Musical Modes and the Circle of Fifths

Contents
Introduction
The Major Scale’s Modes & the Circle of Fifths
Beyond the Major Scale
Endnotes

Introduction

I’ve analyzed the familiar diatonic major scale (whole step, whole step, half step, whole step, whole step, whole step, half step, e.g., C-D-E-F-G-A-B-C), its seven modes, and their interrelationships. In brief, raising a specific pattern of notes in the scale results in cycling through not merely all twelve notes in the chromatic scale, but all eighty-four possible sets of base notes and modes, in a manner inextricably linked with the circle of fifths.

(Before we proceed: As always, please contact me if you notice any errors or omissions.)

A brief (lol, lmao) explanation is perhaps in order: The seven modes I’m analyzing are arrangements of the diatonic major scale’s notes, traditionally numbered by which one they use as their root key, or start.

Following this legend, I’ll list the C Ionian (major) scale’s seven modes.

I use thin, blue, lowercase letters for minor or diminished intervals a semitone below Ionian.
I use a bold, orange, uppercase letter for the augmented interval a semitone above Ionian.
- An augmented fourth and a diminished fifth are both tritones; they just use different notations.

The Seven Modes of the C Major Scale
Mode		1	2	3	4	5	6	7	Comment
1	Ionian	C	D	E	F	G	A	B	The traditional major scale.
2	Dorian	D	E	f	G	A	B	c
3	Phrygian	E	f	g	A	B	c	d
4	Lydian	F	G	A	B	C	D	E	The only mode that raises a note above Ionian.
5	Mixolydian	G	A	B	C	D	E	f
6	Aeolian	A	B	c	D	E	f	g	The natural minor scale.
7	Locrian	B	c	d	E	f	g	a	Rarely used due to its unsettling diminished root chord; most pieces that use it modulate out of it at times, creating a sense that we never truly arrive ‘home’.

These are just examples; it could also be something much better

Immediately recognizable examples of each mode include:

Ionian: The major scale is so ubiquitous I almost feel naming examples would insult readers’ intelligence, but I have to give special mention to one for its historical importance as a pedagogical tool:
- Richard Rodgers & Oscar Hammerstein II – “Do-Re-Mi” (The Sound of Music, 1959) is in B♭ Ionian. If this isn’t how you learned solfège, I’ll eat my hat. (Note: I am not currently wearing a hat.) In fact, you can probably recall this song’s melody to give yourself a vague idea how the first four modes sound:
  - Do = Ionian
  - Re = Dorian
  - Mi = Phrygian
  - Fa = Lydian
  (The lines for “So”, “La”, and “Ti” reuse Ionian, Dorian, and Phrygian respectively.)
Dorian:
- Michael Jackson – “Thriller” (Thriller, 1982) is in C♯ Dorian.
- Tears for Fears – “Mad World” (The Hurting, 1983) is in F♯ Dorian.
  - Gary Jules’ cover, originally from Trading Snakeoil for Wolftickets (2001) but best known for its use in Donnie Darko (2002), lowers it to F Dorian.
- Daft Punk ft. Pharrell Williams – “Get Lucky” (Random Access Memories, 2013) is in B Dorian.
Phrygian:
- Miles Davis & Gil Evans – “Solea” (Sketches of Spain, 1960) is in D Phrygian.
- Jefferson Airplane – “White Rabbit” (Surrealistic Pillow, 1967) is in F♯ Phrygian. Notably, this was directly inspired by Sketches of Spain, though there is some disagreement about whether it was “Solea” or “Concierto de Aranjuez (Adagio)” that was the specific inspiration.
- Metallica – “Wherever I May Roam” (Metallica, 1991) is in E Phrygian.
Lydian:
- Rush – “Freewill” (Permanent Waves, 1980) is mostly in F Lydian (the chorus is in D Ionian).
- The Police – “Every Little Thing She Does Is Magic” (Ghost in the Machine, 1981) has verses in G Lydian.
- Kōji Kondō – “Saria’s Song” (The Legend of Zelda: Ocarina of Time, 1998), the ocarina melody that opens the Lost Woods theme, is in F Lydian. It could be argued that the Lost Woods theme as a whole is mostly in C Ionian, but “Saria’s Song” specifically refers to the six-note ocarina melody the game teaches the player, which is unambiguously in F Lydian; in fact, I doubt a more immediately recognizable example of Lydian’s major third and augmented fourth exists in anything that qualifies as popular music. (The Lost Woods theme’s official name is, shockingly, “Lost Woods”.)
Mixolydian:
- Guns N’ Roses – “Sweet Child o’ Mine” (Appetite for Destruction, 1987) is in C♯ Mixolydian.
- Nirvana – “All Apologies” (In Utero, 1993) is also in C♯ Mixolydian.
Aeolian: Examples of natural minor are almost as ubiquitous as examples of Ionian, but here are two songs with chords that unambiguously emphasize the Aeolian scale’s characteristics:
- Phil Collins – “In the Air Tonight” (Face Value, 1981) is in D Aeolian.
- R.E.M. – “Losing My Religion” (Out of Time, 1991) is in A Aeolian.
Locrian:
- Bobby Prince – “At Doom’s Gate” (Doom, 1993) is primarily in E Locrian; it also harmonizes some of its chords over (in order) A Locrian, C♯ Locrian, and B Locrian scales (though it also throws in numerous accidentals to make its riffs sound even more dissonant). Mick Gordon’s remake for Doom (2016) contains a riff near the end that purely uses the D Locrian scale. (I was going to write that I couldn’t remember any well-known examples of this mode until I remembered this one.)
- Nobuo Uematsu – “The Mines of Narshe” (Final Fantasy VI, 1994) opens in B Locrian, though as is common with pieces in Locrian mode, it arguably modulates away from it for part of its running time. Nonetheless, it is an unambiguous example of Locrian, given that it only uses the B half-diminished 7th chord for about twelve measures straight and provides a good example of using the mode for a mysterious, film noir-type vibe that doesn’t feel dissonant. This is probably the least famous example on this list, but I felt it to be an especially good example of Locrian’s tonality since it contains numerous uses of F♮ and C♮, which emphasizes that the piece is, in fact, in B Locrian.

