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 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.)
• 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:

• 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 (1860-1904) 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 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:

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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?

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:

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.

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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:

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:

1. Lydian (in our example, F)
2. Ionian (in our example, C)
3. Mixolydian (in our example, G)
4. Dorian (in our example, D)
5. Aeolian (in our example, A)
6. Phrygian (in our example, E)
7. 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:

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.

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Interval distribution analysis

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. 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.

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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
  Mode		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. (§9.2 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
  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 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.
• 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 (“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 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 pentatonic5t..
4. We didn’t delete this note; its mode, Dorian, becomes neutral pentatonic5t..
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 pentatonic5t..
1. We didn’t delete this note; its mode, Aeolian, becomes minor pentatonic5t..
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: