 256 scales are chiral, and that 51 scales have reflectively symmetrical modes. If you’ve done the math, you’ve already figured out that I haven’t accounted for 44 scales.
The scale complement pattern I noted above is only so straightforward with odd-numbered scales – even-numbered scales are more complicated. The reason is that reflective symmetry is a subset of a broader type of symmetry called translational symmetry. The 44 remaining scales are translationally symmetrical (and thus achiral), but do not have any reflectively symmetrical modes.
A scale possesses translational symmetry if it can be transformed into its inverse by rotation. Reflective symmetry occurs on notes (like Dorian’s symmetry); however, another form of translational symmetry occurs between notes. This can only exist if a scale’s note count is a multiple of 2. I’ll henceforth call such scales even scales (antonym: odd scales) for brevity’s sake.
In 12-TET, all achiral scales have achiral complements (caveat: the chromatic scale’s complement is… um… er… what‽), but not all reflectively symmetrical scales have reflectively symmetrical complements. However, for the same reason that translational symmetry between notes can only occur in even scales, all reflectively symmetrical odd scales have reflectively symmetrical complements.
Among scales with translational but not reflective symmetry, five have two notes, twelve are tetratonic, fourteen hexatonic, ten octatonic, and three decatonic. Since literally all two-note and decatonic scales are translationally symmetrical, I see little point in printing them here, but the tetratonics, hexatonics, and octatonics are:
| Tetratonic translational symmetry (twelve additional scales) | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | Intervals | SC | ||||
| C♮ | C♯ | D♮ | D♯ | ½ | ½ | ½ | 4½ | H01 | ||
| C♮ | C♯ | D♮ | B♮ | ½ | ½ | 4½ | ½ | |||
| C♮ | C♯ | A♯ | B♮ | ½ | 4½ | ½ | ½ | |||
| C♮ | A♮ | A♯ | B♮ | 4½ | ½ | ½ | ½ | |||
| C♮ | C♯ | D♯ | E♮ | ½ | 1 | ½ | 4 | H03 | ||
| C♮ | D♮ | D♯ | B♮ | 1 | ½ | 4 | ½ | |||
| C♮ | C♯ | A♮ | A♯ | ½ | 4 | ½ | 1 | |||
| C♮ | G♯ | A♮ | B♮ | 4 | ½ | 1 | ½ | |||
| C♮ | D♮ | D♯ | F♮ | 1 | ½ | 1 | 3½ | H07 | ||
| C♮ | C♯ | D♯ | A♯ | ½ | 1 | 3½ | 1 | |||
| C♮ | D♮ | A♮ | B♮ | 1 | 3½ | 1 | ½ | |||
| C♮ | G♮ | A♮ | A♯ | 3½ | 1 | ½ | 1 | |||
| C♮ | C♯ | E♮ | F♮ | ½ | 1½ | ½ | 3½ | H08 | ||
| C♮ | D♯ | E♮ | B♮ | 1½ | ½ | 3½ | ½ | |||
| C♮ | C♯ | G♯ | A♮ | ½ | 3½ | ½ | 1½ | |||
| C♮ | G♮ | G♯ | B♮ | 3½ | ½ | 1½ | ½ | |||
| C♮ | C♯ | F♮ | F♯ | ½ | 2 | ½ | 3 | H17 | ||
| C♮ | E♮ | F♮ | B♮ | 2 | ½ | 3 | ½ | |||
| C♮ | C♯ | G♮ | G♯ | ½ | 3 | ½ | 2 | |||
| C♮ | F♯ | G♮ | B♮ | 3 | ½ | 2 | ½ | |||
| C♮ | C♯ | F♯ | G♮ | ½ | 2½ | ½ | 2½ | H31 | ||
| C♮ | F♮ | F♯ | B♮ | 2½ | ½ | 2½ | ½ | |||
| C♮ | D♮ | E♮ | F♯ | 1 | 1 | 1 | 3 | H15 | ||
| C♮ | D♮ | E♮ | A♯ | 1 | 1 | 3 | 1 | |||
| C♮ | D♮ | G♯ | A♯ | 1 | 3 | 1 | 1 | |||
| C♮ | F♯ | G♯ | A♯ | 3 | 1 | 1 | 1 | |||
| C♮ | D♮ | F♮ | G♮ | 1 | 1½ | 1 | 2½ | H28 | ||
| C♮ | D♯ | F♮ | A♯ | 1½ | 1 | 2½ | 1 | |||
| C♮ | D♮ | G♮ | A♮ | 1 | 2½ | 1 | 1½ | |||
| C♮ | F♮ | G♮ | A♯ | 2½ | 1 | 1½ | 1 | |||
| C♮ | D♮ | F♯ | G♯ | 1 | 2 | 1 | 2 | H41 | ||
| C♮ | E♮ | F♯ | A♯ | 2 | 1 | 2 | 1 | |||
| C♮ | D♯ | E♮ | G♮ | 1½ | ½ | 1½ | 2½ | H26 | ||
| C♮ | C♯ | E♮ | A♮ | ½ | 1½ | 2½ | 1½ | |||
| C♮ | D♯ | G♯ | B♮ | 1½ | 2½ | 1½ | ½ | |||
| C♮ | F♮ | G♯ | A♮ | 2½ | 1½ | ½ | 1½ | |||
| C♮ | D♯ | F♮ | G♯ | 1½ | 1 | 1½ | 2 | H40 | ||
| C♮ | D♮ | F♮ | A♮ | 1 | 1½ | 2 | 1½ | |||
| C♮ | D♯ | G♮ | A♯ | 1½ | 2 | 1½ | 1 | |||
| C♮ | E♮ | G♮ | A♮ | 2 | 1½ | 1 | 1½ | |||
| C♮ | E♮ | F♮ | A♮ | 2 | ½ | 2 | 1½ | H38 | ||
| C♮ | C♯ | F♮ | G♯ | ½ | 2 | 1½ | 2 | |||
| C♮ | E♮ | G♮ | B♮ | 2 | 1½ | 2 | ½ | |||
| C♮ | D♯ | G♮ | G♯ | 1½ | 2 | ½ | 2 | |||
| Hexatonic translational symmetry (fourteen additional scales) | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | Intervals | SC | ||||||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | ½ | ½ | ½ | ½ | ½ | 3½ | F01 | ||
| C♮ | C♯ | D♮ | D♯ | E♮ | B♮ | ½ | ½ | ½ | ½ | 3½ | ½ | |||
| C♮ | C♯ | D♮ | D♯ | A♯ | B♮ | ½ | ½ | ½ | 3½ | ½ | ½ | |||
| C♮ | C♯ | D♮ | A♮ | A♯ | B♮ | ½ | ½ | 3½ | ½ | ½ | ½ | |||
| C♮ | C♯ | G♯ | A♮ | A♯ | B♮ | ½ | 3½ | ½ | ½ | ½ | ½ | |||
| C♮ | G♮ | G♯ | A♮ | A♯ | B♮ | 3½ | ½ | ½ | ½ | ½ | ½ | |||
| C♮ | C♯ | D♮ | E♮ | F♮ | F♯ | ½ | ½ | 1 | ½ | ½ | 3 | F04 | ||
| C♮ | C♯ | D♯ | E♮ | F♮ | B♮ | ½ | 1 | ½ | ½ | 3 | ½ | |||
| C♮ | D♮ | D♯ | E♮ | A♯ | B♮ | 1 | ½ | ½ | 3 | ½ | ½ | |||
| C♮ | C♯ | D♮ | G♯ | A♮ | A♯ | ½ | ½ | 3 | ½ | ½ | 1 | |||
| C♮ | C♯ | G♮ | G♯ | A♮ | B♮ | ½ | 3 | ½ | ½ | 1 | ½ | |||
| C♮ | F♯ | G♮ | G♯ | A♯ | B♮ | 3 | ½ | ½ | 1 | ½ | ½ | |||
| C♮ | D♮ | D♯ | E♮ | F♮ | G♮ | 1 | ½ | ½ | ½ | 1 | 2½ | F11 | ||
| C♮ | C♯ | D♮ | D♯ | F♮ | A♯ | ½ | ½ | ½ | 1 | 2½ | 1 | |||
| C♮ | C♯ | D♮ | E♮ | A♮ | B♮ | ½ | ½ | 1 | 2½ | 1 | ½ | |||
| C♮ | C♯ | D♯ | G♯ | A♯ | B♮ | ½ | 1 | 2½ | 1 | ½ | ½ | |||
| C♮ | D♮ | G♮ | A♮ | A♯ | B♮ | 1 | 2½ | 1 | ½ | ½ | ½ | |||
| C♮ | F♮ | G♮ | G♯ | A♮ | A♯ | 2½ | 1 | ½ | ½ | ½ | 1 | |||
| C♮ | C♯ | D♯ | E♮ | F♯ | G♮ | ½ | 1 | ½ | 1 | ½ | 2½ | F14 | ||
| C♮ | D♮ | D♯ | F♮ | F♯ | B♮ | 1 | ½ | 1 | ½ | 2½ | ½ | |||
| C♮ | C♯ | D♯ | E♮ | A♮ | A♯ | ½ | 1 | ½ | 2½ | ½ | 1 | |||
| C♮ | D♮ | D♯ | G♯ | A♮ | B♮ | 1 | ½ | 2½ | ½ | 1 | ½ | |||
| C♮ | C♯ | F♯ | G♮ | A♮ | A♯ | ½ | 2½ | ½ | 1 | ½ | 1 | |||
| C♮ | F♮ | F♯ | G♯ | A♮ | B♮ | 2½ | ½ | 1 | ½ | 1 | ½ | |||
| C♮ | C♯ | D♮ | F♮ | F♯ | G♮ | ½ | ½ | 1½ | ½ | ½ | 2½ | F16 | ||
| C♮ | C♯ | E♮ | F♮ | F♯ | B♮ | ½ | 1½ | ½ | ½ | 2½ | ½ | |||
| C♮ | D♯ | E♮ | F♮ | A♯ | B♮ | 1½ | ½ | ½ | 2½ | ½ | ½ | |||
| C♮ | C♯ | D♮ | G♮ | G♯ | A♮ | ½ | ½ | 2½ | ½ | ½ | 1½ | |||
| C♮ | C♯ | F♯ | G♮ | G♯ | B♮ | ½ | 2½ | ½ | ½ | 1½ | ½ | |||
| C♮ | F♮ | F♯ | G♮ | A♯ | B♮ | 2½ | ½ | ½ | 1½ | ½ | ½ | |||
| C♮ | C♯ | D♯ | F♮ | G♮ | G♯ | ½ | 1 | 1 | 1 | ½ | 2 | F42 | ||
| C♮ | D♮ | E♮ | F♯ | G♮ | B♮ | 1 | 1 | 1 | ½ | 2 | ½ | |||
| C♮ | D♮ | E♮ | F♮ | A♮ | A♯ | 1 | 1 | ½ | 2 | ½ | 1 | |||
| C♮ | D♮ | D♯ | G♮ | G♯ | A♯ | 1 | ½ | 2 | ½ | 1 | 1 | |||
| C♮ | C♯ | F♮ | F♯ | G♯ | A♯ | ½ | 2 | ½ | 1 | 1 | 1 | |||
| C♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 2 | ½ | 1 | 1 | 1 | ½ | |||
| C♮ | D♮ | D♯ | F♮ | F♯ | G♯ | 1 | ½ | 1 | ½ | 1 | 2 | F33 | ||
| C♮ | C♯ | D♯ | E♮ | F♯ | A♯ | ½ | 1 | ½ | 1 | 2 | 1 | |||
| C♮ | D♮ | D♯ | F♮ | A♮ | B♮ | 1 | ½ | 1 | 2 | 1 | ½ | |||
| C♮ | C♯ | D♯ | G♮ | A♮ | A♯ | ½ | 1 | 2 | 1 | ½ | 1 | |||
| C♮ | D♮ | F♯ | G♯ | A♮ | B♮ | 1 | 2 | 1 | ½ | 1 | ½ | |||
| C♮ | E♮ | F♯ | G♮ | A♮ | A♯ | 2 | 1 | ½ | 1 | ½ | 1 | |||
| C♮ | E♮ | F♮ | F♯ | G♮ | B♮ | 2 | ½ | ½ | ½ | 2 | ½ | F37 | ||
| C♮ | C♯ | D♮ | D♯ | G♮ | G♯ | ½ | ½ | ½ | 2 | ½ | 2 | |||
| C♮ | C♯ | D♮ | F♯ | G♮ | B♮ | ½ | ½ | 2 | ½ | 2 | ½ | |||
| C♮ | C♯ | F♮ | F♯ | A♯ | B♮ | ½ | 2 | ½ | 2 | ½ | ½ | |||
| C♮ | E♮ | F♮ | A♮ | A♯ | B♮ | 2 | ½ | 2 | ½ | ½ | ½ | |||
| C♮ | C♯ | F♮ | F♯ | G♮ | G♯ | ½ | 2 | ½ | ½ | ½ | 2 | |||
| C♮ | D♯ | E♮ | F♮ | F♯ | A♮ | 1½ | ½ | ½ | ½ | 1½ | 1½ | F55 | ||
| C♮ | C♯ | D♮ | D♯ | F♯ | A♮ | ½ | ½ | ½ | 1½ | 1½ | 1½ | |||
| C♮ | C♯ | D♮ | F♮ | G♯ | B♮ | ½ | ½ | 1½ | 1½ | 1½ | ½ | |||
| C♮ | C♯ | E♮ | G♮ | A♯ | B♮ | ½ | 1½ | 1½ | 1½ | ½ | ½ | |||
| C♮ | D♯ | F♯ | A♮ | A♯ | B♮ | 1½ | 1½ | 1½ | ½ | ½ | ½ | |||
| C♮ | D♯ | F♯ | G♮ | G♯ | A♮ | 1½ | 1½ | ½ | ½ | ½ | 1½ | |||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | 1 | 1 | ½ | 1 | 1 | 1½ | F71 | ||
| C♮ | D♮ | D♯ | F♮ | G♮ | A♯ | 1 | ½ | 1 | 1 | 1½ | 1 | |||
| C♮ | C♯ | D♯ | F♮ | G♯ | A♯ | ½ | 1 | 1 | 1½ | 1 | 1 | |||
| C♮ | D♮ | E♮ | G♮ | A♮ | B♮ | 1 | 1 | 1½ | 1 | 1 | ½ | |||
| C♮ | D♮ | F♮ | G♮ | A♮ | A♯ | 1 | 1½ | 1 | 1 | ½ | 1 | |||
| C♮ | D♯ | F♮ | G♮ | G♯ | A♯ | 1½ | 1 | 1 | ½ | 1 | 1 | |||
| C♮ | D♯ | E♮ | F♯ | G♮ | A♯ | 1½ | ½ | 1 | ½ | 1½ | 1 | F65 | ||
| C♮ | C♯ | D♯ | E♮ | G♮ | A♮ | ½ | 1 | ½ | 1½ | 1 | 1½ | |||
| C♮ | D♮ | D♯ | F♯ | G♯ | B♮ | 1 | ½ | 1½ | 1 | 1½ | ½ | |||