The Modes in “Circle of Fifths” Order

I won’t be analyzing the modes in their traditional order, since I’m be analyzing how lowering a regular pattern of notes by a half-step each enables us to walk through every mode on every key. A few notes (pun intended):

A♯ and B♭ represent the same pitch. Likewise B and C♭, C♯ and D♭, and so on. However, in terms of music theory, A♯ and B♭ can be semantically quite different in some contexts – be careful.
Key signatures with seven sharps or seven flats exist, but I’ve omitted them. (There are even monstrosities with double-sharps [𝄪] and double-flats [𝄫] … which I’ll mostly be ignoring.) I feel it was pretty extravagant just to include six-flat and six-sharp key signatures alongside each other.
If we start from Lydian mode, lowering the correct pitches one at a time by a half-step each gives us the next mode in the cycle. I’ll use C Lydian as an example to demonstrate the principle:

Modes Descending from Lydian⁽¹⁾
#	Root & mode		Pitch lowered	1	2	3	4	5	6	7	1
4	C	Lydian	–	C	D	E	F♯	G	A	B	C
1	C	Ionian	4	C	D	E	F	G	A	B	C
5	C	Mixolydian	7	C	D	E	F	G	A	B♭	C
2	C	Dorian	3	C	D	E♭	F	G	A	B♭	C
6	C	Aeolian	6	C	D	E♭	F	G	A♭	B♭	C
3	C	Phrygian	2	C	D♭	E♭	F	G	A♭	B♭	C
7	C	Locrian	5	C	D♭	E♭	F	G♭	A♭	B♭	C
4	C♭	Lydian	1	C♭	D♭	E♭	F	G♭	A♭	B♭	C♭

In other words:
- Lowering C Lydian’s F♯ to F gives us C Ionian (i.e., C major).
- Lowering C Ionian’s B to B♭ gives us C Mixolydian.
- And so on, until we reach C Locrian – whence the pattern repeats for C♭/B, the key below C on the chromatic scale.
Lowering Locrian’s root gives us Lydian mode in the key below it on the chromatic scale.
- In other words, lowering C Locrian’s C to C♭/B gives us C♭/B Lydian.
The sequence repeats from there:
- Lowering B Lydian’s E♯ gives us B Ionian.
- Lowering B Ionian’s A♯ gives us B Mixolydian.
This sequence will repeat 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.

The Principles of Inverse Operations

Now, note that we’ve effectively been subtracting 1 from a regular pattern of values in a sequence:
- 4, 7, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 1…
As a result, if we consult the mathematical properties of inverse operations, we can infer that 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:
- Raising C♭ Lydian’s C♭ gives us C Locrian.
- Raising C Locrian’s G♭ gives us C Phrygian.
- Raising C Phrygian’s D♭ gives us C Aeolian.
- And so on.

An Audio Demonstration

None of this will mean much to readers in isolation, so I put together a crude demonstration of the above principle 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 the following, in order:
1. the scale in ascending order
2. arpeggiated and block versions of:
  1. the root (I) chord
  2. the subdominant (IV) chord
  3. the dominant (V) chord
  4. the root (I) chord
(I could make a version going down the entire chromatic scale, but the entire thing would take thirteen and a half minutes to play through. Even I think that’s overkill, and I submit this exact page as exhibit A for what my definition of “overkill” must entail.)

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” above, the mode number and the pitch we lower to get it are exact mirrors of each other. This is not coincidental – it’s the direct result of a mathematical pattern.

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 that the third columns of “Diabolus in mūsicā” and of “Modes Descending from Lydian” are identical. This is also not coincidental.

The Major Scale’s Modes & the Circle of Fifths

12 Major Scales × 7 Modes = 84 Combinations

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). The differences between their numbers are always 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 may write a JavaScript add-on to reverse the order, but I’d have to learn JavaScript first, so no promises.
“RM” is an initialism for “Relative Major”, and “KS” is for “Key Signature”. A cheat sheet for what modes use what key signatures can be found in the section immediately following this one.
I’ve used zero-based indexing for these tables, so they’re indexed from zero to eighty-three. This is partly because I prefer zero-based indexing, but to be honest, it was mostly to give C Ionian an index of one.

C (B♯)
#	Root	Mode	RM	KS	1	2	3	4	5	6	7	1
0	C♮	4 – Lydian	G	1♯	C♮	D♮	E♮	F♯	G♮	A♮	B♮	C♮
1	C♮	1 – Ionian	C	♮	C♮	D♮	E♮	F♮	G♮	A♮	B♮	C♮
2	C♮	5 – Mixolydian	F	1♭	C♮	D♮	E♮	F♮	G♮	A♮	B♭	C♮
3	C♮	2 – Dorian	B♭	2♭	C♮	D♮	E♭	F♮	G♮	A♮	B♭	C♮
4	C♮	6 – Aeolian	E♭	3♭	C♮	D♮	E♭	F♮	G♮	A♭	B♭	C♮
5	C♮	3 – Phrygian	A♭	4♭	C♮	D♭	E♭	F♮	G♮	A♭	B♭	C♮
6	C♮	7 – Locrian	D♭	5♭	C♮	D♭	E♭	F♮	G♭	A♭	B♭	C♮

B (C♭)
#	Root	Mode	RM	KS	1	2	3	4	5	6	7	1
7	C♭	4 – Lydian	G♭	6♭	C♭	D♭	E♭	F♮	G♭	A♭	B♭	C♭
7	B♮	4 – Lydian	F♯	6♯	B♮	C♯	D♯	E♯	F♯	G♯	A♯	B♮
8	B♮	1 – Ionian	B	5♯	B♮	C♯	D♯	E♮	F♯	G♯	A♯	B♮
9	B♮	5 – Mixolydian	E	4♯	B♮	C♯	D♯	E♮	F♯	G♯	A♮	B♮
10	B♮	2 – Dorian	A	3♯	B♮	C♯	D♮	E♮	F♯	G♯	A♮	B♮
11	B♮	6 – Aeolian	D	2♯	B♮	C♯	D♮	E♮	F♯	G♮	A♮	B♮
12	B♮	3 – Phrygian	G	1♯	B♮	C♮	D♮	E♮	F♯	G♮	A♮	B♮
13	B♮	7 – Locrian	C	♮	B♮	C♮	D♮	E♮	F♮	G♮	A♮	B♮

