The modes in circle of fifths order

I mostly won’t be analyzing the modes in their traditional order, since 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 become readily apparent. 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 2^¹⁄₁₂ 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.)
• Key signatures with seven sharps or flats exist, but I’ve mostly omitted them from my analysis. (There are even monstrosities with double-sharps [𝄪] and double-flats [𝄫] … which I mostly ignore. I can at least tolerate the double-flat symbol, but the double-sharp is a monstrosity that doesn’t even slightly resemble what it’s meant to double.) I feel using both six-flat and six-sharp key signatures was extravagant enough as it was.
• 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:

• 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 highlightedin boxes:

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:

The Russian author Anton Chekhov might have a few words to say about this table. This is my roundabout way of advising you to remember it.

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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 tells 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:

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An audio demonstration

• None of this will mean much to readers in isolation, so I put together a crude demonstration of the above principle ⟨aaronfreed.github.io/c_lydian_to_b_lydian.flac⟩ in Logic Pro. For the following modes, in order:

  1. C Lydian
  2. C Ionian
  3. C Mixolydian
  4. C Dorian
  5. C Aeolian
  6. C Phrygian
  7. C Locrian
  8. C♭/B Lydian

  you’ll hear, in order:

  1. the scale, in ascending order
  2. the root or tonic (I, i, or iᵒ) chord, in arpeggiated and block forms
  3. the subdominant (IV, iv, or ivᵒ) chord, in arpeggiated and block forms
  4. the dominant (V, v, or vᵒ) chord, in arpeggiated and block forms
  5. 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.)

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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:

    Diabolus in mūsicā
    #	Mode	Tritone
    4	Lydian	4	1
    1	Ionian	7	4
    5	Mixolydian	3	7
    2	Dorian	6	3
    6	Aeolian	2	6
    3	Phrygian	5	2
    7	Locrian	1	5

    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.

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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.

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Back to top · Complete 12-TET scale list · My discography · Marathon soundtracks · Contact me · Website index

Back to top · Complete 12-TET scale list · My discography · Marathon soundtracks · Contact me · Website index

Back to top · Complete 12-TET scale list · My discography · Marathon soundtracks · Contact me · Website index

Back to top · Complete 12-TET scale list · My discography · Marathon soundtracks · Contact me · Website index

Back to top · Complete 12-TET scale list · My discography · Marathon soundtracks · Contact me · Website index

Back to top · Complete 12-TET scale list · My discography · Marathon soundtracks · Contact me · Website index

Back to top · Complete 12-TET scale list · My discography · Marathon soundtracks · Contact me · Website index

Back to top · Complete 12-TET scale list · My discography · Marathon soundtracks · Contact me · Website index

Back to top · Complete 12-TET scale list · My discography · Marathon soundtracks · Contact me · Website index

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Key signature cheat sheet

• Twelve-tone equal temperament has fifteen key signatures, 𝄪/𝄫 atrocities notwithstanding:

• 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).

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Why is this happening?

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:

Apart from C Ionian, these modes each rearrange different major scales, as we can see by reshuffling them back to Ionian:

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.

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 keyboard. (The grand piano spans from A0 to C8.) What’s a perfect fifth above that? C2. What’s a perfect fifth above C2? G2. Move up another perfect fifth. We’re at D3. Up another perfect fifth. A3. Up another perfect fifth. E4. Another perfect fifth. B4. So, to recap, we have the notes:

1. F1
2. C2
3. G2
4. D3
5. A3
6. E4
7. 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. In our example, this would be:

6. Lydian
3. Ionian
7. Mixolydian
4. Dorian
1. Aeolian
5. Phrygian
2. Locrian

So, start on the desired mode’s note, flatten the other six notes into linear order, and voilà, there’s your scale.

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 we disregard octaves. 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.

The last part of the explanation has to do with interval distributions:

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. Interval l should always equal 6 − the sum of intervals 1 through 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 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 interval pair, and therefore moves only one note. If more intervals changed, the pattern would break.