| C♮ | C♯ | E♮ | F♯ | A♮ | A♯ | ½ | 1½ | 1 | 1½ | ½ | 1 | |||
| C♮ | D♯ | F♮ | G♯ | A♮ | B♮ | 1½ | 1 | 1½ | ½ | 1 | ½ | |||
| C♮ | D♮ | F♮ | F♯ | G♯ | A♮ | 1 | 1½ | ½ | 1 | ½ | 1½ | |||
| C♮ | D♮ | F♮ | F♯ | A♮ | B♮ | 1 | 1½ | ½ | 1½ | 1 | ½ | F74 | ||
| C♮ | D♯ | E♮ | G♮ | A♮ | A♯ | 1½ | ½ | 1½ | 1 | ½ | 1 | |||
| C♮ | C♯ | E♮ | F♯ | G♮ | A♮ | ½ | 1½ | 1 | ½ | 1 | 1½ | |||
| C♮ | D♯ | F♮ | F♯ | G♯ | B♮ | 1½ | 1 | ½ | 1 | 1½ | ½ | |||
| C♮ | D♮ | D♯ | F♮ | G♯ | A♮ | 1 | ½ | 1 | 1½ | ½ | 1½ | |||
| C♮ | C♯ | D♯ | F♯ | G♮ | A♯ | ½ | 1 | 1½ | ½ | 1½ | 1 | |||
| C♮ | C♯ | D♯ | F♯ | G♯ | A♮ | ½ | 1 | 1½ | 1 | ½ | 1½ | F73 | ||
| C♮ | D♮ | F♮ | G♮ | G♯ | B♮ | 1 | 1½ | 1 | ½ | 1½ | ½ | |||
| C♮ | D♯ | F♮ | F♯ | A♮ | A♯ | 1½ | 1 | ½ | 1½ | ½ | 1 | |||
| C♮ | D♮ | D♯ | F♯ | G♮ | A♮ | 1 | ½ | 1½ | ½ | 1 | 1½ | |||
| C♮ | C♯ | E♮ | F♮ | G♮ | A♯ | ½ | 1½ | ½ | 1 | 1½ | 1 | |||
| C♮ | D♯ | E♮ | F♯ | A♮ | B♮ | 1½ | ½ | 1 | 1½ | 1 | ½ | |||
| C♮ | C♯ | E♮ | F♮ | G♯ | A♮ | ½ | 1½ | ½ | 1½ | ½ | 1½ | F76 | ||
| C♮ | D♯ | E♮ | G♮ | G♯ | B♮ | 1½ | ½ | 1½ | ½ | 1½ | ½ | |||
| Octatonic translational symmetry (ten additional scales) | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Intervals | SC | ||||||||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | 2½ | D01 | ||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | F♯ | B♮ | ½ | ½ | ½ | ½ | ½ | ½ | 2½ | ½ | |||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | A♯ | B♮ | ½ | ½ | ½ | ½ | ½ | 2½ | ½ | ½ | |||
| C♮ | C♯ | D♮ | D♯ | E♮ | A♮ | A♯ | B♮ | ½ | ½ | ½ | ½ | 2½ | ½ | ½ | ½ | |||
| C♮ | C♯ | D♮ | D♯ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | ½ | 2½ | ½ | ½ | ½ | ½ | |||
| C♮ | C♯ | D♮ | G♮ | G♯ | A♮ | A♯ | B♮ | ½ | ½ | 2½ | ½ | ½ | ½ | ½ | ½ | |||
| C♮ | C♯ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | ½ | 2½ | ½ | ½ | ½ | ½ | ½ | ½ | |||
| C♮ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | B♮ | 2½ | ½ | ½ | ½ | ½ | ½ | ½ | ½ | |||
| C♮ | C♯ | D♮ | D♯ | F♮ | F♯ | G♮ | G♯ | ½ | ½ | ½ | 1 | ½ | ½ | ½ | 2 | D05 | ||
| C♮ | C♯ | D♮ | E♮ | F♮ | F♯ | G♮ | B♮ | ½ | ½ | 1 | ½ | ½ | ½ | 2 | ½ | |||
| C♮ | C♯ | D♯ | E♮ | F♮ | F♯ | A♯ | B♮ | ½ | 1 | ½ | ½ | ½ | 2 | ½ | ½ | |||
| C♮ | D♮ | D♯ | E♮ | F♮ | A♮ | A♯ | B♮ | 1 | ½ | ½ | ½ | 2 | ½ | ½ | ½ | |||
| C♮ | C♯ | D♮ | D♯ | G♮ | G♯ | A♮ | A♯ | ½ | ½ | ½ | 2 | ½ | ½ | ½ | 1 | |||
| C♮ | C♯ | D♮ | F♯ | G♮ | G♯ | A♮ | B♮ | ½ | ½ | 2 | ½ | ½ | ½ | 1 | ½ | |||
| C♮ | C♯ | F♮ | F♯ | G♮ | G♯ | A♯ | B♮ | ½ | 2 | ½ | ½ | ½ | 1 | ½ | ½ | |||
| C♮ | E♮ | F♮ | F♯ | G♮ | A♮ | A♯ | B♮ | 2 | ½ | ½ | ½ | 1 | ½ | ½ | ½ | |||
| C♮ | C♯ | D♮ | D♯ | F♯ | G♮ | G♯ | A♮ | ½ | ½ | ½ | 1½ | ½ | ½ | ½ | 1½ | D27 | ||
| C♮ | C♯ | D♮ | F♮ | F♯ | G♮ | G♯ | B♮ | ½ | ½ | 1½ | ½ | ½ | ½ | 1½ | ½ | |||
| C♮ | C♯ | E♮ | F♮ | F♯ | G♮ | A♯ | B♮ | ½ | 1½ | ½ | ½ | ½ | 1½ | ½ | ½ | |||
| C♮ | D♯ | E♮ | F♮ | F♯ | A♮ | A♯ | B♮ | 1½ | ½ | ½ | ½ | 1½ | ½ | ½ | ½ | |||
| C♮ | D♮ | D♯ | E♮ | F♮ | F♯ | G♮ | A♮ | 1 | ½ | ½ | ½ | ½ | ½ | 1 | 1½ | D15 | ||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | G♮ | A♯ | ½ | ½ | ½ | ½ | ½ | 1 | 1½ | 1 | |||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♯ | A♮ | B♮ | ½ | ½ | ½ | ½ | 1 | 1½ | 1 | ½ | |||
| C♮ | C♯ | D♮ | D♯ | F♮ | G♯ | A♯ | B♮ | ½ | ½ | ½ | 1 | 1½ | 1 | ½ | ½ | |||
| C♮ | C♯ | D♮ | E♮ | G♮ | A♮ | A♯ | B♮ | ½ | ½ | 1 | 1½ | 1 | ½ | ½ | ½ | |||
| C♮ | C♯ | D♯ | F♯ | G♯ | A♮ | A♯ | B♮ | ½ | 1 | 1½ | 1 | ½ | ½ | ½ | ½ | |||
| C♮ | D♮ | F♮ | G♮ | G♯ | A♮ | A♯ | B♮ | 1 | 1½ | 1 | ½ | ½ | ½ | ½ | ½ | |||
| C♮ | D♯ | F♮ | F♯ | G♮ | G♯ | A♮ | A♯ | 1½ | 1 | ½ | ½ | ½ | ½ | ½ | 1 | |||
| C♮ | C♯ | D♯ | E♮ | F♮ | F♯ | G♯ | A♮ | ½ | 1 | ½ | ½ | ½ | 1 | ½ | 1½ | D20 | ||
| C♮ | D♮ | D♯ | E♮ | F♮ | G♮ | G♯ | B♮ | 1 | ½ | ½ | ½ | 1 | ½ | 1½ | ½ | |||
| C♮ | C♯ | D♮ | D♯ | F♮ | F♯ | A♮ | A♯ | ½ | ½ | ½ | 1 | ½ | 1½ | ½ | 1 | |||
| C♮ | C♯ | D♮ | E♮ | F♮ | G♯ | A♮ | B♮ | ½ | ½ | 1 | ½ | 1½ | ½ | 1 | ½ | |||
| C♮ | C♯ | D♯ | E♮ | G♮ | G♯ | A♯ | B♮ | ½ | 1 | ½ | 1½ | ½ | 1 | ½ | ½ | |||
| C♮ | D♮ | D♯ | F♯ | G♮ | A♮ | A♯ | B♮ | 1 | ½ | 1½ | ½ | 1 | ½ | ½ | ½ | |||
| C♮ | C♯ | E♮ | F♮ | G♮ | G♯ | A♮ | A♯ | ½ | 1½ | ½ | 1 | ½ | ½ | ½ | 1 | |||
| C♮ | D♯ | E♮ | F♯ | G♮ | G♯ | A♮ | B♮ | 1½ | ½ | 1 | ½ | ½ | ½ | 1 | ½ | |||
| C♮ | C♯ | D♮ | E♮ | F♮ | G♮ | G♯ | A♮ | ½ | ½ | 1 | ½ | 1 | ½ | ½ | 1½ | D24 | ||
| C♮ | C♯ | D♯ | E♮ | F♯ | G♮ | G♯ | B♮ | ½ | 1 | ½ | 1 | ½ | ½ | 1½ | ½ | |||
| C♮ | D♮ | D♯ | F♮ | F♯ | G♮ | A♯ | B♮ | 1 | ½ | 1 | ½ | ½ | 1½ | ½ | ½ | |||
| C♮ | C♯ | D♯ | E♮ | F♮ | G♯ | A♮ | A♯ | ½ | 1 | ½ | ½ | 1½ | ½ | ½ | 1 | |||
| C♮ | D♮ | D♯ | E♮ | G♮ | G♯ | A♮ | B♮ | 1 | ½ | ½ | 1½ | ½ | ½ | 1 | ½ | |||
| C♮ | C♯ | D♮ | F♮ | F♯ | G♮ | A♮ | A♯ | ½ | ½ | 1½ | ½ | ½ | 1 | ½ | 1 | |||
| C♮ | C♯ | E♮ | F♮ | F♯ | G♯ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1 | ½ | 1 | ½ | |||
| C♮ | D♯ | E♮ | F♮ | G♮ | G♯ | A♯ | B♮ | 1½ | ½ | ½ | 1 | ½ | 1 | ½ | ½ | |||
| C♮ | D♯ | E♮ | F♮ | F♯ | G♮ | G♯ | B♮ | 1½ | ½ | ½ | ½ | ½ | ½ | 1½ | ½ | D16 | ||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | G♯ | A♮ | ½ | ½ | ½ | ½ | ½ | 1½ | ½ | 1½ | |||
| C♮ | C♯ | D♮ | D♯ | E♮ | G♮ | G♯ | B♮ | ½ | ½ | ½ | ½ | 1½ | ½ | 1½ | ½ | |||
| C♮ | C♯ | D♮ | D♯ | F♯ | G♮ | A♯ | B♮ | ½ | ½ | ½ | 1½ | ½ | 1½ | ½ | ½ | |||
| C♮ | C♯ | D♮ | F♮ | F♯ | A♮ | A♯ | B♮ | ½ | ½ | 1½ | ½ | 1½ | ½ | ½ | ½ | |||
| C♮ | C♯ | E♮ | F♮ | G♯ | A♮ | A♯ | B♮ | ½ | 1½ | ½ | 1½ | ½ | ½ | ½ | ½ | |||
| C♮ | D♯ | E♮ | G♮ | G♯ | A♮ | A♯ | B♮ | 1½ | ½ | 1½ | ½ | ½ | ½ | ½ | ½ | |||
| C♮ | C♯ | E♮ | F♮ | F♯ | G♮ | G♯ | A♮ | ½ | 1½ | ½ | ½ | ½ | ½ | ½ | 1½ | |||
| C♮ | C♯ | D♯ | F♮ | F♯ | G♯ | A♯ | B♮ | ½ | 1 | 1 | ½ | 1 | 1 | ½ | ½ | D38 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | A♯ | B♮ | 1 | 1 | ½ | 1 | 1 | ½ | ½ | ½ | |||
| C♮ | D♮ | D♯ | F♮ | G♮ | G♯ | A♮ | A♯ | 1 | ½ | 1 | 1 | ½ | ½ | ½ | 1 | |||
| C♮ | C♯ | D♯ | F♮ | F♯ | G♮ | G♯ | A♯ | ½ | 1 | 1 | ½ | ½ | ½ | 1 | 1 | |||
| C♮ | D♮ | E♮ | F♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | ½ | ½ | ½ | 1 | 1 | ½ | |||
| C♮ | D♮ | D♯ | E♮ | F♮ | G♮ | A♮ | A♯ | 1 | ½ | ½ | ½ | 1 | 1 | ½ | 1 | |||
| C♮ | C♯ | D♮ | D♯ | F♮ | G♮ | G♯ | A♯ | ½ | ½ | ½ | 1 | 1 | ½ | 1 | 1 | |||
| C♮ | C♯ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | ½ | ½ | 1 | 1 | ½ | 1 | 1 | ½ | |||
| C♮ | C♯ | D♯ | E♮ | F♯ | G♮ | A♮ | A♯ | ½ | 1 | ½ | 1 | ½ | 1 | ½ | 1 | D43 | ||
| C♮ | D♮ | D♯ | F♮ | F♯ | G♯ | A♮ | B♮ | 1 | ½ | 1 | ½ | 1 | ½ | 1 | ½ | |||
| C♮ | D♮ | D♯ | E♮ | F♯ | G♮ | G♯ | A♯ | 1 | ½ | ½ | 1 | ½ | ½ | 1 | 1 | D36 | ||
| C♮ | C♯ | D♮ | E♮ | F♮ | F♯ | G♯ | A♯ | ½ | ½ | 1 | ½ | ½ | 1 | 1 | 1 | |||
| C♮ | C♯ | D♯ | E♮ | F♮ | G♮ | A♮ | B♮ | ½ | 1 | ½ | ½ | 1 | 1 | 1 | ½ | |||
| C♮ | D♮ | D♯ | E♮ | F♯ | G♯ | A♯ | B♮ | 1 | ½ | ½ | 1 | 1 | 1 | ½ | ½ | |||
| C♮ | C♯ | D♮ | E♮ | F♯ | G♯ | A♮ | A♯ | ½ | ½ | 1 | 1 | 1 | ½ | ½ | 1 | |||
| C♮ | C♯ | D♯ | F♮ | G♮ | G♯ | A♮ | B♮ | ½ | 1 | 1 | 1 | ½ | ½ | 1 | ½ | |||
| C♮ | D♮ | E♮ | F♯ | G♮ | G♯ | A♯ | B♮ | 1 | 1 | 1 | ½ | ½ | 1 | ½ | ½ | |||
| C♮ | D♮ | E♮ | F♮ | F♯ | G♯ | A♮ | A♯ | 1 | 1 | ½ | ½ | 1 | ½ | ½ | 1 | |||
Counting achiral scales
As I mentioned, all achiral scales have achiral complements. I also mentioned a caveat: the chromatic scale’s complement, the null set, shares some characteristics of a divide by zero error. In the context of scale complements, however, I’m willing to count it as one. Perhaps more importantly, Forte counts it as a pitch class. Bearing that in mind, the achiral scales break down as follows:
| Achiral scales by size | ||||
|---|---|---|---|---|
| Notes | Scales | |||
| 0 | 1 | 11 | 12 | 1 |
| 2 | 10 | 6 | ||
| 3 | 9 | 5 | ||
| 4 | 8 | 15 | ||
| 5 | 7 | 10 | ||
| 6 | 20 | |||
| Total | 96 | |||
Forte grouped pitch classes into 224 discrete sets. 128 of these are the 256 chiral scales; the other 96 are the achiral scales, for a total 352 scales (or 351, if we exclude the null set).