A♯ / B♭
#	Root	Mode	RM	KS	1	2	3	4	5	6	7	1
14	B♭	4 – Lydian	F	1♭	B♭	C♮	D♮	E♮	F♮	G♮	A♮	B♭
15	B♭	1 – Ionian	B♭	2♭	B♭	C♮	D♮	E♭	F♮	G♮	A♮	B♭
16	B♭	5 – Mixolydian	E♭	3♭	B♭	C♮	D♮	E♭	F♮	G♮	A♭	B♭
17	B♭	2 – Dorian	A♭	4♭	B♭	C♮	D♭	E♭	F♮	G♮	A♭	B♭
18	B♭	6 – Aeolian	E♭	5♭	B♭	C♮	D♭	E♭	F♮	G♭	A♭	B♭
19	B♭	3 – Phrygian	G♭	6♭	B♭	C♭	D♭	E♭	F♮	G♭	A♭	B♭
19	A♯	3 – Phrygian	F♯	6♯	A♯	B♮	C♯	D♯	E♯	F♯	G♯	A♯
20	A♯	7 – Locrian	B	5♯	A♯	B♮	C♯	D♯	E♮	F♯	G♯	A♯

A
#	Root	Mode	RM	KS	1	2	3	4	5	6	7	1
21	A♮	4 – Lydian	E	4♯	A♮	B♮	C♯	D♯	E♮	F♯	G♯	A♮
22	A♮	1 – Ionian	A	3♯	A♮	B♮	C♯	D♮	E♮	F♯	G♯	A♮
23	A♮	5 – Mixolydian	D	2♯	A♮	B♮	C♯	D♮	E♮	F♯	G♮	A♮
24	A♮	2 – Dorian	G	1♯	A♮	B♮	C♮	D♮	E♮	F♯	G♮	A♮
25	A♮	6 – Aeolian	C	♮	A♮	B♮	C♮	D♮	E♮	F♮	G♮	A♮
26	A♮	3 – Phrygian	F	1♭	A♮	B♭	C♮	D♮	E♮	F♮	G♮	A♮
27	A♮	7 – Locrian	B♭	2♭	A♮	B♭	C♮	D♮	E♭	F♮	G♮	A♮

G♯ / A♭
#	Root	Mode	RM	KS	1	2	3	4	5	6	7	1
28	A♭	4 – Lydian	E♭	3♭	A♭	B♭	C♮	D♮	E♭	F♮	G♮	A♭
29	A♭	1 – Ionian	A♭	4♭	A♭	B♭	C♮	D♭	E♭	F♮	G♮	A♭
30	A♭	5 – Mixolydian	D♭	5♭	A♭	B♭	C♮	D♭	E♭	F♮	G♭	A♭
31	A♭	2 – Dorian	G♭	6♭	A♭	B♭	C♭	D♭	E♭	F♮	G♭	A♭
31	G♯	2 – Dorian	F♯	6♯	G♯	A♯	B♮	C♯	D♯	E♯	F♯	G♯
32	G♯	6 – Aeolian	B	5♯	G♯	A♯	B♮	C♯	D♯	E♮	F♯	G♯
33	G♯	3 – Phrygian	E	4♯	G♯	A♮	B♮	C♯	D♯	E♮	F♯	G♯
34	G♯	7 – Locrian	A	3♯	G♯	A♮	B♮	C♯	D♮	E♮	F♯	G♯

G
#	Root	Mode	RM	KS	1	2	3	4	5	6	7	1
35	G♮	4 – Lydian	D	2♯	G♮	A♮	B♮	C♯	D♮	E♮	F♯	G♮
36	G♮	1 – Ionian	G	1♯	G♮	A♮	B♮	C♮	D♮	E♮	F♯	G♮
37	G♮	5 – Mixolydian	C	♮	G♮	A♮	B♮	C♮	D♮	E♮	F♮	G♮
38	G♮	2 – Dorian	F	1♭	G♮	A♮	B♭	C♮	D♮	E♮	F♮	G♮
39	G♮	6 – Aeolian	B♭	2♭	G♮	A♮	B♭	C♮	D♮	E♭	F♮	G♮
40	G♮	3 – Phrygian	E♭	3♭	G♮	A♭	B♭	C♮	D♮	E♭	F♮	G♮
41	G♮	7 – Locrian	A♭	4♭	G♮	A♭	B♭	C♮	D♭	E♭	F♮	G♮

F♯ / G♭
#	Root	Mode	RM	KS	1	2	3	4	5	6	7	1
42	G♭	4 – Lydian	D♭	5♭	G♭	A♭	B♭	C♮	D♭	E♭	F♮	G♭
43	G♭	1 – Ionian	G♭	6♭	G♭	A♭	B♭	C♭	D♭	E♭	F♮	G♭
43	F♯	1 – Ionian	F♯	6♯	F♯	G♯	A♯	B♮	C♯	D♯	E♯	F♯
44	F♯	5 – Mixolydian	B	5♯	F♯	G♯	A♯	B♮	C♯	D♯	E♮	F♯
45	F♯	2 – Dorian	E	4♯	F♯	G♯	A♮	B♮	C♯	D♯	E♮	F♯
46	F♯	6 – Aeolian	A	3♯	F♯	G♯	A♮	B♮	C♯	D♮	E♮	F♯
47	F♯	3 – Phrygian	D	2♯	F♯	G♮	A♮	B♮	C♯	D♮	E♮	F♯
48	F♯	7 – Locrian	G	1♯	F♯	G♮	A♮	B♮	C♮	D♮	E♮	F♯

F (E♯)
#	Root	Mode	RM	KS	1	2	3	4	5	6	7	1
49	F♮	4 – Lydian	C	♮	F♮	G♮	A♮	B♮	C♮	D♮	E♮	F♮
50	F♮	1 – Ionian	F	1♭	F♮	G♮	A♮	B♭	C♮	D♮	E♮	F♮
51	F♮	5 – Mixolydian	B♭	2♭	F♮	G♮	A♮	B♭	C♮	D♮	E♭	F♮
52	F♮	2 – Dorian	E♭	3♭	F♮	G♮	A♭	B♭	C♮	D♮	E♭	F♮
53	F♮	6 – Aeolian	A♭	4♭	F♮	G♮	A♭	B♭	C♮	D♭	E♭	F♮
54	F♮	3 – Phrygian	D♭	5♭	F♮	G♭	A♭	B♭	C♮	D♭	E♭	F♮
55	F♮	7 – Locrian	G♭	6♭	F♮	G♭	A♭	B♭	C♭	D♭	E♭	F♮
55	E♯	7 – Locrian	F♯	6♯	E♯	F♯	G♯	A♯	B♮	C♯	D♯	E♯