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Chord analysis by mode

• 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 Ionian	C maj	D min	E min	F maj	G maj	A min	B dim
  2	C Dorian	C min	D min	E♭ maj	F maj	G min	A dim	B♭ maj
  3	C Phrygian	C min	D♭ maj	E♭ maj	F min	G dim	A♭ maj	B♭ min
  4	C Lydian	C maj	D maj	E min	F♯ dim	G maj	A min	B min
  5	C Mixolydian	C maj	D min	E dim	F maj	G min	A min	B♭ maj
  6	C Aeolian	C min	D dim	E♭ maj	F min	G min	A♭ maj	B♭ maj
  7	C Locrian	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 Ionian	C maj	D min	E min	F maj	G maj	A min	B dim
  2	D Dorian	D min	E min	F maj	G maj	A min	B dim	C maj
  3	E Phrygian	E min	F maj	G maj	A min	B dim	C maj	D min
  4	F Lydian	F maj	G maj	A min	B dim	C maj	D min	E min
  5	G Mixolydian	G maj	A min	B dim	C maj	D min	E min	F maj
  6	A Aeolian	A min	B dim	C maj	D min	E min	F maj	G maj
  7	B Locrian	B dim	C maj	D min	E min	F maj	G maj	A min

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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
  Scale		1	2	3	4	5	6	7	Intervals
  Chromatic Dorian	Kanakāngi	C♮	D♭	E𝄫	F♮	G♮	A♭	B𝄫	½	½	1½	1	½	½	1½
  Mixolydian ♯1	Super-Locrian ♭7	C♮	D♭	E♭	F♭	G♭	A♭	B𝄫	½	1	½	1	1	½	1½
  Locrian ♭4	Super-Locrian	C♮	D♭	E♭	F♭	G♭	A♭	B♭	½	1	½	1	1	1	1
  Locrian ♯6	Maqam Tarznauyn	C♮	D♭	E♭	F♮	G♭	A♮	B♭	½	1	1	½	1½	½	1
  Dorian ♭2	Jazz minor inverse	C♮	D♭	E♭	F♮	G♮	A♮	B♭	½	1	1	1	1	½	1
  Phrygian ♯3	Phrygian dominant	C♮	D♭	E♮	F♮	G♮	A♭	B♭	½	1½	½	1	½	1	1
  Major Phrygian		C♮	D♭	E♮	F♮	G♮	A♭	B♮	½	1½	½	1	½	1½	½
  Aeolian ♭5	Half-diminished	C♮	D♮	E♭	F♮	G♭	A♭	B♭	1	½	1	½	1	1	1
  Harmonic minor ♭5		C♮	D♮	E♭	F♮	G♭	A♭	B♮	1	½	1	½	1	1½	½
  Aeolian ♯7	Harmonic minor	C♮	D♮	E♭	F♮	G♮	A♭	B♮	1	½	1	1	½	1½	½
  Ionian ♭3	Melodic minor	C♮	D♮	E♭	F♮	G♮	A♮	B♮	1	½	1	1	1	1	½
  Double harmonic minor		C♮	D♮	E♭	F♯	G♮	A♭	B♮	1	½	1½	½	½	1½	½
  Dorian ♯4	Lydian diminished	C♮	D♮	E♭	F♯	G♮	A♮	B♭	1	½	1½	½	1	½	1
  Mixolydian ♭6	Aeolian dominant	C♮	D♮	E♮	F♮	G♮	A♭	B♭	1	1	½	1	½	1	1
  Ionian ♯5	Ionian augmented	C♮	D♮	E♮	F♮	G♯	A♮	B♮	1	1	½	1½	½	1	½
  Lydian ♭7	Lydian dominant	C♮	D♮	E♮	F♯	G♮	A♮	B♭	1	1	1	½	1	½	1
  Phrygian ♭1	Lydian augmented	C♮	D♮	E♮	F♯	G♯	A♮	B♮	1	1	1	1	½	1	½
  Chromatic Dorian inverse		C♮	D♯	E♮	F♮	G♮	A♯	B♮	1½	½	½	1	1½	½	½
  Lydian ♯2	Aeolian harmonic	C♮	D♯	E♮	F♯	G♮	A♮	B♮	1½	½	1	½	1	1	½
  Chromatic Hypophrygian		C♮	D♯	E♯	F♯	G♮	A♯	B♮	1½	1	½	½	1½	½	½
  Superlydian augmented		C♮	D♯	E♯	F♯	G♯	A♯	B♮	1½	1	½	1	1	½	½