Self-complementary scales, symmetry, and scale generators
Only six scales in 12-TET are their own complements. In the table below, “RS” means “rotation size”, namely how many semitones we must rotate the scale after flipping its bits to wind up with its original note composition. Importantly, in each case, rotating the scale up and rotating it down will produce the same mode.
| Self-complementing scales | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ID | Forte | Notes (prime form) | Intervals | Binary form | RS | ||||||||||
| C♮ | C♯ | D♮ | D♯ | E♮ | F♮ | ½ | ½ | ½ | ½ | ½ | 3½ | 111111000000 | 6 | ||
| C♮ | C♯ | D♮ | F♯ | G♮ | G♯ | ½ | ½ | 2 | ½ | ½ | 2 | 111000111000 | 3 | ||
| C♮ | C♯ | E♮ | F♮ | G♯ | A♮ | ½ | 1½ | ½ | 1½ | ½ | 1½ | 110011001100 | 2 | ||
| C♮ | D♮ | E♮ | F♯ | G♯ | A♯ | 1 | 1 | 1 | 1 | 1 | 1 | 101010101010 | 1 | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | 1 | 1 | ½ | 1 | 1 | 1½ | 101011010100 | 6 | ||
| C♮ | D♮ | D♯ | E♮ | F♮ | G♮ | 1 | ½ | ½ | ½ | 1 | 2½ | 101111010000 | 6 | ||
For presumably obvious reasons, such scales must be hexatonic, and for less obvious reasons, they must also be achiral. With only twenty achiral hexatonic scales, it’s little surprise that so few self-complementing scales exist. We’ll explore why self-complements must be achiral once we’ve examined some other interesting commonalities between the self-complements:
- Four (F01, F47, F76, F80) subdivide the octave into sections of equal length (respectively, six notes, three notes, two notes, and one note), then place notes on every tone of the odd-numbered subdivisions and no tones of the even-numbered ones.
- Three of these (F47, F76, and F80) are also modes of limited transposition.
Three also have scale generators:
- F01 consists of five stacked minor seconds (or major sevenths). Generator step size: one semitone.
- F80 consists of five stacked major seconds (or minor sevenths). Generator step size: two semitones.
- F71 consists of five stacked perfect fourths (or perfect fifths). Generator step size: five semitones.
This means that every generator that produces a hexatonic scale also produces a self-complementing scale.
Scale F11 is the oddball here: the only property I’ve identified that makes it stand out from the non-self-complementary scales is that it moves a single note of two other self-complements by a tritone:
- It moves F01’s C♯ to G.
- It moves F71’s A to D♯.
This is the only such transformation I’ve found that produces an achiral scale (for instance, produces a chiral scale), although I haven’t exhausted all the possibilities yet.
As long as we’re returning to the subject of generators, let’s also observe that F71 truncates Ionian by a single note. We’ve already analyzed how the pentatonic scale both truncates and complements Ionian, so this should surprise no one. In fact, I hypothesize that, in any t-tone equal temperament, for a given note count of c:
- For every generator that produces both a scale of length c and a scale of length (t − c) in that tonality:
- For t = 2c:
- That scale will complement itself. (Examples: All three hexatonic scale generators in 12-TET.)
- For t ≠ 2c:
- The two scales will complement each other.
- Removing the generator’s outer |2c − t| notes from the longer scale will produce the shorter one.
- Examples:
- We remove C Ionian (Forte 7-35)’s F and B to get C major pentatonic (Forte 5-35).
- We remove C bebop Mixolydian (Forte 8-23)’s B♭, F, B, and E to get C Genus Primum (Forte 4-23). (The former adds B♭ to C Ionian; the latter subtracts A from C Scottish pentatonic).
- For t = 2c:
- Finally, if the generator has an interval size s (in units of 1/t octave) such that c, (t − c), s, and (t − s) are all coprime with t, then both the scale and its complement will have single-note mode transformations that progress through every mode on every root.
Since 12-TET contains a very obvious dearth of test cases, I’ll need to write better tools for analysis in other equal temperaments to ascertain whether I’m correct.
Now, why do self-complementing scales need to be achiral? Let’s take Rāga Syamalam as an example. (I’ll produce more detailed representations of this data soon™.)
- Notes: C, D, E♭, F♯, G, A♭
- Intervals: 1, ½, 1½, ½, ½, 2
- Binary representation: 101100111000101100111000
If we flip its bits, we get 010011000111. Arbitrarily, I’m going to rotate that five semitones to the left, producing Rāga Gangatarangini.
- Notes: C, E, F, G♭, A♭, B
- Intervals: 2, ½, ½, 1, ½, 1½
- Binary representation: 100011101001100011101001
Clearly, not the same scale. This is an inevitable result of XORing an asymmetrical sequence of ones and zeroes.
Perhaps confusingly, we haven’t wound up with Rāga Syamalam’s enantiomorph, although it is quite similar:
- Notes: C, E, F, G♭, A, B♭
- Intervals: 2, ½, ½, 1½, ½, 1
- Binary representation: 100011100110100011100110
Effectively, the 0011 and the 01 swap positions between the two scales. For what it’s worth, Forte considers these scales’ set classes to be “zygotic”, or twinned: their interval distribution is so similar that they share a large number of characteristics. Forte assigned Rāga Gangatarangini the number 6-Z17 and Rāga Syamalam (and its enantiomorph) the number 6-Z43, where Z stands for “zygotic”.
A crash course in Ancient Greek harmony
Etymology
Ionian and its modes are named for places in or near ancient Greece and/or ancient Greek tribes:
|
Αἱ ἐτῠμολογῐ́αι τῶν ἑπτᾰ́ τόνων (Hai etumologíai tô heptá tónōn) [The seven modes’ etymologies] |
||||
|---|---|---|---|---|
| # | Mode | Greek | Romanized | Reference |
| 1 | Ἰωνία | Iōnía | region on the western coast of Anatolia (modern Turkey) | |
| 2 | Δωρῐεύς | Dōrieús | one of the four major Hellenic tribes | |
| 3 | Φρῠγῐ́ᾱ | Phrugíā | kingdom in west-central Anatolia | |
| 4 | Λῡδῐ́ᾱ | Lūdíā | Anatolian kingdom most famously ruled by Croesus | |
| 5 | μιξο-Λῡ́δῐος | mixo-Lū́dios | literally “mixed Lydian” | |
| 6 | Αἰολῐ́ᾱ | Aiolíā | region of northwestern Anatolia | |
| 7 | Λοκρῐ́ς | Lokrís | ⟨en | |
However, they have little to do with the regions or tribes they were named after, or even the ancient Greek tonoi (singular: tonos) that in many cases shared their names; it was more a case of “medieval Europeans thought it’d be cool to name their modes after aspects of Ancient Greece.”
Tonos comes from the ancient Greek ὁ τόνος, ho tónos, meaning cord, chord, note, tone, or tension; its dual nominative form was τὼ τόνω, tṑ tónō, and its plural nominative form was οἱ τόνοι, hoi tónoi. As for the word diatonic, that comes from the Ancient Greek διατονικός (diatonikós), literally meaning two tones, in reference to the diatonic tetrachord, which… well, keep reading.
Ancient Greek Harmony: The Cliffs Notes
This is an oversimplification by necessity, as ancient Greek music theory wasn’t unified; (Φιλόλαος, Philólaos), (Ἀρχύτας), (Ἀριστόξενος ὁ Ταραντῖνος, Aristóxenos ho Tarantínos), (Πτολεμαῖος, Ptolemaios), and others had quite different conceptions of it. I will list modern sources in an acknowledgement section below.
I should first clarify that very little music of ancient Greece actually survives to this day; the earliest known piece that can reasonably be described as complete is ⟨en
I’ll be using numbers to represent the intervals of ancient Greek harmony within ⟨en
| Interval key | ||||||||
|---|---|---|---|---|---|---|---|---|
| # | Interval | Tone | Exact | Approximate | ||||
| ¼ | Infra second | Quarter-tone | ²⁴√2 | : | 1 | ≈ | 1.02930223664 | |
| ½ | Minor second | Semitone | ¹²√2 | : | 1 | ≈ | 1.05946309436 | |
| 1 | Major second | Whole tone | ⁶√2 | : | 1 | ≈ | 1.12246204831 | |
| 1½ | Minor third | Three semitones | ⁴√2 | : | 1 | ≈ | 1.18920711500 | |
| 2 | Major third | Two whole tones | ³√2 | : | 1 | ≈ | 1.25992104989 | |
Note that in scales with only whole-steps and half-steps, I’ll use H (i.e., Half) interchangeably with ½, and W (i.e., Whole) interchangeably with 1. In all other scales, I’ll only use the numbers.
24-TET is by necessity an oversimplification as well, not least since different authors defined the same interval differently. For instance, here are how Philolaus, Archytas, and 24-TET would define the intervals within a diatonic tetrachord. (A tetrachord is a major element of Greek harmony consisting of four notes.)
Also, I’ve taken the liberty of reversing what they called the first and third intervals – ancient Greek harmony defined scales in what we would consider descending order. But then, the ancient Greek metaphor for time literally inverted the modern one: they saw the past as receding away in front of us, continually getting ever more distant, and the future as creeping up from behind us. I actually find their metaphor far more apt than ours: who can actually see the future? And our memories of the past get more distant every day.
I don’t know if this metaphor affected how they described changes over time. I may be overthinking this, but if they thought of the past as in front of them, they may not have perceived this as a descent. I don’t have enough information to know if concrete proof exists one way or the other. Certainly, where the ancient Greeks refer to time, translators must be aware of their metaphor, and anyone who reads translated Greek writing that concerns time should take the differences into account (and even ask if the translator knew of them).
| Interval ratios of a diatonic tetrachord | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Source | Low interval | Middle interval | High interval | ||||||||||||
| Philolaus | 256 | : | 243 | ≈ | 1.05349794239 | 9 | : | 8 | = | 1.125 | 9 | : | 8 | = | 1.125 |
| Archytas | 28 | : | 27 | = | 1.0370370370… | 8 | : | 7 | = | 1.142857142857… | 9 | : | 8 | = | 1.125 |
| 24-TET | ¹²√2 | : | 1 | ≈ | 1.05946309436 | ⁶√2 | : | 1 | ≈ | 1.12246204831 | ⁶√2 | : | 1 | ≈ | 1.12246204831 |
Thus, representing ancient Greek harmony using 24-TET is a substantial oversimplification, and in point of fact, we’d get closer to Archytas’ definitions in 24-TET by using a quarter tone for the low interval and 1¼-tone (that is, the interval between a major second and a minor third, known as an infra third or ultra second) for the middle interval. However, the intervals I’ve selected (a minor second and two major seconds) are closer to Philolaus’ definitions and, crucially, can be represented in modern Western music’s 12-TET.
As one further example, both Philolaus and Archytas define a diatonic tetrachord’s high and low notes as having 4:3 (1.33333…) ratios, which 24-TET would represent using a perfect fourth (≈1.33483985417). Most people’s ears are insufficiently trained to tell the difference – even trained singers with perfect pitch frequently wander off-key by larger differences than that (at least when they’re not subjected to pitch alteration).
The same ancient Greek tonos could actually use any of three genera (singular: genus). Each tonos contained two tetrachords, four-note sets spanning a 4:3 ratio, or perfect fourth. The note spacing varied between genera, but all three used one interval once per tetrachord and another interval twice, in the following order:
| Interval Genera: A Feed from Cloud Mountain | |||
|---|---|---|---|
| Genus | Low interval | Middle interval | High interval |
| Enharmonic | Infra second | Infra second | Major third |
| Chromatic | Minor second | Minor second | Minor third |
| Diatonic | Minor second | Major second | Major second |
The tetrachords and synaphai’s positions, meanwhile, varied between tonoi, with notable consequences:
- In most tonoi, the synaphe occurred between the tetrachords.
- In Hypodorian and Mixolydian, respectively, it occurred before and after them.
- Also, Lydian, Phrygian, Hypolydian, and Hypophrygian split one tetrachord.
Since tetrachords spanned 4:3 ratios, synaphai were mathematically constrained to 9:8, or major seconds:
- 4² × 9 = 144
- 3² × 8 = 72
- 144 ÷ 72 = 2
Note that enharmonic has a different meaning in twelve-tone equal temperament than it has in ancient Greek harmony (or any tuning system that uses microtonality). In 12-TET, enharmonic means two tones have the same pitch. In ancient Greek harmony, it refers to an interval spacing smaller than a semitone and to the tuning system that used it. As stated, I’ll approximate this interval in 24-TET with quarter-tones.
Pythagoras with the looking glass: Comparing interval ratios
One point in twelve-tone equal temperament’s favor is how closely it approximates every foundational interval of Pythagorean tuning. In the chart headers below, P stands for Pythagorean, M for Modern (i.e., 12-TET), R for Ratio, and Q for Quotient. The final column shows 12-TET’s difference from Pythagorean tuning.
(The name “Pythagorean” is a misnomer – their eponymous tuning system is actually much older. The oldest known description of it appears on a Mesopotamian clay tablet from the 19th century BCE.)
| A comparison of Pythagorean tuning & twelve-tone equal temperament | ||||||
|---|---|---|---|---|---|---|
| Interval | PR | PQ | MR | MQ | MQ − PQ | |
| Octave | 2:1 | =2 | 2 | 2 | 0 | |
| Perfect fifth | 3:2 | =1.5 | 2⁷⁄₁₂ | ≈1.49830707688 | ≈−0.00169292312 | |
| Perfect fourth | 4:3 | =1.333333333… | 2⁵⁄₁₂ | ≈1.33483985417 | ≈0.00150652083 | |
| Major third | 81:64 | =1.265625 | 2¹⁄₃ | ≈1.25992104989 | ≈−0.00570395011 | |
| Diminished fourth | 8192:6561 | ≈1.24859015394 | 2¹⁄₃ | ≈1.25992104989 | ≈0.01133089595 | |
| Minor third | 32:27 | =1.185185185… | 2¹⁄₄ | ≈1.18920711500 | ≈0.00402192982 | |
| Major second | 9:8 | =1.125 | 2¹⁄₆ | ≈1.12246204831 | ≈−0.00253795169 | |
| Minor second | 256:243 | ≈1.05349794239 | 2¹⁄₁₂ | ≈1.05946309436 | ≈0.00596515197 | |
So 12-TET is within about 0.0015 of the Pythagorean ratios for perfect fourths and fifths; its major second is off by about 0.0025; its minor third is off by about 0.004; and its major third and minor second are off by about 0.006. Only an incredibly well-trained ear could discern any of these differences. (The difference becomes vastly more noticeable when we stack the interval several times; 12-TET ultimately became standard because it avoids dissonances that are inherent to Pythagorean tuning, notably its so-called wolf intervals).