E (F♭)
#	Root	Mode	RM	KS	1	2	3	4	5	6	7	1
56	E♮	4 – Lydian	B	5♯	E♮	F♯	G♯	A♯	B♮	C♯	D♯	E♮
57	E♮	1 – Ionian	E	4♯	E♮	F♯	G♯	A♮	B♮	C♯	D♯	E♮
58	E♮	5 – Mixolydian	A	3♯	E♮	F♯	G♯	A♮	B♮	C♯	D♮	E♮
59	E♮	2 – Dorian	D	2♯	E♮	F♯	G♮	A♮	B♮	C♯	D♮	E♮
60	E♮	6 – Aeolian	G	1♯	E♮	F♯	G♮	A♮	B♮	C♮	D♮	E♮
61	E♮	3 – Phrygian	C	♮	E♮	F♮	G♮	A♮	B♮	C♮	D♮	E♮
62	E♮	7 – Locrian	F	1♭	E♮	F♮	G♮	A♮	B♭	C♮	D♮	E♮

D♯ / E♭
#	Root	Mode	RM	KS	1	2	3	4	5	6	7	1
63	E♭	4 – Lydian	B♭	2♭	E♭	F♮	G♮	A♮	B♭	C♮	D♮	E♭
64	E♭	1 – Ionian	E♭	3♭	E♭	F♮	G♮	A♭	B♭	C♮	D♮	E♭
65	E♭	5 – Mixolydian	A♭	4♭	E♭	F♮	G♮	A♭	B♭	C♮	D♭	E♭
66	E♭	2 – Dorian	D♭	5♭	E♭	F♮	G♭	A♭	B♭	C♮	D♭	E♭
67	E♭	6 – Aeolian	G♭	6♭	E♭	F♮	G♭	A♭	B♭	C♭	D♭	E♭
67	D♯	6 – Aeolian	F♯	6♯	D♯	E♯	F♯	G♯	A♯	B♮	C♯	D♯
68	D♯	3 – Phrygian	B	5♯	D♯	E♮	F♯	G♯	A♯	B♮	C♯	D♯
69	D♯	7 – Locrian	E	4♯	D♯	E♮	F♯	G♯	A♮	B♮	C♯	D♯

D
#	Root	Mode	RM	KS	1	2	3	4	5	6	7	1
70	D♮	4 – Lydian	A	3♯	D♮	E♮	F♯	G♯	A♮	B♮	C♯	D♮
71	D♮	1 – Ionian	D	2♯	D♮	E♮	F♯	G♮	A♮	B♮	C♯	D♮
72	D♮	5 – Mixolydian	G	1♯	D♮	E♮	F♯	G♮	A♮	B♮	C♮	D♮
73	D♮	2 – Dorian	C	♮	D♮	E♮	F♮	G♮	A♮	B♮	C♮	D♮
74	D♮	6 – Aeolian	F	1♭	D♮	E♮	F♮	G♮	A♮	B♭	C♮	D♮
75	D♮	3 – Phrygian	B♭	2♭	D♮	E♭	F♮	G♮	A♮	B♭	C♮	D♮
76	D♮	7 – Locrian	E♭	3♭	D♮	E♭	F♮	G♮	A♭	B♭	C♮	D♮

C♯ / D♭
#	Root	Mode	RM	KS	1	2	3	4	5	6	7	1
77	D♭	4 – Lydian	A♭	4♭	D♭	E♭	F♮	G♮	A♭	B♭	C♮	D♭
78	D♭	1 – Ionian	D♭	5♭	D♭	E♭	F♮	G♭	A♭	B♭	C♮	D♭
79	D♭	5 – Mixolydian	G♭	6♭	D♭	E♭	F♮	G♭	A♭	B♭	C♭	D♭
79	C♯	5 – Mixolydian	F♯	6♯	C♯	D♯	E♯	F♯	G♯	A♯	B♮	C♯
80	C♯	2 – Dorian	B	5♯	C♯	D♯	E♮	F♯	G♯	A♯	B♮	C♯
81	C♯	6 – Aeolian	E	4♯	C♯	D♯	E♮	F♯	G♯	A♮	B♮	C♯
82	C♯	3 – Phrygian	A	3♯	C♯	D♮	E♮	F♯	G♯	A♮	B♮	C♯
83	C♯	7 – Locrian	D	2♯	C♯	D♮	E♮	F♯	G♮	A♮	B♮	C♯

Key Signature Cheat Sheet

The twelve-note chromatic scale has fifteen key signatures (𝄪 and 𝄫 monstrosities notwithstanding):

Key Signatures of the Seven Modes for the Twelve-Note Chromatic Scale
Lyd	Maj	Mix	Dor	Min	Phr	Loc	KS	A	B	C	D	E	F	G
F♯	C♯	D♯	D♯	A♯	E♯	B♯	7♯	♯	♯	♯	♯	♯	♯	♯
B	F♯	C♯	G♯	D♯	A♯	E♯	6♯	♯		♯	♯	♯	♯	♯
E	B	F♯	C♯	G♯	D♯	A♯	5♯	♯		♯	♯		♯	♯
A	E	B	F♯	C♯	G♯	D♯	4♯			♯	♯		♯	♯
D	A	E	B	F♯	C♯	G♯	3♯			♯			♯	♯
G	D	A	E	B	F♯	C♯	2♯			♯			♯
C	G	D	A	E	B	F♯	1♯						♯
F	C	G	D	A	E	B	♮
B♭	F	C	G	D	A	E	1♭		♭
E♭	B♭	F	C	G	D	A	2♭		♭			♭
A♭	E♭	B♭	F	C	G	D	3♭	♭	♭			♭
D♭	A♭	E♭	B♭	F	C	G	4♭	♭	♭		♭	♭
G♭	D♭	A♭	E♭	B♭	F	C	5♭	♭	♭		♭	♭		♭
C♭	G♭	D♭	A♭	E♭	B♭	F	6♭	♭	♭	♭	♭	♭		♭
F♭	C♭	G♭	D♭	A♭	E♭	B♭	7♭	♭	♭	♭	♭	♭	♭	♭

Legend for the above:
- Lyd = Lydian
- Maj = major (Ionian)
- Mix = Mixolydian
- Dor = Dorian (if you’re not sure why it’s grey, ask your English teacher about the picture)
- Min = natural minor (Aeolian)
- Phr = Phrygian
- Loc = Locrian
- KS = key signature (highlighted because it’s the key to the table – pun coincidental, though I definitely didn’t even try to avoid it)
Additional notes:
- 5♯ and 7♭ express the same pitches.
- 6♯ and 6♭ express the same pitches.
- 7♯ and 5♭ express the same pitches.
- As in the tables above, I’ve printed C major’s modes in blue and F♯/G♭ major’s modes in orange.
Accidentals also fall onto the circle of fifths (e.g., B♭ to E♭ and E♭ to A♭ are both perfect fifths).