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  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 Carnatic Numbered Mēḷakartā, which effectively alphabetize the scales themselves: sort intervals in ascending order, with the lowest intervals counting as most significant. (§9 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
  Scale		1	2	3	4	5	6	7	Intervals
  Aeolian ♯7	Harmonic minor	C♮	D♮	E♭	F♮	G♮	A♭	B♮	1	½	1	1	½	1½	½
  Locrian ♯6	Maqam Tarznauyn	C♮	D♭	E♭	F♮	G♭	A♮	B♭	½	1	1	½	1½	½	1
  Ionian ♯5	Ionian augmented	C♮	D♮	E♮	F♮	G♯	A♮	B♮	1	1	½	1½	½	1	½
  Dorian ♯4	Lydian diminished	C♮	D♮	E♭	F♯	G♮	A♮	B♭	1	½	1½	½	1	½	1
  Phrygian ♯3	Phrygian dominant	C♮	D♭	E♮	F♮	G♮	A♭	B♭	½	1½	½	1	½	1	1
  Lydian ♯2	Aeolian harmonic	C♮	D♯	E♮	F♯	G♮	A♮	B♮	1½	½	1	½	1	1	½
  Mixolydian ♯1	Super-Locrian ♭7	C♮	D♭	E♭	F♭	G♭	A♭	B𝄫	½	1	½	1	1	½	1½
  Ionian ♭3	Melodic minor	C♮	D♮	E♭	F♮	G♮	A♮	B♮	1	½	1	1	1	1	½
  Dorian ♭2	Jazz minor inverse	C♮	D♭	E♭	F♮	G♮	A♮	B♭	½	1	1	1	1	½	1
  Phrygian ♭1	Lydian augmented	C♮	D♮	E♮	F♯	G♯	A♮	B♮	1	1	1	1	½	1	½
  Lydian ♭7	Lydian dominant	C♮	D♮	E♮	F♯	G♮	A♮	B♭	1	1	1	½	1	½	1
  Mixolydian ♭6	Aeolian dominant	C♮	D♮	E♮	F♮	G♮	A♭	B♭	1	1	½	1	½	1	1
  Aeolian ♭5	Half-diminished	C♮	D♮	E♭	F♮	G♭	A♭	B♭	1	½	1	½	1	1	1
  Locrian ♭4	Super-Locrian	C♮	D♭	E♭	F♭	G♭	A♭	B♭	½	1	½	1	1	1	1

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  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:

  1. 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).
  2. 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.

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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 neither five nor seven divide evenly into the parent temperament). 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.
• Twelve-tone equal temperament (12-TET) ⟨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 twenty-four-note scale ⟨en.wikipedia.org/wiki/Quarter_tone⟩ is common in Arabic and Turkic music.
• Indian rāgas ⟨en.wikipedia.org/wiki/Raga⟩⁽²⁾ (similar to scales) typically use four to seven svaras⁽³⁾ (similar to scale tones) of the twenty-two śrútis ⟨en.wikipedia.org/wiki/Shruti_(music)⟩⁽⁴⁾ (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:
  • Flying Microtonal Banana ⟨kinggizzard.bandcamp.com/album/flying-microtonal-banana⟩ by King Giz­zard & the Lizard Wizard (24-TET)
  • Nowherer ⟨votsband.bandcamp.com/album/nowherer⟩ by Victory over the Sun (17-TET)
  • Every metal album by Jute Gyte ⟨jutegyte.bandcamp.com/music⟩ from Discontinuities onward (24-TET)
• 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:

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?

1. 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.
2. 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.
3. Tritonic means spanning an interval of three whole tones, i.e., a tritone.