The only interval where 12-TET is off by more than 0.01 is the diminished fourth – which doesn’t really exist in 12-TET. It’s the result of subtracting two Pythagorean minor thirds from an octave. Since Pythagorean note spacing wasn’t even, an octave minus two minor thirds produced a different interval than a major third. In 12-TET, it’s just a major third. Unlike many diminished intervals, the Pythagorean diminished fourth is quite consonant – in fact it’s actually closer to a 5:4 ratio than the major third is, so this may not be surprising. In any case, both the modern and Pythagorean major third and minor third closely approximate 5:4 (1.25) and 6:5 (1.2) ratios, which may be part of why our ears find them so harmonically pleasing. (In just intonation, a minor third would use a 6:5 ratio and a major third would use a 5:4 ratio; however, Pythagorean tuning does not use any numbers that are not exact powers of three or exact powers of two.)
Incidentally, all Pythagorean interval ratios are based on powers of two and three, and Pythagorean intervals smaller than perfect fifths can be derived through a sequence of ratio division. To wit:
| Pythagorean interval division (or is it subtraction?) | ||||||||
| Dividend | Divisor | Quotient | ||||||
|---|---|---|---|---|---|---|---|---|
| – | – | Octave | 2¹:3⁰ | 2:1 | ||||
| – | – | Perfect fifth | 3¹:2¹ | 3:2 | ||||
| Octave | 2¹:3⁰ | 2:1 | Perfect fifth | 3¹:2¹ | 3:2 | Perfect fourth | 2²:3¹ | 4:3 |
| Perfect fifth | 3¹:2¹ | 3:2 | Perfect fourth | 2²:3¹ | 4:3 | Major second | 3²:2³ | 9:8 |
| Perfect fourth | 2²:3¹ | 4:3 | Major second | 3²:2³ | 9:8 | Minor third | 2⁵:3³ | 32:27 |
| Minor third | 2⁵:3³ | 32:27 | Major second | 3²:2³ | 9:8 | Minor second | 2⁸:3⁵ | 256:243 |
| Minor third | 2⁵:3³ | 32:27 | Minor second | 2⁸:3⁵ | 256:243 | Diminished fourth | 2¹³:3⁸ | 8192:6561 |
| Perfect fourth | 2²:3¹ | 4:3 | Minor second | 2⁸:3⁵ | 256:243 | Major third | 3⁴:2⁶ | 81:64 |
Note that dividing pitch ratios effectively equates to subtracting intervals. Since pitch is a binary logarthmic scale, raising a note an octave doubles its pitch. In mathematical terms, log₂ (ᵅ⁄ᵦ) equals the number of octaves separating the notes with pitches α and β, which will be fractional if they’re not separated by an exact multiple of an octave, zero if α = β, and negative if β > α. As a result, dividing 2:1 by 3:2 gives us a 4:3 ratio, but it subtracts a perfect fifth from an octave, giving us a perfect fourth. Try not to think about it too hard and you might not get a headache.
One interesting footnote to the above table: Remember the Fibonacci sequence? Start with 0 and 1, and repeatedly add the previous two numbers together. Thus, the sequence’s next numbers are:
- 0 + 1 = 1
- 1 + 1 = 2
- 1 + 2 = 3
- 2 + 3 = 5
- 5 + 3 = 8
- 8 + 5 = 13
- 13 + 8 = 21
- 21 + 13 = 34
- 34 + 21 = 55
Now, look again at the ratios above. 0, 1, 2, 3, 5, 8, and 13 – the sequence’s first eight numbers – appear both as exponents and, in 2 and 3’s cases, as bases. The major third is, in fact, the only interval in the above table that doesn’t use Fibonacci numbers as exponents. The Fibonacci spiral truly is everywhere – although the Indian mathematician Acharya Pingala (Sanskrit: आचार्य पिङ्गल) is the first writer known to have explicitly described the sequence (ca. 200 BCE), there it is in Pythagorean tuning. (In fact, it’s even older than that: the system dates back to the Mesopotamians.)
It may seem rather haphazard as to whether the ratio starts with a power of 3 or a power of 2; it may also seem rather haphazard as to whether we’re dividing the most recent interval by the second-most recent interval or vice versa. In both these cases, the answer simply depends on which is larger: the larger number appears first in the ratio, and the larger ratio serves as the dividend. Either way, the power of 2 is always one Fibonacci number ahead of the power of 3. Another way to think of the sequence is as follows:
| The Fibonacci sequence in Pythagorean tuning | ||||
|---|---|---|---|---|
| Ratio | Interval jump | Example | ||
| 2¹:3⁰♭ | ♭an octave higher♭ | ♭C3 | ♭to♭ | C4♭ |
| 2¹:3¹♭ | ♭a perfect fifth lower♭ | ♭C4 | ♭to♭ | F3♭ |
| 2²:3¹♭ | ♭a perfect fourth higher♭ | ♭F3 | ♭to♭ | B♭3 |
| 2³:3²♭ | ♭a major second lower♭ | B♭3 | ♭to♭ | A♭3 |
| 2⁵:3³♭ | ♭a minor third higher♭ | A♭3 | ♭to♭ | B3♭ |
| 2⁸:3⁵♭ | ♭a minor second higher♭ | ♭B3 | ♭to♭ | C4♭ |
| 2¹³:3⁸♭ | ♭a diminshed fourth higher♭ | ♭C4 | ♭to♭ | E4♭ |
If the sequence continued, the next interval would be 2²¹:3¹³, or 2,097,152:1,594,323 (≈1.31538715806). As far as I know, this was not a Pythagorean interval, which probably won’t surprise anyone. Nor is it likely to surprise anyone that Fibonacci had his own tuning system, which I’ll undoubtedly write about soon™.
Ancient Greek tonoi & modern modes
- As stated above, ancient Greek harmony defined tonoi in descending order; I’ve listed them in our more familiar ascending order to keep them consistent with the other scales on this page.
- I’ve printed intervals rather than notes because the notation for microtones is extremely confusing if you’re not already familiar with it. Hopefully, the interval notation will be slightly easier to understand for the enharmonic tonoi. Below this table, I’ll present examples of the chromatic tonoi based on C.
- A thicker font denotes a larger interval size: 2, 1½, 1, ½, ¼.
- I’ve placed borders between tetrachords (again, four tonoi split one of their tetrachords) and around the extra whole step (whose background I’ve also highlighted).
- I’ve printed the modern modes with blue text and brighter borders, with a thicker border below since the next tonos will not be related.
- I’ve printed the tonoi in the order ancient Greek harmony assigned them: Mixolydian (modern Locrian’s equivalent) first, Hypodorian (modern Aeolian’s equivalent) last.
- Our Dorian, Phrygian, Lydian, and Mixolydian modes don’t match their eponymous diatonic tonoi, apparently due to medieval Europeans’ misconception that the ancient Greeks defined tetrachords’ intervals in ascending order. (Swiss poet Heinrich Glarean appears to have popularized the error.) I’ll explain this further below.
Here are all three genera of all seven tonoi, followed by the diatonic genus’ modern equivalent:
| Approximate intervals of Ancient Greek tonoi & modern diatonic modes | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Tonos | Genus | 1–2 | 2–3 | 3–4 | 4–5 | 5–6 | 6–7 | 7–8 | |
| Mixolydian | Enharmonic | ¼ | ¼ | 2 | ¼ | ¼ | 2 | 1 | |
| Chromatic | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | ||
| Diatonic | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| Modern | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
| Lydian | Enharmonic | ¼ | 2 | ¼ | ¼ | 2 | 1 | ¼ | |
| Chromatic | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | ||
| Diatonic | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| Modern | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| Phrygian | Enharmonic | 2 | ¼ | ¼ | 2 | 1 | ¼ | ¼ | |
| Chromatic | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | ||
| Diatonic | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| Modern | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| Dorian | Enharmonic | ¼ | ¼ | 2 | 1 | ¼ | ¼ | 2 | |
| Chromatic | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | ||
| Diatonic | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| Modern | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| Hypolydian | Enharmonic | ¼ | 2 | 1 | ¼ | ¼ | 2 | ¼ | |
| Chromatic | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | ||
| Diatonic | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| Modern | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| Hypophrygian | Enharmonic | 2 | 1 | ¼ | ¼ | 2 | ¼ | ¼ | |
| Chromatic | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | ||
| Diatonic | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| Modern | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| Hypodorian | Enharmonic | 1 | ¼ | ¼ | 2 | ¼ | ¼ | 2 | |
| Chromatic | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | ||
| Diatonic | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| Modern | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
The above table is quite abstract, so to follow it up, here are the actual scales. There is no particularly well-established standard for 24-TET notation. I’ve chosen to use ʌ to mean “raise this note by a quarter tone” and v to mean “lower this note by a quarter tone.” As in the previous table, I’ve separated the tetrachords in the interval listing to make it clear where they occur, and I’ve highlighted the synaphe (a bit more so, even, because it will become a bit less legible shortly).
| Greek enharmonic tonoi (C roots, linear order) | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Enharmonic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| Mixolydian | C♮ | Cʌ | C♯ | F♮ | Fʌ | F♯ | B♭ | ¼ | ¼ | 2 | ¼ | ¼ | 2 | 1 |
| Lydian | C♮ | Cʌ | Fv | F♮ | Fʌ | Aʌ | Cv | ¼ | 2 | ¼ | ¼ | 2 | 1 | ¼ |
| Phrygian | C♮ | E♮ | Fv | F♮ | A♮ | B♮ | Cv | 2 | ¼ | ¼ | 2 | 1 | ¼ | ¼ |
| Dorian | C♮ | Cʌ | C♯ | F♮ | G♮ | Gʌ | G♯ | ¼ | ¼ | 2 | 1 | ¼ | ¼ | 2 |
| Hypolydian | C♮ | Cʌ | Fv | Gv | G♮ | Gʌ | Cv | ¼ | 2 | 1 | ¼ | ¼ | 2 | ¼ |
| Hypophrygian | C♮ | E♮ | F♯ | Gv | G♮ | B♮ | Cv | 2 | 1 | ¼ | ¼ | 2 | ¼ | ¼ |
| Hypodorian | C♮ | D♮ | Dʌ | D♯ | G♮ | Gʌ | G♯ | 1 | ¼ | ¼ | 2 | ¼ | ¼ | 2 |
Here are the chromatic tonoi rooted in C and, for the sake of representing what medieval Europeans might have thought they were, their inversions.
| Greek chromatic tonoi & their inversions (C roots, linear order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B♭ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | ||
| C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | ||
| C♮ | D♯ | E♮ | F♮ | G♯ | A♯ | B♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | ||
| C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B𝄫 | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | ||
| C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | ||
| C♮ | D♯ | E♯ | F♯ | G♮ | A♯ | B♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | ||
| C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B𝄫 | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | ||
| C♮ | D♮ | E♯ | F♯ | G♮ | A♯ | B♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | ||
| C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♮ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | ||
| C♮ | D♭ | E𝄫 | F♭ | G♮ | A♭ | B𝄫 | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | ||
| C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♮ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | ||
| C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B𝄫 | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | ||
| C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♭ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 | ||
Remember how I said above that Chromatic Dorian was directly relevant to this section? Well, there you go.
Scale-based transposition now. My base scales are Chromatic Lydian and Chromatic Hypophrygian inverse; this is an admittedly arbitrary choice that I made purely because they use the fewest accidentals on C. This also creates a neat pattern in the table below:
| Greek chromatic tonoi & their inversions (mode-based roots, linear order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| B♮ | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | ||
| C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | ||
| D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | C♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | ||
| E♮ | F♮ | G♭ | A♮ | B♮ | C♮ | D♭ | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | ||
| F♮ | G♭ | A♮ | B♮ | C♮ | D♭ | E♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | ||
| G♭ | A♮ | B♮ | C♮ | D♭ | E♮ | F♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | ||
| A♮ | B♮ | C♮ | D♭ | E♮ | F♮ | G♭ | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | ||
| F♮ | G♮ | A♯ | B♮ | C♮ | D♯ | E♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | ||
| E♮ | F♮ | G♮ | A♯ | B♮ | C♮ | D♯ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | ||
| D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | C♮ | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | ||
| C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | ||
| B♮ | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | ||
| A♯ | B♮ | C♮ | D♯ | E♮ | F♮ | G♮ | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | ||
| G♮ | A♯ | B♮ | C♮ | D♯ | E♮ | F♮ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 | ||
The ancient Greek tonoi’s “circle of fifths” order is:
- Hypolydian
- Lydian
- Hypophrygian
- Phrygian
- Hypodorian
- Dorian
- Mixolydian
This may help explain how the Greeks got the names Hypolydian, Hypophrygian, and Hypodorian in the first place: ὑπό (hupó) is literally Ancient Greek for under, and remember, the ancient Greeks’ scales went in what we consider descending order.
I’m reversing the inverted scales’ order in the next table, since as its predecessor clearly demonstrates, they’re actually moving in the opposite direction from their namesakes. Also, I’m reintroducing Major Phrygian and its modes here, since they’re the midway point between the chromatic scales and their inversions.
| Greek chromatic tonoi & their variants (mode-based roots, “circle of fifths” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| F♮ | G♭ | A♮ | B♮ | C♮ | D♭ | E♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | ||
| G♭ | A♮ | B♮ | C♮ | D♭ | E♮ | F♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | ||
| D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | C♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | ||
| A♮ | B♮ | C♮ | D♭ | E♮ | F♮ | G♭ | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | ||
| E♮ | F♮ | G♭ | A♮ | B♮ | C♮ | D♭ | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | ||
| B♮ | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | ||
| E♮ | F♮ | G♮ | A♭ | B♮ | C♮ | D♭ | ½ | 1 | ½ | 1½ | ½ | ½ | 1½ | ||
| B♮ | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | ½ | ½ | 1½ | ½ | 1 | ½ | 1½ | ||
| F♮ | G♮ | A♭ | B♮ | C♮ | D♭ | E♮ | 1 | ½ | 1½ | ½ | ½ | 1½ | ½ | ||
| C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | ||
| G♮ | A♭ | B♮ | C♮ | D♭ | E♮ | F♮ | ½ | 1½ | ½ | ½ | 1½ | ½ | 1 | ||
| D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | C♮ | 1½ | ½ | 1 | ½ | 1½ | ½ | ½ | ||
| A♭ | B♮ | C♮ | D♭ | E♮ | F♮ | G♮ | 1½ | ½ | ½ | 1½ | ½ | 1 | ½ | ||
| F♮ | G♮ | A♯ | B♮ | C♮ | D♯ | E♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | ||
| C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | ||
| G♮ | A♯ | B♮ | C♮ | D♯ | E♮ | F♮ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 | ||
| D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | C♮ | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | ||
| A♯ | B♮ | C♮ | D♯ | E♮ | F♮ | G♮ | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | ||
| E♮ | F♮ | G♮ | A♯ | B♮ | C♮ | D♯ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | ||
| B♮ | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | ||
So effectively, Lydian and Hypolydian keep one tetrachord in the same place; the other tetrachord just swaps places with the extra whole-step. This actually continues to be true throughout the rest of the chromatic scales. Effectively, the extra whole-step either moves three places forward or four places back. Dividing the Ionian scale in this way shows us the same thing occurring with it:
| Greek diatonic tonoi (C roots, circle of fifths order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Modern | Ancient | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | ||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | ||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | ||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | ||
Every scale in this table lowers its predecessor’s extra whole-step to a half-step – and because the extra whole-step is always followed by a half-step, that half-step subsequently becomes a whole-step. This is, in fact, exactly the source of most of the patterns we’ve observed throughout our analysis of the Ionian scale. I don’t know how much the ancient Greeks mapped this out and how much of it simply stemmed from intuition, but if it was by design, the designer was a genius, and I’m sad that their name has been lost to history.