Why is this happening?

The simple answer: whether we realized it or not, we’ve been traversing the circle of fifths this entire time. In the introduction, I mentioned that traveling from C Lydian to C Ionian was, in a sense, traveling from G major to C major. Here’s the C table again. Note how “relative major” traverses the circle of fifths downward:

C++
#	Root	Mode	RM	KS	1	2	3	4	5	6	7	1
0	C♮	4 – Lydian	G	1♯	C♮	D♮	E♮	F♯	G♮	A♮	B♮	C♮
1	C♮	1 – Ionian	C	♮	C♮	D♮	E♮	F♮	G♮	A♮	B♮	C♮
2	C♮	5 – Mixolydian	F	1♭	C♮	D♮	E♮	F♮	G♮	A♮	B♭	C♮
3	C♮	2 – Dorian	B♭	2♭	C♮	D♮	E♭	F♮	G♮	A♮	B♭	C♮
4	C♮	6 – Aeolian	E♭	3♭	C♮	D♮	E♭	F♮	G♮	A♭	B♭	C♮
5	C♮	3 – Phrygian	A♭	4♭	C♮	D♭	E♭	F♮	G♮	A♭	B♭	C♮
6	C♮	7 – Locrian	D♭	5♭	C♮	D♭	E♭	F♮	G♭	A♭	B♭	C♮

The explanation is that each of these modes, apart from C Ionian, has been rearranging a different major scale. Reshuffling each mode back into its Ionian form may explain the cause:

You were expecting modes, but it was me, ~~Dio~~ the circle of fifths!
#	Original Mode	Root	KS	1	2	3	4	5	6	7	1
0	4 – Lydian	G	1♯	G♮	A♮	B♮	C♮	D♮	E♮	F♯	G♮
1	1 – Ionian	C	♮	C♮	D♮	E♮	F♮	G♮	A♮	B♮	C♮
2	5 – Mixolydian	F	1♭	F♮	G♮	A♮	B♭	C♮	D♮	E♮	F♮
3	2 – Dorian	B♭	2♭	B♭	C♮	D♮	E♭	F♮	G♮	A♮	B♭
4	6 – Aeolian	E♭	3♭	E♭	F♮	G♮	A♭	B♭	C♮	D♮	E♭
5	3 – Phrygian	A♭	4♭	A♭	B♭	C♮	D♭	E♭	F♮	G♮	A♭
6	7 – Locrian	D♭	5♭	D♭	E♭	F♮	G♭	A♭	B♭	C♮	D♭

Every note of every scale in this table is, in fact, a perfect fifth below its counterpart in its predecessor. Almost every other pattern we’ve observed that follows the circle of fifths in some way is a direct consequence of this.

Chord Analysis by Mode

Chord Tonalities by Scale Position & Mode
Mode		I	II	III	IV	V	VI	VII
1	Ionian	Maj	min	min	Maj	Maj	min	dim
2	Dorian	min	min	Maj	Maj	min	dim	Maj
3	Phrygian	min	Maj	Maj	min	dim	Maj	min
4	Lydian	Maj	Maj	min	dim	Maj	min	min
5	Mixolydian	Maj	min	dim	Maj	min	min	Maj
6	Aeolian	min	dim	Maj	min	min	Maj	Maj
7	Locrian	dim	Maj	min	min	Maj	Maj	min

I’ve printed the root (I), dominant (V), and subdominant (IV) chords of each mode in bold – they are especially important to the mode’s tonality.
I’ve printed the seventh chord in lighter text because it’s diminished in the major scale.
Minor chords are blue; diminished chords are orange.
Printing the chords this way makes it clearer why Ionian became the standard major scale and Aeolian became the natural minor scale: the root, dominant, and subdominant are all major in Ionian mode and all minor in Aeolian mode. 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.

Beyond the Major Scale

Other Scales & Tonalities

Although this page focuses on modes of the major scale, numerous possible scales (and modes thereof) don’t fit its pattern, such as:

The twelve-note chromatic scale (though works that use the entire twelve-note scale for non-colorative purposes are quite rare and are mostly considered avant-garde).

Numerous seven-note scales that aren’t modes of the major scale. I’ve rooted the following examples (many of which are common in jazz) in C:

Other Seven-Note Scales
Scale		1	2	3	4	5	6	7
Ascending melodic minor		C♮	D	E♭	F	G	A	B
Dorian ♭2	Phrygian ♯6	C♮	D♭	E♭	F	G	A	B♭
Lydian augmented		C♮	D	E	F♯	G♯	A	B
Lydian dominant	Mixolydian ♯4	C♮	D	E	F♯	G	A	B♭
Aeolian dominant	Mixolydian ♭6	C♮	D	E	F	G	A♭	B♭
Aeolian ♭5	Locrian ♯2	C♮	D	E♭	F	G♭	A♭	B♭
Super-Locrian	Altered dominant	C♮	D♭	E♭	F♭	G♭	A♭	B♭
Harmonic minor	Aeolian ♯7	C♮	D	E♭	F	G	A♭	B
Locrian ♮6		C♮	D♭	E♭	F	G♭	A	B♭
Augmented major	Ionian ♯5	C♮	D	E	F	G♯	A	B
Hungarian minor		C♮	D	E♭	F♯	G	A♭	B
Ukrainian Dorian		C♮	D	E♭	F♯	G	A	B♭
Phrygian dominant		C♮	D♭	E	F	G	A♭	B♭
Lydian ♯2		C♮	D♯	E	F♯	G	A	B
Altered diminished	Super-Locrian 𝄫2	C♮	D♭	E♭	F♭	G♭	A♭	B𝄫

I’ve again printed augmented intervals in bold, orange text and minor or diminished intervals in thin, blue text. Note that this list is far from exhaustive; it doesn’t even delve into scales with different ascents and descents. (I also didn’t list every name of some scales on this list; some have at least four.)