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The pentatonic scale

As complement of Ionian

Pentatonic (five-note, “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. That’s 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 nasty prefer 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.

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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:

1. Major pentatonic deletes Lydian’s fourth and seventh notes: F♯ G♯ A♯ C♯ D♯ (1, 1, 1½, 1, 1½).
2. Scottish pentatonic deletes Ionian’s third and seventh notes: C♯ D♯ F♯ G♯ A♯ (1, 1½, 1, 1, 1½).
3. Neutral pentatonic deletes Mixolydian’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).
4. Minor pentatonic deletes Dorian’s second and sixth notes: D♯ F♯ G♯ A♯ C♯ (1½, 1, 1, 1½, 1).
5. Blues minor deletes Aeolian’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)

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:

(End warning.)

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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?

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?

1. If both scales have symmetrical modes (Dorian and neutral pentatonic), then:

   1. We should compare the symmetrical modes to each other.
   2. Symmetrical modes should be their circle-order comparisons’ center data rows.
   3. Rows whose intervals are mirrors in the base should remain mirrors in the transformation.
   4. 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.

2. 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.
3. 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).

4. 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.

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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:

(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:

3. We didn’t delete this note; its mode, Ionian, becomes major pentatonic 5t..
4. We didn’t delete this note; its mode, Dorian, becomes neutral pentatonic 5t..
5. We didn’t delete this note; its mode, Phrygian, becomes blues minor.
6. Deleting this note deletes its mode; Lydian has no pentatonic equivalent.
7. We didn’t delete this note; its mode, Mixolydian, becomes Scottish pentatonic 5t..
1. We didn’t delete this note; its mode, Aeolian, becomes minor pentatonic 5t..
2. Deleting this note deletes its mode; Locrian has no pentatonic equivalent.

Put another way, recall how Ionian’s modes got their numbering:

1. Ionian starts on its first note.
2. Dorian starts on its second note.
3. Phrygian starts on its third note.
4. Lydian starts on its fourth note.
5. Mixolydian starts on its fifth note.
6. Aeolian starts on its sixth note.
7. Locrian starts on its seventh note.

So, applying the same principle to the pentatonic scale:

1. Major pentatonic 5t. starts on its first note.
2. Neutral pentatonic 5t. starts on its second note.
3. Blues minor starts on its third note.
4. Scottish pentatonic 5t. starts on its fourth note.
5. Minor pentatonic 5t. starts on its fifth note.

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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:

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. Also, bear in mind that:

1. Ionian has only semitones and whole tones

2. We only delete notes that:

   1. follow semitones and precede whole tones
   2. 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 with the above highlighting:

So, let’s correct our original analysis, shall we?

1. Major pentatonic 5t. deletes Ionian’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.
2. Scottish pentatonic 5t. deletes Mixolydian’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.
3. Neutral pentatonic 5t. deletes Dorian’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.
4. Minor pentatonic 5t. deletes Aeolian’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.
5. Blues minor deletes Phrygian’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.

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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).

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.

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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:

1. 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.
2. 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.

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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.

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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.

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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
  Blues minor	C♮	D♯	F♮	G♯	A♯	Aeolian, Phrygian, Locrian	C♮	E♭	F♮	A♭	B♭
  Minor 5t.	C♮	D♯	F♮	G♮	A♯	Dorian, Aeolian, Phrygian	C♮	E♭	F♮	G♮	B♭
  Neutral 5t.	C♮	D♮	F♮	G♮	A♯	Mixolydian, Dorian, Aeolian	C♮	D♮	F♮	G♮	B♭
  Scottish 5t.	C♮	D♮	F♮	G♮	A♮	Ionian, Mixolydian, Dorian	C♮	D♮	F♮	G♮	A♮
  Major 5t.	C♮	D♮	E♮	G♮	A♮	Lydian, Ionian, Mixolydian	C♮	D♮	E♮	G♮	A♮

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  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.)

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.

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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

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• 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:

1. Differ in size from the remainder by only a semitone
2. 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.

1. 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.
2. 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.

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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.