Chromatic tonos analysis: The circle of fifths
There’s no obvious equivalent of the circle of fifths progression for the chromatic genus, though; for reasons explained above, that’s a special property of the diatonic genus’ mathematical regularity. Rotating most scales requires making more changes to their intervals. Let’s see the scales on C again, this time with the tetrachord placement standardized around Major Phrygian’s layout (since it centers the synaphe within the middle row).
I’ve numbered the scales so I can more clearly explain patterns. The Greek letters refer to scales: α denotes chromatic tonoi, β denotes a mode of Major Phrygian, and γ denotes inverse chromatic tonoi. The number refers to Greek mode order (Mixolydian first, Hypodorian last). I used the same synaphe positions for all three, so the inverse chromatic scales reverse the order (Hypodorian first, Mixolydian last). I’ll retain these numbers throughout my analysis of these scales.
| Greek chromatic tonoi & their variants (C roots, linear order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| α.1 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B♭ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | |
| α.2 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | |
| α.3 | C♮ | D♯ | E♮ | F♮ | G♯ | A♯ | B♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | |
| α.4 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B𝄫 | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | |
| α.5 | C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | |
| α.6 | C♮ | D♯ | E♯ | F♯ | G♮ | A♯ | B♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | |
| α.7 | C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B𝄫 | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | |
| β.1 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♭ | ½ | 1½ | ½ | ½ | 1½ | ½ | 1 | |
| β.2 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♮ | 1½ | ½ | ½ | 1½ | ½ | 1 | ½ | |
| β.3 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B𝄫 | ½ | ½ | 1½ | ½ | 1 | ½ | 1½ | |
| β.4 | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | |
| β.5 | C♮ | D♯ | E♮ | F♯ | G♮ | A♯ | B♮ | 1½ | ½ | 1 | ½ | 1½ | ½ | ½ | |
| β.6 | C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B𝄫 | ½ | 1 | ½ | 1½ | ½ | ½ | 1½ | |
| β.7 | C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♮ | 1 | ½ | 1½ | ½ | ½ | 1½ | ½ | |
| γ.1 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♭ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 | |
| γ.2 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B𝄫 | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | |
| γ.3 | C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♮ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | |
| γ.4 | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | |
| γ.5 | C♮ | D♭ | E𝄫 | F♭ | G♮ | A♭ | B𝄫 | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | |
| γ.6 | C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♮ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | |
| γ.7 | C♮ | D♮ | E♯ | F♯ | G♮ | A♯ | B♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | |
Their “circle of fifths” order looks like this:
| Greek chromatic tonoi & their variants (C roots, “circle of fifths” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| α.6 | C♮ | D♯ | E♯ | F♯ | G♮ | A♯ | B♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | |
| α.3 | C♮ | D♯ | E♮ | F♮ | G♯ | A♯ | B♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | |
| α.7 | C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B𝄫 | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | |
| α.4 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B𝄫 | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | |
| α.1 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B♭ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | |
| α.5 | C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | |
| α.2 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | |
| β.6 | C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B𝄫 | ½ | 1 | ½ | 1½ | ½ | ½ | 1½ | |
| β.3 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B𝄫 | ½ | ½ | 1½ | ½ | 1 | ½ | 1½ | |
| β.7 | C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♮ | 1 | ½ | 1½ | ½ | ½ | 1½ | ½ | |
| β.4 | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | |
| β.1 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♭ | ½ | 1½ | ½ | ½ | 1½ | ½ | 1 | |
| β.5 | C♮ | D♯ | E♮ | F♯ | G♮ | A♯ | B♮ | 1½ | ½ | 1 | ½ | 1½ | ½ | ½ | |
| β.2 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♮ | 1½ | ½ | ½ | 1½ | ½ | 1 | ½ | |
| γ.6 | C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♮ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | |
| γ.3 | C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♮ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | |
| γ.7 | C♮ | D♮ | E♯ | F♯ | G♮ | A♯ | B♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | |
| γ.4 | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | |
| γ.1 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♭ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 | |
| γ.5 | C♮ | D♭ | E𝄫 | F♭ | G♮ | A♭ | B𝄫 | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | |
| γ.2 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B𝄫 | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | |
It fascinates me how these three sets of modes complement each other. Note the latter chart’s accidental distributions: the inversions have sharps bunched in the middle, the chromatic scales have flats bunched in the middle, and Major Phrygian has flats bunched above it and sharps below it. Of course, the interval distribution explains why that might have happened:
- The following scales open with minor thirds:
- The first two chromatic scales
- Major Phrygian’s two lowest modes
- The fourth and fifth inverted chromatic scales
- The following scales inevitably (we are discussing inversions and reflectively symmetrical scales) close with minor thirds:
- Major Phrygian’s two highest modes
- The last two chromatic inversions
- The third and fourth chromatic scales
In short, the accidental distribution results directly from the interval distribution. Comparing the same position across sets (i.e., α.1, β.1, γ.1, then α.2, β.2, γ.2, then α.3, β.3, γ.3, etc.) may clarify this:
| Greek chromatic tonoi & their variants (C roots, “aligned synaphai” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| α.1 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B♭ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | |
| β.1 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♭ | ½ | 1½ | ½ | ½ | 1½ | ½ | 1 | |
| γ.1 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♭ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 | |
| α.2 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | |
| β.2 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♮ | 1½ | ½ | ½ | 1½ | ½ | 1 | ½ | |
| γ.2 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B𝄫 | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | |
| α.3 | C♮ | D♯ | E♮ | F♮ | G♯ | A♯ | B♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | |
| β.3 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B𝄫 | ½ | ½ | 1½ | ½ | 1 | ½ | 1½ | |
| γ.3 | C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♮ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | |
| α.4 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B𝄫 | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | |
| β.4 | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | |
| γ.4 | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | |
| α.5 | C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | |
| β.5 | C♮ | D♯ | E♮ | F♯ | G♮ | A♯ | B♮ | 1½ | ½ | 1 | ½ | 1½ | ½ | ½ | |
| γ.5 | C♮ | D♭ | E𝄫 | F♭ | G♮ | A♭ | B𝄫 | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | |
| α.6 | C♮ | D♯ | E♯ | F♯ | G♮ | A♯ | B♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | |
| β.6 | C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B𝄫 | ½ | 1 | ½ | 1½ | ½ | ½ | 1½ | |
| γ.6 | C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♮ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | |
| α.7 | C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B𝄫 | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | |
| β.7 | C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♮ | 1 | ½ | 1½ | ½ | ½ | 1½ | ½ | |
| γ.7 | C♮ | D♮ | E♯ | F♯ | G♮ | A♯ | B♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | |
Chromatic tonos analysis: Tetrachord swap
So let’s run an experiment. Let’s allow ourselves to swap scales with the same Greek letters, while keeping the Arabic numerals the same. That is to say, without moving any tetrachords, let’s swap scales between sets. The first set of scales will front-load the largest intervals, and the third set will back-load the largest intervals; balanced scales will go into the second set. Here’s the result:
| Greek chromatic tonoi & their variants (C roots, “cyclical tetrachord swap” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| α.6 | C♮ | D♯ | E♯ | F♯ | G♮ | A♯ | B♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | |
| α.3 | C♮ | D♯ | E♮ | F♮ | G♯ | A♯ | B♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | |
| γ.7 | C♮ | D♮ | E♯ | F♯ | G♮ | A♯ | B♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | |
| γ.4 | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | |
| γ.1 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♭ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 | |
| β.5 | C♮ | D♯ | E♮ | F♯ | G♮ | A♯ | B♮ | 1½ | ½ | 1 | ½ | 1½ | ½ | ½ | |
| β.2 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♮ | 1½ | ½ | ½ | 1½ | ½ | 1 | ½ | |
| γ.6 | C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♮ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | |
| γ.3 | C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♮ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | |
| β.7 | C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♮ | 1 | ½ | 1½ | ½ | ½ | 1½ | ½ | |
| β.4 | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | |
| β.1 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♭ | ½ | 1½ | ½ | ½ | 1½ | ½ | 1 | |
| α.5 | C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | |
| α.2 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | |
| β.6 | C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B𝄫 | ½ | 1 | ½ | 1½ | ½ | ½ | 1½ | |
| β.3 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B𝄫 | ½ | ½ | 1½ | ½ | 1 | ½ | 1½ | |
| α.7 | C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B𝄫 | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | |
| α.4 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B𝄫 | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | |
| α.1 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B♭ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | |
| γ.5 | C♮ | D♭ | E𝄫 | F♭ | G♮ | A♭ | B𝄫 | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | |
| γ.2 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B𝄫 | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | |
Neat. Now what if we put the scales back into something resembling linear order?
| Greek chromatic tonoi & their variants (C roots, “linear tetrachord swap” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| γ.1 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♭ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 | |
| β.2 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♮ | 1½ | ½ | ½ | 1½ | ½ | 1 | ½ | |
| α.3 | C♮ | D♯ | E♮ | F♮ | G♯ | A♯ | B♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | |
| γ.4 | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | |
| β.5 | C♮ | D♯ | E♮ | F♯ | G♮ | A♯ | B♮ | 1½ | ½ | 1 | ½ | 1½ | ½ | ½ | |
| α.6 | C♮ | D♯ | E♯ | F♯ | G♮ | A♯ | B♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | |
| γ.7 | C♮ | D♮ | E♯ | F♯ | G♮ | A♯ | B♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | |
| β.1 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♭ | ½ | 1½ | ½ | ½ | 1½ | ½ | 1 | |
| α.2 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | |
| γ.3 | C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♮ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | |
| β.4 | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | |
| α.5 | C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | |
| γ.6 | C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♮ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | |
| β.7 | C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♮ | 1 | ½ | 1½ | ½ | ½ | 1½ | ½ | |
| α.1 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B♭ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | |
| γ.2 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B𝄫 | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | |
| β.3 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B𝄫 | ½ | ½ | 1½ | ½ | 1 | ½ | 1½ | |
| α.4 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B𝄫 | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | |
| γ.5 | C♮ | D♭ | E𝄫 | F♭ | G♮ | A♭ | B𝄫 | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | |
| β.6 | C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B𝄫 | ½ | 1 | ½ | 1½ | ½ | ½ | 1½ | |
| α.7 | C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B𝄫 | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | |
The above table clarifies why the group distribution breaks down as it does: we’ve swapped scales in such a way that across all three scale sets, we cycle between γ, β, and α, in order, seven times in a row. This sorts scales 6 and 3 to one group; scales 7, 4, and 1 to another; and scales 5 and 2 to a third.
Grouping like numbers in the above table together produces similar results to our first “aligned synaphai” table, but as one might expect, we’ve consistently front-loaded large intervals in the first scale of each trio and back-loaded them in the third. The minor thirds also move more consistently, always moving two positions earlier when they cross synaphai and one when they don’t.
(Aside: English needs an equivalent of quadrant, quintant, sextant, septant, octant, etc. for the number three.)
| Greek chromatic tonoi & their variants (C roots, “aligned tetrachord swap” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| α.1 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B♭ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | |
| β.1 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♭ | ½ | 1½ | ½ | ½ | 1½ | ½ | 1 | |
| γ.1 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♭ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 | |
| γ.2 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B𝄫 | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | |
| α.2 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | |
| β.2 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♮ | 1½ | ½ | ½ | 1½ | ½ | 1 | ½ | |
| β.3 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B𝄫 | ½ | ½ | 1½ | ½ | 1 | ½ | 1½ | |
| γ.3 | C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♮ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | |
| α.3 | C♮ | D♯ | E♮ | F♮ | G♯ | A♯ | B♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | |
| α.4 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B𝄫 | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | |
| β.4 | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | |
| γ.4 | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | |
| γ.5 | C♮ | D♭ | E𝄫 | F♭ | G♮ | A♭ | B𝄫 | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | |
| α.5 | C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | |
| β.5 | C♮ | D♯ | E♮ | F♯ | G♮ | A♯ | B♮ | 1½ | ½ | 1 | ½ | 1½ | ½ | ½ | |
| β.6 | C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B𝄫 | ½ | 1 | ½ | 1½ | ½ | ½ | 1½ | |
| γ.6 | C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♮ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | |
| α.6 | C♮ | D♯ | E♯ | F♯ | G♮ | A♯ | B♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | |
| α.7 | C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B𝄫 | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | |
| β.7 | C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♮ | 1 | ½ | 1½ | ½ | ½ | 1½ | ½ | |
| γ.7 | C♮ | D♮ | E♯ | F♯ | G♮ | A♯ | B♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | |
So, as we can see, each interval’s relative position has a massive impact on a scale’s accidental distribution. This makes intuitive sense, but it still might be hard to understand how much it underpins a scale’s entire composition without seeing it laid out like this.
I only noticed after numbering the scales that each trio starts on the same letter that closed out the previous trio, then cycles through the others in ascending order (resetting to α after γ). That actually explains a lot.