However, one thing distinguishes these scales from modes of diatonic scales: we can print any mode of any diatonic scale with no accidentals outside the key signature, but every single scale in the above table would require accidentals for at least some notes. Key signatures are strictly based on major scales and the circle of fifths; using three flats that weren’t B♭, E♭, and A♭ in a key signature would just confuse readers.

There are also, naturally, scales that use fewer than seven notes; pentatonic (five-note) scales are quite common in rock, blues, and jazz music, though they’re much older than that and exist in many cultures. Six-note scales are also fairly common (including whole-tone scales).
Scales may also have more than seven notes; eight- and nine-note scales are common in blues and jazz.
And of course, all of the above is based on the twelve-tone equal temperament used in the vast majority of Western music from the last five centuries. Plenty of other tunings have been and still are used, though; there’s nothing even restricting the number of notes per octave to twelve. For instance:
- A twenty-four-note scale is common in Arabic and Turkic music.
- Indian rāgas ⁽²⁾ typically use anywhere from four to seven svaras⁽³⁾ (roughly comparable to the notes of a Western diatonic scale) selected from the twenty-two śrútis⁽⁴⁾ (roughly comparable to the Western twelve-note chromatic scale, except with ten more notes).
- Experimental Western musicians also occasionally use alternate temperaments. Here are a few recent rock and metal examples, in increasing order of heaviness:
  - Flying Microtonal Banana by King Gizzard & the Lizard Wizard (24-tone scale)
  - Nowherer by Victory over the Sun (17-tone scale)
  - any metal album by Jute Gyte since Discontinuities (also 24-tone scale)
- The Ancient Greek enharmonic genus, as we shall see shortly, also used microtonality.

Etymology

The names of all seven modes refer to regions in or near ancient Greece, ancient Greek tribes, or both:

Αἱ ἐτῠμολογῐ́αι τῶν ἑπτᾰ́ τόνων (Hai etumologíai tô heptá tónōn) [The Seven Modes’ Etymologies]
#	Mode	Greek	Romanized	Reference
1	Ionian	Ἰωνία	Iōnía	region on the western coast of Anatolia (modern Turkey)
2	Dorian	Δωρῐεύς	Dōrieús	one of the four major Hellenic tribes
3	Phrygian	Φρῠγῐ́ᾱ	Phrugíā	kingdom in west-central Anatolia
4	Lydian	Λῡδῐ́ᾱ	Lūdíā	Anatolian kingdom most famously ruled by Croesus
5	Mixolydian	μιξο-Λῡ́δῐος	mixo-Lū́dios	literally “mixed Lydian”
6	Aeolian	Αἰολῐ́ᾱ	Aiolíā	region of northwestern Anatolia
7	Locrian	Λοκρῐ́ς	Lokrís	Three separate regions of ancient Greece

However, they really don’t have anything to do with the regions or tribes they were named after, or even the ancient Greek tonoi (singular: tonos) that in many cases shared their names; it was more a case of “medieval Europeans thought it’d be cool to name their modes after aspects of Ancient Greece.”

(Tonos comes from the ancient Greek ὁ τόνος, ho tónos, meaning cord, chord⁽⁵⁾, note, tone, or tension; its dual nominative form was τὼ τόνω, tṑ tónō, and its plural nominative form was οἱ τόνοι, hoi tónoi.)

As for the word diatonic, that comes from the Ancient Greek διατονικός (diatonikós), literally meaning two tones, in reference to the diatonic tetrachord, which… well, keep reading.

Ancient Greek Tonoi & Modern Modes

This is an oversimplification by necessity, as ancient Greek music theory wasn’t unified;⁽⁶⁾ Philolaus (Φιλόλαος, Philólaos), Archytas (Ἀρχύτας), Aristoxenus (Ἀριστόξενος ὁ Ταραντῖνος, Aristóxenos ho Tarantínos), Ptolemy (Πτολεμαῖος, Ptolemaios), and others had quite different conceptions of it. Wikipedia has much, much more on the subject; the Xenharmonic wiki, Midicode, and Feel Your Sound have also been helpful. I consulted several other resources researching this section, but most were too technical to be of interest to non-specialists.

I should first clarify that very little music of ancient Greece actually survives to this day; the earliest known piece that can reasonably be described as complete is the Seikilos epitaph from the first or second century CE, and while we have an idea what its melody sounded like, how it would have been harmonized is a matter of conjecture. (Older pieces still survive in fragments, such as the Hurrian songs, one of which is nearly complete.) Greek authors actually provided fairly complete descriptions of tuning practices, and in at least Archytas’ case, modern scholars believe he was describing the actual practices of his day, but we don’t really know what Greek music sounded like; on some level, we’re taking the word of contemporary authors, only some of whom seem to have been especially concerned with describing actual musical practice.⁽⁷⁾

I’ll be using numbers to represent the intervals of ancient Greek harmony within 24-tone equal temperament (24-TET), which adds an additional 12 notes exactly halfway between each note of the familiar 12-note chromatic scale. In 24-TET, an exact ratio of ²⁴√2:1 determines the spacing of the smallest interval (known as a quarter-tone, downminor second, infra second, or wide unison), thus:

Interval Key
#	Interval	Tone	Exact				Approximate
¼	Infra second	Quarter-tone	²⁴√2	:	1	≈	1.02930223664
½	Minor second	Semitone	¹²√2	:	1	≈	1.05946309436
1	Major second	Whole tone	⁶√2	:	1	≈	1.12246204831
1½	Minor third	Three semitones	⁴√2	:	1	≈	1.18920711500
2	Major third	Two whole tones	³√2	:	1	≈	1.25992104989

24-TET is by necessity an oversimplification as well, not least since different authors defined the same interval differently. For instance, here are how Philolaus, Archytas, and 24-TET would define the intervals within a diatonic tetrachord. (A tetrachord is a major element of Greek harmony consisting of four notes. Also, I’ve taken the liberty of reversing what they called the first and third intervals – ancient Greek harmony defined scales in descending order, where we use ascending order.)

Interval Ratios of a Diatonic Tetrachord
Source	Low interval					Middle interval					High interval
Philolaus	256	:	243	≈	1.05349794239	9	:	8	=	1.125	9	:	8	=	1.125
Archytas	28	:	27	=	1.037037037…	8	:	7	=	1.142857142857…	9	:	8	=	1.125
24-TET	¹²√2	:	1	≈	1.05946309436	⁶√2	:	1	≈	1.12246204831	⁶√2	:	1	≈	1.12246204831

Thus, representing ancient Greek harmony using 24-TET is a substantial oversimplification, and in point of fact, we’d get closer to Archytas’ definitions in 24-TET by using a quarter tone for the low interval and 1¼-tone (that is, the interval in between a major second and a minor third, sometimes called an infra third or ultra second) for the middle interval. However, the intervals I’ve selected (a minor second and two major seconds) are closer to Philolaus’ definitions and, crucially, can be represented in modern Western music’s 12-TET.

As one further example, both Philolaus and Archytas define the ratio of a diatonic tetrachord’s high and low notes as 4:3 (1.33333…), which 24-TET would represent using a perfect fourth (≈1.33483985417). Most people’s ears are insufficiently trained to tell the difference – even trained singers with perfect pitch frequently wander off-key by larger differences than that (at least when they’re not subjected to pitch correction⁽⁸⁾).

The same ancient Greek tonos could actually use any of three genera (singular: genus). Each tonos contained two tetrachords, or sets of four notes, whose spacing differed depending on the genus; a major second would complete the octave (though in some tonoi, it occurred between the tetrachords, and in Hypodorian and Mixolydian, it occurred respectively before and after them; additionally, several tonoi split one tetrachord). All three genera used one interval once per tetrachord and another interval twice, in the following order:

Interval Genera: A Feed from Cloud Mountain
Genus	Low interval	Middle interval	High interval
Enharmonic	Infra second	Infra second	Major third
Chromatic	Minor second	Minor second	Minor third
Diatonic	Minor second	Major second	Major second

Each tonos contained two tetrachords, with what effectively reduces to a major second to complete the octave. A few notes:

As stated above, ancient Greek harmony defined tonoi in descending order; I’ve listed them in our more familiar ascending order to keep them consistent with the other scales on this page.
I’ve printed intervals rather than notes because the notation for microtones is extremely confusing if you’re not already familiar with it. Hopefully, the intervals are slightly easier to understand.
I’ve placed borders between tetrachords (again, four tonoi split one of their tetrachords) and around the extra whole step (whose background I’ve also highlighted).
I’ve printed the modern modes with blue text and brighter borders, with a thicker border below since the next tonos will not be related.
I’ve printed the tonoi in the order ancient Greek harmony assigned them: Mixolydian (modern Locrian’s equivalent) first, Hypodorian (modern Aeolian’s equivalent) last.
Apparently, our Dorian, Phrygian, Lydian, and Mixolydian modes don’t match the eponymous diatonic tonoi because medieval Europeans erroneously believed the ancient Greeks described tonoi in ascending order. (Swiss poet Heinrich Glarean appears to have popularized the misconception.)

Here are all three genera of all seven tonoi, followed by the diatonic genus’ modern equivalent:

Approximate Intervals of Ancient Greek Tonoi & Modern Diatonic Modes
Tonos	Genus	1–2	2–3	3–4	4–5	5–6	6–7	7–8
Mixolydian	Enharmonic	¼	¼	2	¼	¼	2	1
Mixolydian	Chromatic	½	½	1½	½	½	1½	1
Mixolydian	Diatonic	½	1	1	½	1	1	1
Locrian	Modern	½	1	1	½	1	1	1
Lydian	Enharmonic	¼	2	¼	¼	2	1	¼
Lydian	Chromatic	½	1½	½	½	1½	1	½
Lydian	Diatonic	1	1	½	1	1	1	½
Ionian	Modern	1	1	½	1	1	1	½
Phrygian	Enharmonic	2	¼	¼	2	1	¼	¼
Phrygian	Chromatic	1½	½	½	1½	1	½	½
Phrygian	Diatonic	1	½	1	1	1	½	1
Dorian	Modern	1	½	1	1	1	½	1
Dorian	Enharmonic	¼	¼	2	1	¼	¼	2
Dorian	Chromatic	½	½	1½	1	½	½	1½
Dorian	Diatonic	½	1	1	1	½	1	1
Phrygian	Modern	½	1	1	1	½	1	1
Hypolydian	Enharmonic	¼	2	1	¼	¼	2	¼
Hypolydian	Chromatic	½	1½	1	½	½	1½	½
Hypolydian	Diatonic	1	1	1	½	1	1	½
Lydian	Modern	1	1	1	½	1	1	½
Hypophrygian	Enharmonic	2	1	¼	¼	2	¼	¼
Hypophrygian	Chromatic	1½	1	½	½	1½	½	½
Hypophrygian	Diatonic	1	1	½	1	1	½	1
Mixolydian	Modern	1	1	½	1	1	½	1
Hypodorian	Enharmonic	1	¼	¼	2	¼	¼	2
Hypodorian	Chromatic	1	½	½	1½	½	½	1½
Hypodorian	Diatonic	1	½	1	1	½	1	1
Aeolian	Modern	1	½	1	1	½	1	1

Acknowledgements to Marty O’Donnell – yes, that Marty O’Donnell – for pointing out that it wasn’t sufficiently clear what parts of this section I was oversimplifying; his feedback resulted in a much clearer explanation. However, the opinions presented in this section are entirely my own (Marty believes modern scholars have extrapolated more from ancient Greek texts than is merited by their contents).