Chromatic tonos analysis: Minor third position
Next set of comparisons: by shifting Hypophrygian and Phrygian to the end of the chromatic tonoi, and Phrygian inverse and Hypophrygian inverse to the start of the inverse chromatic tonoi, we align the minor thirds instead of the synaphai. This resembles the first set of scales while increasing the similarity of the scale ordering between sets. I also swapped the inverse chromatic and chromatic scales’ positions from the “circle of fifths” order comparisons; I’ll explain why below.
| Greek chromatic tonoi & their variants (C roots, “cyclical aligned minor thirds” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| γ.5 | C♮ | D♭ | E𝄫 | F♭ | G♮ | A♭ | B𝄫 | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | |
| γ.2 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B𝄫 | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | |
| γ.6 | C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♮ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | |
| γ.3 | C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♮ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | |
| γ.7 | C♮ | D♮ | E♯ | F♯ | G♮ | A♯ | B♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | |
| γ.4 | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | |
| γ.1 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♭ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 | |
| β.6 | C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B𝄫 | ½ | 1 | ½ | 1½ | ½ | ½ | 1½ | |
| β.3 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B𝄫 | ½ | ½ | 1½ | ½ | 1 | ½ | 1½ | |
| β.7 | C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♮ | 1 | ½ | 1½ | ½ | ½ | 1½ | ½ | |
| β.4 | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | |
| β.1 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♭ | ½ | 1½ | ½ | ½ | 1½ | ½ | 1 | |
| β.5 | C♮ | D♯ | E♮ | F♯ | G♮ | A♯ | B♮ | 1½ | ½ | 1 | ½ | 1½ | ½ | ½ | |
| β.2 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♮ | 1½ | ½ | ½ | 1½ | ½ | 1 | ½ | |
| α.7 | C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B𝄫 | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | |
| α.4 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B𝄫 | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | |
| α.1 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B♭ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | |
| α.5 | C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | |
| α.2 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | |
| α.6 | C♮ | D♯ | E♯ | F♯ | G♮ | A♯ | B♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | |
| α.3 | C♮ | D♯ | E♮ | F♮ | G♯ | A♯ | B♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | |
Between the three sets of scales, this set of comparisons simply moves each synaphe one position to the left (e.g., Phrygian inverse to Ultra-Phrygian, Ultra-Phrygian to Hypodorian). This becomes especially clear when we rearrange them into linear order:
| Greek chromatic tonoi & their variants (C roots, “linear aligned minor thirds” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| γ.7 | C♮ | D♮ | E♯ | F♯ | G♮ | A♯ | B♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | |
| γ.1 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♭ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 | |
| γ.2 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B𝄫 | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | |
| γ.3 | C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♮ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | |
| γ.4 | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | |
| γ.5 | C♮ | D♭ | E𝄫 | F♭ | G♮ | A♭ | B𝄫 | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | |
| γ.6 | C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♮ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | |
| β.1 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♭ | ½ | 1½ | ½ | ½ | 1½ | ½ | 1 | |
| β.2 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♮ | 1½ | ½ | ½ | 1½ | ½ | 1 | ½ | |
| β.3 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B𝄫 | ½ | ½ | 1½ | ½ | 1 | ½ | 1½ | |
| β.4 | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | |
| β.5 | C♮ | D♯ | E♮ | F♯ | G♮ | A♯ | B♮ | 1½ | ½ | 1 | ½ | 1½ | ½ | ½ | |
| β.6 | C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B𝄫 | ½ | 1 | ½ | 1½ | ½ | ½ | 1½ | |
| β.7 | C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♮ | 1 | ½ | 1½ | ½ | ½ | 1½ | ½ | |
| α.2 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | |
| α.3 | C♮ | D♯ | E♮ | F♮ | G♯ | A♯ | B♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | |
| α.4 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B𝄫 | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | |
| α.5 | C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | |
| α.6 | C♮ | D♯ | E♯ | F♯ | G♮ | A♯ | B♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | |
| α.7 | C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B𝄫 | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | |
| α.1 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B♭ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | |
Now let’s compare across sets with the minor thirds aligned. This is why I swapped the chromatic scales and their inversions: if I hadn’t, we’d be moving the synaphai right rather than left.
| Greek chromatic tonoi & their variants (C roots, “doubly aligned minor thirds” order) | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
| γ.7 | C♮ | D♮ | E♯ | F♯ | G♮ | A♯ | B♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | |
| β.1 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♭ | ½ | 1½ | ½ | ½ | 1½ | ½ | 1 | |
| α.2 | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | |
| γ.1 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♭ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 | |
| β.2 | C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♮ | 1½ | ½ | ½ | 1½ | ½ | 1 | ½ | |
| α.3 | C♮ | D♯ | E♮ | F♮ | G♯ | A♯ | B♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | |
| γ.2 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B𝄫 | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | |
| β.3 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A♭ | B𝄫 | ½ | ½ | 1½ | ½ | 1 | ½ | 1½ | |
| α.4 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B𝄫 | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | |
| γ.3 | C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♮ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | |
| β.4 | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | |
| α.5 | C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | |
| γ.4 | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | |
| β.5 | C♮ | D♯ | E♮ | F♯ | G♮ | A♯ | B♮ | 1½ | ½ | 1 | ½ | 1½ | ½ | ½ | |
| α.6 | C♮ | D♯ | E♯ | F♯ | G♮ | A♯ | B♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | |
| γ.5 | C♮ | D♭ | E𝄫 | F♭ | G♮ | A♭ | B𝄫 | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | |
| β.6 | C♮ | D♭ | E♭ | F♭ | G♮ | A♭ | B𝄫 | ½ | 1 | ½ | 1½ | ½ | ½ | 1½ | |
| α.7 | C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B𝄫 | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | |
| γ.6 | C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♮ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | |
| β.7 | C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♮ | 1 | ½ | 1½ | ½ | ½ | 1½ | ½ | |
| α.1 | C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B♭ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | |
Now, we’ve effectively just reversed the first “aligned synaphai” table’s pattern. That table moved the minor third one degree left twice in a row, then moved it one degree right and the synaphe one degree left. This one moves the synaphe one degree left twice in a row, then moves it one degree right and the minor third one degree left. Another way to say this is that by rotating Mixolydian to the end of the chromatic tonoi and Mixolydian inverse to the front of the inverse chromatic tonoi, we’ve swapped the synaphai and tetrachords’ movement patterns.
One curiosity here is how much Mixolydian and Mixolydian inverse differ from the two scales immediately below and above them, respectively. Most other scale trios remain fairly consistent in note composition; these two are the exceptions. In fact, Mixolydian really behaves more like the third and sixth trios, and Mixolydian inverse really behaves more like the second and fifth. I understand why, but it’s still slightly surreal to see it laid out like this. I can think of at least three explanations that clarify why this occurs:
- Mixolydian inverse is one of this table’s only two scales that start with a consecutive minor third and major second, in either order, and Mixolydian is one of its only two scales that end with those intervals.
- We can just refer back to the “linear tetrachord swap” table and see that its first seven scales consist of this table’s second trio, fifth trio, and Mixolydian inverse; its last seven scales consist of Mixolydian followed by this table’s third trio and sixth trio; and its middle seven scales consist of this table’s second scale, third scale, middle three scales, third-from-last scale, and second-from-last scale;.
- We can just look at the interval distributions here. The first and last scale trios are the only ones where a major second moves from the front of the scale to the end, or vice versa. Thus, all the other scale trios have much more consistent note distributions.
Why our modes have historically inaccurate names
Our Ionian mode’s chromatic counterpart is actually Chromatic Hypolydian, and our Aeolian mode’s counterpart is Chromatic Hypodorian. Why is that? Well, as I remarked above, medieval Europeans were confused about some aspects of Greek harmony. There are actually multiple possible sources of this, and I’m not totally sure which one was at fault, but I’ll present a couple of ways a person could wind up with the scale names they got.
One possibility is that they erroneously thought the Greeks described their tetrachords in ascending order. Four of our modern modes also had multiple names, and they borrowed three of these from the Ancient Greek tonoi: their Hypodorian was our Aeolian, their Hypophrygian was our Locrian, and their Hypolydian was our Ionian.
| Medieval names for the Greek diatonic tonoi | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ancient | Medieval | Modern | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | |||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | |||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |||
So, what if we were to purchase fast food and disguise it as our own cooking reverse the order of the notes within each tetrachord?
| Inverting the Greek diatonic tetrachord | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Original | Reversed | Modern | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | |||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |||
| C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | |||
| C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |||
| C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |||
| C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |||
| C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |||
So, effectively, for Mixolydian, they put the tetrachords in the right parts of the scale, but they put the intervals within each tetrachord in the wrong order. Then, they put the remaining modes in the opposite of the Ancient Greeks’ order, likely assuming that they were rotating the scale in the opposite direction.
Remember, the ancient Greeks’ metaphor for time inverted the modern one: they thought of the past as being in front and the future as being behind them. I don’t know if this was the source of medieval Europeans’ confusion, but it wouldn’t entirely shock me if it were.
Now, remember Chekhov’s table near the start of this document? This is another potential source of their confusion. I’ll reprint a variant, this time depicting both the blueshifted (ascending) and redshifted (descending) scale transformations. In this case, we care about the redshift, since Greek harmony went in descending order.
| Mode transformations re-re-revisited | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Mode | Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |||
| 1 | ♮1 | |||||||||||
| 2 | ♭3 | ♭7 | ||||||||||
| 3 | ♭2 | ♭3 | ♭6 | ♭7 | ||||||||
| 4 | ♯4 | |||||||||||
| 5 | ♭7 | |||||||||||
| 6 | ♭3 | ♭6 | ♭7 | |||||||||
| 7 | ♭2 | ♭3 | ♭5 | ♭6 | ♭7 | |||||||
The renamed modes aside, each modern name is the Greek name for the mode of the note we redshift in circle of fifths order to get the first mode. That is:
| Modes and the notes they redshift | |||||
|---|---|---|---|---|---|
| # | Modern | Medieval | Greek | Redshift | |
| 1 | Ionian | Hypolydian | Lydian | 4 | Lydian |
| 2 | Dorian | Dorian | Phrygian | 3 | Phrygian |
| 3 | Phrygian | Phrygian | Dorian | 2 | Dorian |
| 4 | Lydian | Lydian | Hypolydian | 1 | Ionian |
| 5 | Mixolydian | Mixolydian | Hypophrygian | 7 | Locrian |
| 6 | Aeolian | Hypodorian | Hypodorian | 6 | Aeolian |
| 7 | Mixolydian | Locrian | Hypophrygian | 5 | Mixolydian |
Perhaps medieval scholars were confused on this point; perhaps it’s just coincidence.
Applied Greek harmony: Tetrachords in modern scales
We’ve already seen how the Ionian scale is a variant of a scale in which a tetrachord is repeated with a whole-tone separation (though in our Ionian mode, specifically, the second tetrachord is split midway through). To reiterate, let’s look at D Dorian, whose intervals are W-H-W-W-W-H-W:
- It opens with a W-H-W tetrachord (i.e., D, E, F, G).
- It features another whole step (i.e., G, A).
- It closes by repeating the W-H-W tetrachord (i.e., A, B, C, D).
And because the tetrachord itself is symmetrical, so is Dorian mode itself.
A few scales in Other Scales and Tonalities above are also built on two tetrachords separated by a whole step:
- Chromatic Dorian (½, ½, 1½, 1, ½, ½, 1½) and its modes literally are an ancient Greek genus.
- Chromatic Dorian inverse (1½, ½, ½, 1, 1½, ½, ½) and its modes also qualify.
- Major Phrygian (½, 1½, ½, 1, ½, 1½, ½) uses a ½–1½–½ tetrachord, a whole step, and another ½–1½–½ tetrachord. It also has the same reflective symmetry as our Dorian mode: since its repeated tetrachord is symmetrical and its extra whole-step occurs between them, it too is symmetrical.
Scales built on two tetrachords can be pleasing in their regularity, and they may be helpful starting places when you first write pieces that stray from the Ionian scale’s familiarity. Symmetrical scales built on two of the same tetrachord (e.g., double harmonic minor or modes thereof) may be especially ideal starting places. I’d suggest inventing your own, but there aren’t any others.
But you can be creative in varying how the ancient Greeks constructed their harmony. One possibility: a nine-note scale featuring two of the same pentachord (five-note sequence) separated by a whole-step. To fit these criteria, your pentachord must span a perfect fourth (2½ steps), which unfortunately prevents it from being rotationally symmetrical - your options are W-H-H-H, H-W-H-H, H-H-W-H, or H-H-H-W.
If you want rotational symmetry, though, you could invert the second pentachord:
- W-H-H-H-W-H-H-H-W ()
- H-W-H-H-W-H-H-W-H ()
- H-H-W-H-W-H-W-H-H ()
- H-H-H-W-W-W-H-H-H (; this one seems especially daring)
The second option, H-W-H-H-W-H-H-W-H, can also be constructed by repeating the same trichord (three-note sequence), H-W, with a half-step separation each time, which is another interesting variation on the ancient Greek idea. Within it, each trichord spans a minor third; the added half-step above it means that the same interval pattern repeats every major third. Above its root key, it also includes a minor third, a major third, a perfect fourth, and a perfect fifth above its root key. These make it potentially a very versatile scale. (Since its dominant chord is diminished and it excludes the major second above its root, it also shares some harmonic characteristics with Phrygian mode and Phrygian dominant.) But these aren’t the only possible variants – be creative!
Then again, you may prefer harmonic minor or melodic minor, which respectively only lower one note of Aeolian mode and raise one note of Ionian mode; both are also so ubiquitous in Western music that they may be intuitive. Neither, however, possess the repeated tetrachord of the ancient Greek genera. (Melodic minor does possess a symmetrical mode, Aeolian dominant [W-W-H-W-H-W-W]; harmonic minor does not.)
Acknowledgements & sources
I first wish to thank Marty O’Donnell (yes, that Marty O’Donnell) for pointing out that it wasn’t sufficiently clear what parts of this section I was oversimplifying; his feedback resulted in a much clearer explanation. However, the opinions presented in this section are entirely my own. In particular, Marty believes modern scholars have extrapolated more from ancient Greek texts than is merited by their contents – and in the interest of fairness, I must point out that he has a degree in music theory, and I don’t. However, I must also be fair to myself: Marty got his degree decades ago, and a lot of music scholarship has been done since then.
But even then, I’ll be the first person to admit that I’m by no means an infallible source, so here are some starting places for readers wishing to learn more about this subject. ⟨en
I consulted several other resources researching this section; many were too technical to be of interest to non-specialists, but those seeking more detailed technical analysis of ancient Greek tuning systems may be interested in Robert Erickson’s ⟨ex-tempore
But I think it’s most helpful to quote the ancient Greeks in their own words (or as close to their words as English speakers without educations in Attic Greek will understand), so, via Cris Forester’s ⟨chrysalis-foundation
The magnitude of harmonia is syllaba and di’oxeian. The di’oxeian is greater than the syllaba in epogdoic ratio. From hypate [E] to mese [A] is a syllaba, from mese [A] to neate [or nete, E¹] is a di’oxeian, from neate [E¹] to trite [later paramese, B] is a syllaba, and from trite [B] to hypate [E] is a di’oxeian. The interval between trite [B] and mese [A] is epogdoic [9:8], the syllaba is epitritic [4:3], the di’oxeian hemiolic [3:2], and the dia pason is duple [2:1]. Thus harmonia consists of five epogdoics and two dieses; di’oxeian is three epogdoics and a diesis, and syllaba is two epogdoics and a diesis.—Philolaus, translated by Andrew Barker, Greek Musical Writings, Vol. 2 (1989:
Cambridge University Press). [Text and ratios in brackets are Cris Forester’s.]
Difficult as this is to parse, a close reading reveals Philolaus to be describing the Ionian scale:
- 3:2 is 1.5; our perfect fifth rounds to 1.49830707688.
- 4:3 is 1.333…; our perfect fourth rounds to 1.33483985417.
- 9:8 is 1.125; our whole step rounds to 1.12246204831.
- Philolaus’ dieses are thus 256:243, or 1.05349794239; our half step rounds to 1.05946309436.
In other words:
- A dia pason (or harmonia) is exactly an octave.
- A di’oxean is almost exactly a perfect fifth.
- A syllaba is almost exactly a perfect fourth.
- An epogdoic is almost exactly a whole step.
- A diesis is almost exactly a half-step.
Plugging those in gives us:
The magnitude of an octave is a perfect fourth and a perfect fifth. The perfect fifth is greater than the perfect fourth in whole-step ratio. From hypate [E] to mese [A] is a perfect fourth, from mese [A] to neate [or nete, E¹] is a perfect fifth, from neate [E¹] to trite [later paramese, B] is a perfect fourth, and from trite [B] to hypate [E] is a perfect fifth. The interval between trite [B] and mese [A] is a whole step [9:8], the perfect fourth is epitritic [4:3], the perfect fifth hemiolic [3:2], and the octave is duple [2:1]. Thus an octave consists of five whole steps and two half-steps; a perfect fifth is three whole steps and a half-step, and a perfect fourth is two whole steps and a half-step.
Plus ça change, plus c’est la même chose.
Philolaus’ description is so exact that I believe we can conclude from it that the ancient Greeks routinely used a direct ancestor of our Ionian scale. However, I must reiterate: “ancient Greek harmony” refers to over a millennium of musical practices that were by no means uniform; various authors described other tonoi and indeed entirely different tuning systems. I’ve focused on the tonoi described above for two reasons:
- They’re easy to equate to modern tuning systems.
- They clearly inspired (four of) our modern modes’ names.
I also wish to acknowledge a few resources for the Greek language itself. Wiktionary is low key one of the best online resources for learning languages; it contains a wealth of information on Greek declensions, conjugations, and vocabulary. Λογεῖον and the Liddell, Scott, Jones wiki capably filled gaps in Wiktionary’s coverage. I’m by no means fluent in Attic Greek, but I’ve managed to write lyrics in it that don’t completely embarrass me. (Here’s the song itself if you want to listen to it.) I’d never have managed that without such comprehensive lexicons.
(Keep an eye on this page – I still intend to add more information on the medieval church modes that served as the precursors to our modern modes.)
Appendix 1: Greek musical terminology
It might seem like overkill to include a table this repetitive, but Google Translate is not great at parsing Ancient Greek. My hope is that this will help.
This table focuses exclusively on musical meanings of terms; many have other meanings as well. For instance, the lyre’s three strings are named after the three Muses.
| Λεξικογραφία Ἑλληνῐκῶν μουσῐκῶν ῐ̓́δῐογλῶσσῐῶν Lexikographía Hellēnĭkôn mousĭkôn ĭ́dĭóglôssĭôn A lexicography of Ancient Greek musical idioglossia |
||
|---|---|---|
| Ἀττικός Ἑλληνική Attĭkós Hellēnĭkḗ Attic Greek |
Ῥωμᾰῐ̈σμένη Rhṓmăĭ̈sméni Romanized |
Μετάφρασις Metáphrasis Translation |
| ἐναρμόνιος μιξολῡ́δῐος τόνος | enarmónios mixolū́dĭos tónos | enharmonic Mixolydian tonos |
| ἐναρμόνιος Λῡ́δῐος τόνος | enarmónios Lū́dĭos tónos | enharmonic Lydian tonos |
| ἐναρμόνιος Φρῠ́γῐος τόνος | enarmónios Phrŭ́gios tónos | enharmonic Phrygian tonos |
| ἐναρμόνιος Δώριος τόνος | enarmónios Dṓrios tónos | enharmonic Dorian tonos |
| ἐναρμόνιος ὑπολύδῐος τόνος | enarmónios hŭpolū́dĭos tónos | enharmonic Hypolydian tonos |
| ἐναρμόνιος ὑποφρῠ́γῐος τόνος | enarmónios hŭpophrŭ́gios tónos | enharmonic Hypophrygian tonos |
| ἐναρμόνιος ὑποδώριος τόνος | enarmónios hŭpodṓrios tónos | enharmonic Hypodorian tonos |
| χρωμᾰτῐκός μιξολῡ́δῐος τόνος | khrōmătĭkós mixolū́dĭos tónos | chromatic Mixolydian tonos |
| χρωμᾰτῐκός Λῡ́δῐος τόνος | khrōmătĭkós Lū́dĭos tónos | chromatic Lydian tonos |
| χρωμᾰτῐκός Φρῠ́γῐος τόνος | khrōmătĭkós Phrŭ́gios tónos | chromatic Phrygian tonos |
| χρωμᾰτῐκός Δώριος τόνος | khrōmătĭkós Dṓrios tónos | chromatic Dorian tonos |
| χρωμᾰτῐκός ὑπολύδῐος τόνος | khrōmătĭkós hŭpolū́dĭos tónos | chromatic Hypolydian tonos |
| χρωμᾰτῐκός ὑποφρῠ́γῐος τόνος | khrōmătĭkós hŭpophrŭ́gios tónos | chromatic Hypophrygian tonos |
| χρωμᾰτῐκός ὑποδώριος τόνος | khrōmătĭkós hŭpodṓrios tónos | chromatic Hypodorian tonos |
| διατονικός μιξολῡ́δῐος τόνος | diatonikós mixolū́dĭos tónos | diatonic Mixolydian tonos |
| διατονικός Λῡ́δῐος τόνος | diatonikós Lū́dĭos tónos | diatonic Lydian tonos |
| διατονικός Φρῠ́γῐος τόνος | diatonikós Phrŭ́gios tónos | diatonic Phrygian tonos |
| διατονικός Δώριος τόνος | diatonikós Dṓrios tónos | diatonic Dorian tonos |
| διατονικός ὑπολύδῐος τόνος | diatonikós hŭpolū́dĭos tónos | diatonic Hypolydian tonos |
| διατονικός ὑποφρῠ́γῐος τόνος | diatonikós hŭpophrŭ́gios tónos | diatonic Hypophrygian tonos |
| διατονικός ὑποδώριος τόνος | diatonikós hŭpodṓrios tónos | diatonic Hypodorian tonos |
| μουσικά | mousiká | music |
| μουσικός | mousikós | musically skilled, musical |
| ἁρμονίᾱ | harmoníā | harmony |
| διαπασῶν | diapasôn | octave (lit. “through all”) |
| διπλόος | diplóos | double, 2:1 ratio |
| δῐοξειῶν | dĭoxeiôn | perfect fifth |
| ἡμιόλιος | hēmiólios | 1½, 3:2 ratio |
| σῠλλᾰβή | sŭllăbḗ | perfect fourth |
| ἐπίτριτος | epítritos | 1⅓, 4:3 ratio |
| τρῐ́τος, τρῐ́τη | trĭ́tos, trĭ́tē | third |
| ἐπόγδοος | epógdoos | 1⅛, 9:8 ratio |
| δίεσις | díesis | a scale’s smallest interval |
| ὑπάτη | hupátē | lyre’s lowest-pitched string |
| παραμέση | paramésē | second-lowest-pitched string |
| μέση | mésē | lyre’s middle string |
| νήτη, νεάτη | nḗtē, neátē | lyre’s highest-pitched string |
…OK, fine, I completely made up the declension of «ῐ̓́δῐογλῶσσῐῶν», but to be fair, it wouldn’t have sufficed at all to have used a modern declension when all the surrounding language is Attic.
(For the time being, a complete explanation of declensions is beyond my scope, but I may eventually find myself unable to resist writing one.)
Appendix 2: Interval ratios of 12- and 24-tone equal temperament
As an appendix to the section on tonoi, I’ve also created this table of every possible interval in 24-tone equal temperament. The column “LPT” means “Lowest Possible Temperament” – in other words, to contain an interval, a temperament must be a multiple of its LPT; e.g., if the LPT is 8, the interval will appear in 16-TET, 24-TET, 32-TET, and so on, but will not appear in 12-TET. The lower the LPT, the bolder the font used to print the interval. Intervals printed in blue also appear in 12-TET (our familiar 12-note chromatic scale).
| 24-tone equal temperament’s interval ratios | |||||||
|---|---|---|---|---|---|---|---|
| # | Interval | Exact | Approximate | LPT | |||
| 1 | Quarter tone, infra second | 2¹⁄₂₄ | = | ²⁴√ | 2 | 1.02930223664 | 24 |
| 2 | Minor second | 2²⁄₂₄ | = | ¹²√ | 2 | 1.05946309436 | 12 |
| 3 | Neutral second | 2³⁄₂₄ | = | ⁸√ | 2 | 1.09050773267 | 8 |
| 4 | Major second | 2⁴⁄₂₄ | = | ⁶√ | 2 | 1.12246204831 | 6 |
| 5 | Ultra second, infra third | 2⁵⁄₂₄ | = | ²⁴√ | 32 | 1.15535269687 | 24 |
| 6 | Minor third | 2⁶⁄₂₄ | = | ⁴√ | 2 | 1.18920711500 | 4 |
| 7 | Neutral third | 2⁷⁄₂₄ | = | ²⁴√ | 128 | 1.22405354330 | 24 |
| 8 | Major third | 2⁸⁄₂₄ | = | ³√ | 2 | 1.25992104989 | 3 |
| 9 | Ultra third, narrow fourth | 2⁹⁄₂₄ | = | ⁸√ | 8 | 1.29683955465 | 8 |
| 10 | Perfect fourth | 2¹⁰⁄₂₄ | = | ¹²√ | 32 | 1.33483985417 | 12 |
| 11 | Wide fourth | 2¹¹⁄₂₄ | = | ²⁴√ | 2,048 | 1.37395364746 | 24 |
| 12 | Tritone | 2¹²⁄₂₄ | = | √ | 2 | 1.41421356237 | 2 |
| 13 | Narrow fifth | 2¹³⁄₂₄ | = | ²⁴√ | 8,192 | 1.45565318284 | 24 |
| 14 | Perfect fifth | 2¹⁴⁄₂₄ | = | ¹²√ | 128 | 1.49830707688 | 12 |
| 15 | Wide fifth, infra sixth | 2¹⁵⁄₂₄ | = | ⁸√ | 32 | 1.54221082541 | 8 |
| 16 | Minor sixth | 2¹⁶⁄₂₄ | = | ³√ | 4 | 1.58740105197 | 3 |
| 17 | Neutral sixth | 2¹⁷⁄₂₄ | = | ²⁴√ | 131,072 | 1.63391545324 | 24 |
| 18 | Major sixth | 2¹⁸⁄₂₄ | = | ⁴√ | 8 | 1.68179283051 | 4 |
| 19 | Ultra sixth, infra seventh | 2¹⁹⁄₂₄ | = | ²⁴√ | 524,288 | 1.73107312201 | 24 |
| 20 | Minor seventh | 2²⁰⁄₂₄ | = | ⁶√ | 32 | 1.78179743628 | 6 |
| 21 | Neutral seventh | 2²¹⁄₂₄ | = | ⁸√ | 128 | 1.83400808641 | 8 |
| 22 | Major seventh | 2²²⁄₂₄ | = | ¹²√ | 2,048 | 1.88774862536 | 12 |
| 23 | Ultra seventh, narrow octave | 2²³⁄₂₄ | = | ²⁴√ | 8,388,608 | 1.94306388231 | 24 |
| 24 | Octave | 2²⁴⁄₂₄ | = | 2 | 2 | 1 | |
Embryonic surveys of other musical traditions
Back to Babylon, or, the Mesopotamian major scale
The Greeks didn’t even invent the Ionian scale: the Hurrian songs from ca. 1400 BCE Mesopotamia (which survive in fragments) conform to its interval spacing, and surviving Mesopotamian literature on tuning (from 1800 BCE and earlier) is so voluminous that scholars even believe they know the Mesopotamian modes’ correspondence to ours. Musicologist and Assyriologist Anne Draffkorn Kilmer (1931-2023), who deciphered the oldest and most complete surviving Hurrian song, identified the modes in 2014 as:
| Mesopotamian modes | |||
|---|---|---|---|
| Akkadian | Translation | Greek | Modern |
| Išartu | Normal | Lydian | Ionian |
| Embūbu | Reed pipe | Phrygian | Dorian |
| Nīd qabli | Fall of the middle | Dorian | Phrygian |
| Qablītu | Middle | Hypolydian | Lydian |
| Kitmu | Closed | Hypophrygian | Mixolydian |
| Pītu | Open | Hypodorian | Aeolian |
| Nīš tuhri | Rise of the Achilles tendon | Mixolydian | Locrian |
Ironically, until roughly 1990, scholars had apparently repeated medieval Europeans’ Greek scale order mistake: as Kilmer explained, Mesopotamians, like the Greeks, defined scales in descending order, and scholars had once again assumed they were in ascending order. (I understand a fair amount of Greek, but not a single word of Akkadian, so I can do little but summarize others’ scholarship in this subsection.) Kilmer writes that scholars had previously read a fragmented treatise on tuning to mean that strings should be loosened, then tightened, but it was later reread to mean they should be tightened, then loosened, which clarified that it was describing notes in descending order. Thus:
The resulting changes in translation […] turned Gurney’s (1968) “If the harp is (tuned as) X, and the interval Y is not clear; you alter the (string) N, and then Y will be…” into his new (Gurney 1994) rendering (lines 1–12), “If the instrument is (tuned as) X, and the (interval) Y is not clear, you tighten the (string) N, and then Y will be clear.” [Editor’s note: The Mesopotamians called the tritone impure or unclear (Akkadian: la zakû).] The preceding procedures were summed up as “tightening.” The second tuning section of the same text is now translated as follows: (lines 13–20) “If the instrument is (tuned as) X, and you have played an (unclear) interval Y, you loosen the string N and the instrument will be (in the tuning) Z.” The second section was presumably and logically summed up as: “[loosening]”. This newer interpretation is generally accepted today.
Ironically, this means that what caused me to start writing this entire book was independently rediscovering what had been the cornerstone of Mesopotamian tuning practice four millennia earlier. Kilmer continues:
That the seven Mesopotamian musical scales (at least as early as ca. 1800 bce) were heptatonic-diatonic scales has been proven to the satisfaction of cuneiformists and musicologists alike. It should be noted here that, thanks to the observations of Wulstan (1968) and Kümmel (1970), it was recognized that the ancient Mesopotamian musicians/“musicologists” knew what we call today the Pythagorean series of fifths, and that the series could be accomplished within a single octave by means of “inversion.” Kümmel taught us that the names given to the seven tunings/scales were derived from the specific intervals on which the tuning procedure started. For example, if the tuning procedure started with the interval išartu “normal”, the resulting scale was called išartu.
However, the change in our recognizing of the scales as moving downward rather than upward means that išartu 2–6 was no longer string 2 (RE) up to string 6 (LA), but rather 2–6 as 2 (TI) down to 6 (MI). As Crocker (1997: 195) emphasized, “The principal difference brought about by Krispijn’s restoration (of nu-su-h[u-um]) is that the seven octave segments (or intervals) receive different names” (than those they bore earlier). Thus, nīd qabli “fall from the middle” was, before Krispijn (1990), the scale from C-C (ascending). After 1990, it is E-E (descending).
Note that, as of this writing (2025-10-09), Wikipedia’s article on Mesopotamian music theory still relies on sources that predate this re-reading, so it uses the incorrect ascending modes (and nīš tuhri was still read as nīš gabarî, “rise of the duplicate”). I plan to fix it once I’ve had the chance to read Kilmer’s sources.
The Chinese independently invented a twelve-note chromatic scale called Shi’er lü (十二律) somewhere between 600 BCE and 250 BCE; it uses the same ratios as the Mesopotamian and Pythagorean scales (3:2, 4:3, 9:8, 32:27, 81:64, etc.) Shi’er lü was not used as a scale in its own right, but it was used as a basis on which to construct other scales, more comparable to the way in which Indian classical music constructs rāgas. On which note…
The carnatic numbered mēḷakartā
A numbered set of fundamental rāgasa (musical scales) that originated in carnatic music (South Indian classical music). They must obey a few rules:
- They must be sampūrṇa, or complete: i.e., they must contain all seven svaras.
- The ascending and descending notes must be the same.
- They must end with the same note they start with. (I have omitted the upper C here for the sake of space.)
- The third and fifth scale degrees may not be sharp.
- The fourth and fifth scale degrees may not be flat.
Many of these correspond exactly to frequently used Western scales (e.g., #8 is Phrygian, #20 is Aeolian, #21 is harmonic minor, #22 is Dorian, #23 is melodic minor, #28 is Mixolydian, #29 is Ionian, #65 is Lydian); however, many others are virtually unique to Indian music, and some fundamental modes of Western music (e.g., Locrian mode) are absent here, as they break some of the fundamental rules of mēḷakartā.
Indian music uses multiple scales, and although there are commonly held to be 22 shruti per octave, this remains a matter of some debate, since in practice, pitch tends to vary somewhat. To avoid confusion, I’m therefore printing the swaras’ Western names.
| The carnatic numbered mēḷakartā | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Mode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
| 1 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B𝄫 | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | |
| 2 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B♭ | ½ | ½ | 1½ | 1 | ½ | 1 | 1 | |
| 3 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B♮ | ½ | ½ | 1½ | 1 | ½ | 1½ | ½ | |
| 4 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♮ | B♭ | ½ | ½ | 1½ | 1 | 1 | ½ | 1 | |
| 5 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♮ | B♮ | ½ | ½ | 1½ | 1 | 1 | 1 | ½ | |
| 6 | C♮ | D♭ | E𝄫 | F♮ | G♮ | A♯ | B♮ | ½ | ½ | 1½ | 1 | 1½ | ½ | ½ | |
| 7 | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B𝄫 | ½ | 1 | 1 | 1 | ½ | ½ | 1½ | |
| 8 | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |
| 9 | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♮ | ½ | 1 | 1 | 1 | ½ | 1½ | ½ | |
| 10 | C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | |
| 11 | C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♮ | ½ | 1 | 1 | 1 | 1 | 1 | ½ | |
| 12 | C♮ | D♭ | E♭ | F♮ | G♮ | A♯ | B♮ | ½ | 1 | 1 | 1 | 1½ | ½ | ½ | |
| 13 | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B𝄫 | ½ | 1½ | ½ | 1 | ½ | ½ | 1½ | |
| 14 | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♭ | ½ | 1½ | ½ | 1 | ½ | 1 | 1 | |
| 15 | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♮ | ½ | 1½ | ½ | 1 | ½ | 1½ | ½ | |
| 16 | C♮ | D♭ | E♮ | F♮ | G♮ | A♮ | B♭ | ½ | 1½ | ½ | 1 | 1 | ½ | 1 | |
| 17 | C♮ | D♭ | E♮ | F♮ | G♮ | A♮ | B♮ | ½ | 1½ | ½ | 1 | 1 | 1 | ½ | |
| 18 | C♮ | D♭ | E♮ | F♮ | G♮ | A♯ | B♮ | ½ | 1½ | ½ | 1 | 1½ | ½ | ½ | |
| 19 | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B𝄫 | 1 | ½ | 1 | 1 | ½ | ½ | 1½ | |
| 20 | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |
| 21 | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | 1 | ½ | 1 | 1 | ½ | 1½ | ½ | |
| 22 | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |
| 23 | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | |
| 24 | C♮ | D♮ | E♭ | F♮ | G♮ | A♯ | B♮ | 1 | ½ | 1 | 1 | 1½ | ½ | ½ | |
| 25 | C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B𝄫 | 1 | 1 | ½ | 1 | ½ | ½ | 1½ | |
| 26 | C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | ½ | 1 | ½ | 1 | 1 | |
| 27 | C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♮ | 1 | 1 | ½ | 1 | ½ | 1½ | ½ | |
| 28 | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | |
| 29 | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |
| 30 | C♮ | D♮ | E♮ | F♮ | G♮ | A♯ | B♮ | 1 | 1 | ½ | 1 | 1½ | ½ | ½ | |
| 31 | C♮ | D♯ | E♮ | F♮ | G♮ | A♭ | B𝄫 | 1½ | ½ | ½ | 1 | ½ | ½ | 1½ | |
| 32 | C♮ | D♯ | E♮ | F♮ | G♮ | A♭ | B♭ | 1½ | ½ | ½ | 1 | ½ | 1 | 1 | |
| 33 | C♮ | D♯ | E♮ | F♮ | G♮ | A♭ | B♮ | 1½ | ½ | ½ | 1 | ½ | 1½ | ½ | |
| 34 | C♮ | D♯ | E♮ | F♮ | G♮ | A♮ | B♭ | 1½ | ½ | ½ | 1 | 1 | ½ | 1 | |
| 35 | C♮ | D♯ | E♮ | F♮ | G♮ | A♮ | B♮ | 1½ | ½ | ½ | 1 | 1 | 1 | ½ | |
| 36 | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | |
| 37 | C♮ | D♭ | E𝄫 | F♯ | G♮ | A♭ | B𝄫 | ½ | ½ | 2 | ½ | ½ | ½ | 1½ | |
| 38 | C♮ | D♭ | E𝄫 | F♯ | G♮ | A♭ | B♭ | ½ | ½ | 2 | ½ | ½ | 1 | 1 | |
| 39 | C♮ | D♭ | E𝄫 | F♯ | G♮ | A♭ | B♮ | ½ | ½ | 2 | ½ | ½ | 1½ | ½ | |
| 40 | C♮ | D♭ | E𝄫 | F♯ | G♮ | A♮ | B♭ | ½ | ½ | 2 | ½ | 1 | ½ | 1 | |
| 41 | C♮ | D♭ | E𝄫 | F♯ | G♮ | A♮ | B♮ | ½ | ½ | 2 | ½ | 1 | 1 | ½ | |
| 42 | C♮ | D♭ | E𝄫 | F♯ | G♮ | A♯ | B♮ | ½ | ½ | 2 | ½ | 1½ | ½ | ½ | |
| 43 | C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B𝄫 | ½ | 1 | 1½ | ½ | ½ | ½ | 1½ | |
| 44 | C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♭ | ½ | 1 | 1½ | ½ | ½ | 1 | 1 | |
| 45 | C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♮ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | |
| 46 | C♮ | D♭ | E♭ | F♯ | G♮ | A♮ | B♭ | ½ | 1 | 1½ | ½ | 1 | ½ | 1 | |
| 47 | C♮ | D♭ | E♭ | F♯ | G♮ | A♮ | B♮ | ½ | 1 | 1½ | ½ | 1 | 1 | ½ | |
| 48 | C♮ | D♭ | E♭ | F♯ | G♮ | A♯ | B♮ | ½ | 1 | 1½ | ½ | 1½ | ½ | ½ | |
| 49 | C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B𝄫 | ½ | 1½ | 1 | ½ | ½ | ½ | 1½ | |
| 50 | C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B♭ | ½ | 1½ | 1 | ½ | ½ | 1 | 1 | |
| 51 | C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | |
| 52 | C♮ | D♭ | E♮ | F♯ | G♮ | A♮ | B♭ | ½ | 1½ | 1 | ½ | 1 | ½ | 1 | |
| 53 | C♮ | D♭ | E♮ | F♯ | G♮ | A♮ | B♮ | ½ | 1½ | 1 | ½ | 1 | 1 | ½ | |
| 54 | C♮ | D♭ | E♮ | F♯ | G♮ | A♯ | B♮ | ½ | 1½ | 1 | ½ | 1½ | ½ | ½ | |
| 55 | C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B𝄫 | 1 | ½ | 1½ | ½ | ½ | ½ | 1½ | |
| 56 | C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♭ | 1 | ½ | 1½ | ½ | ½ | 1 | 1 | |
| 57 | C♮ | D♮ | E♭ | F♯ | G♮ | A♭ | B♮ | 1 | ½ | 1½ | ½ | ½ | 1½ | ½ | |
| 58 | C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♭ | 1 | ½ | 1½ | ½ | 1 | ½ | 1 | |
| 59 | C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♮ | 1 | ½ | 1½ | ½ | 1 | 1 | ½ | |
| 60 | C♮ | D♮ | E♭ | F♯ | G♮ | A♯ | B♮ | 1 | ½ | 1½ | ½ | 1½ | ½ | ½ | |
| 61 | C♮ | D♮ | E♮ | F♯ | G♮ | A♭ | B𝄫 | 1 | 1 | 1 | ½ | ½ | ½ | 1½ | |
| 62 | C♮ | D♮ | E♮ | F♯ | G♮ | A♭ | B♭ | 1 | 1 | 1 | ½ | ½ | 1 | 1 | |
| 63 | C♮ | D♮ | E♮ | F♯ | G♮ | A♭ | B♮ | 1 | 1 | 1 | ½ | ½ | 1½ | ½ | |
| 64 | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 | |
| 65 | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |
| 66 | C♮ | D♮ | E♮ | F♯ | G♮ | A♯ | B♮ | 1 | 1 | 1 | ½ | 1½ | ½ | ½ | |
| 67 | C♮ | D♯ | E♮ | F♯ | G♮ | A♭ | B𝄫 | 1½ | ½ | 1 | ½ | ½ | ½ | 1½ | |
| 68 | C♮ | D♯ | E♮ | F♯ | G♮ | A♭ | B♭ | 1½ | ½ | 1 | ½ | ½ | 1 | 1 | |
| 69 | C♮ | D♯ | E♮ | F♯ | G♮ | A♭ | B♮ | 1½ | ½ | 1 | ½ | ½ | 1½ | ½ | |
| 70 | C♮ | D♯ | E♮ | F♯ | G♮ | A♮ | B♭ | 1½ | ½ | 1 | ½ | 1 | ½ | 1 | |
| 71 | C♮ | D♯ | E♮ | F♯ | G♮ | A♮ | B♮ | 1½ | ½ | 1 | ½ | 1 | 1 | ½ | |
| 72 | C♮ | D♯ | E♮ | F♯ | G♮ | A♯ | B♮ | 1½ | ½ | 1 | ½ | 1½ | ½ | ½ | |
The numbering consistently obeys several patterns:
- Adding 36 to numbers below 36 changes F to F♯; subtracting 36 from numbers above 36 changes F♯ to F♮.
The scale’s first three notes depend on the range of the scale number, while its fifth, sixth, and seventh depend on the scale number modulo 6 (“%” column):
Mēḷakartā note ranges Range 1 Range 2 Notes % Notes 1–6 37–42 C D♭ E𝄫 1 G A♭ B𝄫 7–12 43–48 C D♭ E♭ 2 G A♭ B♭ 13–18 49–54 C D♭ E 3 G A♭ B 19–24 55–60 C D E♭ 4 G A B♭ 25–30 61–66 C D E 5 G A B 31–36 67–72 C D♯ E 0 G A♯ B Note that the two “Notes” columns’ accidentals correspond exactly.
Scale counts in 12-TET by scale size
How to count scales
In twelve-tone equal temperament, by definition, heptatonic scales contain the root and six of the eleven other pitches. Since (as we proved above) heptatonic and pentatonic scales can’t be modes of limited transposition in 12-TET, we can thus calculate the total number of scales using the formula (11
6) = 11!
6!5! = 11·10·9·8·7
5·4·3·2 = 11·7·3·2 = 462. Since 462 / 7, there are 66 discrete scales with seven modes each.
Meanwhile, the number of pentatonic scales in 12-TET is (11
4) = 11!
4!7! = 11·10·9·8
4·3·2 = 11·5·3·2 = 330. If we discount modes, 330 / 5 also leaves us with 66 discrete pentatonic scales in 12-TET. This is no coincidence: since 12-TET’s heptatonic and pentatonic scales can’t be modes of limited transposition, every pentatonic scale in 12-TET has five modes, every heptatonic scale in 12-TET has seven modes, and every pentatonic scale in 12-TET is, by definition, the scale complement of a heptatonic scale.
As it turns out, 12-tone equal temperament contains the following mode counts for each scale size:
| 12-TET mode counts | ||
|---|---|---|
| Notes | Modes | |
| 1 | 12 | 1 |
| 2 | 11 | 11 |
| 3 | 10 | 55 |
| 4 | 9 | 165 |
| 5 | 8 | 330 |
| 6 | 7 | 462 |
| Total | 2,048 | |
Because sixteen scales are modes of limited transposition, however, we can’t simply divide each of those scale counts by the number of notes in the scale. The following table’s “−” column denotes the number of missing modes. As mentioned above, counting truncations and the chromatic scale, 12-TET contains 16 discrete modes of limited transposition with 38 modes between them; if they were not modes of limited transposition, they’d have a total of 102 modes, so they’re short by 64. (2,048 + 64 = ⟨youtu.
| All modes of limited transposition | Modes of limited transposition | |||||||
|---|---|---|---|---|---|---|---|---|
| # | Intervals | − | # | Intervals | − | |||
| 12 | 11 | 6 | 5 | |||||
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| 2 | 1 | |||||||
| 12 | 11 | |||||||
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| 4 | 3 | |||||||
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| 3 | 2 | |||||||
| 2 | 1 | |||||||
Thus, with modes excluded, 12-tone equal temperament’s discrete scales break down as follows:
| 12-TET discrete scales | ||||
|---|---|---|---|---|
| Notes | Scales | |||
| 1 | 11 | 12 | 1 | |
| 2 | 10 | 6 | ||
| 3 | 9 | 19 | ||
| 4 | 8 | 43 | ||
| 5 | 7 | 66 | ||
| 6 | 80 | |||
| Total | 351 | |||
I’m not yet sure if it’s coincidental that modes of limited transposition are responsible for 64 missing modes and that 64 scale modes possess internal reflective symmetry. I have a very strong hunch that it is not.
All 2,048 modes of all 351 scales in 12-TET
I’ve listed all 2,048 modes of 12-tone equal temperament’s 351 scales. Due to the sheer quantity of data, I’ve put this on its own page, but I consider it an extension of this book. (As of 2025-10-09, the scale list is ≈2,200 lines, ≈34,000 words, and ≈135,000 characters; this page is ≈4,750 lines, ≈67,000 words, and ≈350,000 characters. This adds up to ≈6,950 lines, ≈101,000 words, and ≈485,000 characters between both.)