Those seeking more detailed technical analysis of ancient Greek tuning systems may find Robert Erickson’s analysis of Archytas (who provided a detailed and apparently accurate description of what modern scholars believe to have been the actual musical practices of his era) to be of interest. Our knowledge of Archytas’ musical writings apparently comes from Ptolemy’s Harmonics, whose author comments in depth on the former’s writings⁽⁷⁾; large fragments of Aristoxenus’ Elements of Harmony and smaller fragments of Philolaus’ musical writings survive to this day. Here, via Cris Forester’s book on the subject, is an excerpt from Philolaus, whom I quote less for his comprehensibility than for his technical detail:

The magnitude of harmonia is syllaba and di’oxeian. The di’oxeian is greater than the syllaba in epogdoic ratio. From hypate [E] to mese [A] is a syllaba, from mese [A] to neate [or nete, E¹] is a di’oxeian, from neate [E¹] to trite [later paramese, B] is a syllaba, and from trite [B] to hypate [E] is a di’oxeian. The interval between trite [B] and mese [A] is epogdoic [9:8], the syllaba is epitritic [4:3], the di’oxeian hemiolic [3:2], and the dia pason is duple [2:1]. Thus harmonia consists of five epogdoics and two dieses; di’oxeian is three epogdoics and a diesis, and syllaba is two epogdoics and a diesis. [Text and ratios in brackets are Cris Forester’s.]
–Philolaus, translated by Andrew Barker, Greek Musical Writings, Vol. 2 (1989: Cambridge University Press).

Difficult as this is to parse, a close reading reveals that Philolaus has just described the diatonic major scale:

3:2 is 1.5; our perfect fifth rounds to 1.49830707688.
4:3 is 1.333…; our perfect fourth rounds to 1.33483985417.
9:8 is 1.125; our whole step rounds to 1.12246204831.
Philolaus’ dieses are therefore 256:243, or 1.05349794239; our half step rounds to 1.05946309436.

In other words:

A dia pason (or harmonia) is exactly an octave.
A di’oxean is almost exactly a perfect fifth.
A syllaba is almost exactly a perfect fourth.
An epogdoic is almost exactly a whole step.
A diesis is almost exactly a half-step.

Plugging those in gives us:

The magnitude of an octave is a perfect fourth and a perfect fifth. The perfect fifth is greater than the perfect fourth in whole-step ratio. From hypate [E] to mese [A] is a perfect fourth, from mese [A] to neate [or nete, E¹] is a perfect fifth, from neate [E¹] to trite [later paramese, B] is a perfect fourth, and from trite [B] to hypate [E] is a perfect fifth. The interval between trite [B] and mese [A] is a whole step [9:8], the perfect fourth is epitritic [4:3], the perfect fifth hemiolic [3:2], and the octave is duple [2:1]. Thus an octave consists of five whole steps and two half-steps; a perfect fifth is three whole steps and a half-step, and a perfect fourth is two whole steps and a half-step.

Plus ça change, plus c’est la même chose.

In closing, I should reiterate that “ancient Greek harmony” was not uniform; various authors described other tonoi and indeed entirely different tuning systems. I’ve focused on the tonoi described above because they were the clear inspiration for the modern modes’ names.

(Keep an eye on this page – I intend to add information on the medieval church modes that served as the precursors to our modern modes someday.)

Interval Ratios of 12- and 24-Tone Equal Temperament

As an appendix to the section on tonoi, I’ve also created this table of every possible interval in 24-tone equal temperament. The column “LPT” means “Lowest Possible Temperament” – in other words, to contain an interval, a temperament must be a multiple of its LPT; e.g., if the LPT is 8, the interval will appear in 16-TET, 24-TET, 32-TET, and so on, but will not appear in 12-TET. The lower the LPT, the bolder the font used to print the interval. Intervals printed in blue also appear in 12-TET (our familiar 12-note chromatic scale).

24-Tone Equal Temperament’s Interval Ratios
#	Interval	Exact				Approximate	LPT
1	Quarter tone, infra second	2^¹⁄₂₄	=	^²⁴√	2	1.02930223664	24
2	Minor second	2^²⁄₂₄	=	^¹²√	2	1.05946309436	12
3	Neutral second	2^³⁄₂₄	=	^⁸√	2	1.09050773267	8
4	Major second	2^⁴⁄₂₄	=	^⁶√	2	1.12246204831	6
5	Ultra second, infra third	2^⁵⁄₂₄	=	^²⁴√	32	1.15535269687	24
6	Minor third	2^⁶⁄₂₄	=	^⁴√	2	1.18920711500	4
7	Neutral third	2^⁷⁄₂₄	=	^²⁴√	128	1.22405354330	24
8	Major third	2^⁸⁄₂₄	=	^³√	2	1.25992104989	3
9	Ultra third, narrow fourth	2^⁹⁄₂₄	=	^⁸√	8	1.29683955465	8
10	Perfect fourth	2^{¹⁰⁄₂₄}	=	^¹²√	32	1.33483985417	12
11	Wide fourth	2^{¹¹⁄₂₄}	=	^²⁴√	2,048	1.37395364746	24
12	Tritone	2^{¹²⁄₂₄}	=	√	2	1.41421356237	2
13	Narrow fifth	2^{¹³⁄₂₄}	=	^²⁴√	8,192	1.45565318284	24
14	Perfect fifth	2^{¹⁴⁄₂₄}	=	^¹²√	128	1.49830707688	12
15	Wide fifth, infra sixth	2^{¹⁵⁄₂₄}	=	^⁸√	32	1.54221082541	8
16	Minor sixth	2^{¹⁶⁄₂₄}	=	^³√	4	1.58740105197	3
17	Neutral sixth	2^{¹⁷⁄₂₄}	=	^²⁴√	131,072	1.63391545324	24
18	Major sixth	2^{¹⁸⁄₂₄}	=	^⁴√	8	1.68179283051	4
19	Ultra sixth, infra seventh	2^{¹⁹⁄₂₄}	=	^²⁴√	524,288	1.73107312201	24
20	Minor seventh	2^{²⁰⁄₂₄}	=	^⁶√	32	1.78179743628	6
21	Neutral seventh	2^{²¹⁄₂₄}	=	^⁸√	128	1.83400808641	8
22	Major seventh	2^{²²⁄₂₄}	=	^¹²√	2,048	1.88774862536	12
23	Ultra seventh, narrow octave	2^{²³⁄₂₄}	=	^²⁴√	8,388,608	1.94306388231	24
24	Octave	2^{²⁴⁄₂₄}	=		2	2	1

Yes, but, why?

I gazed into an eldritch dimension by writing a piece in Locrian mode⁽⁹⁾. This was the result. I should’ve heeded Nietzsche’s warning about abysses.⁽¹⁰⁾

(More seriously, my use of Locrian mode occasioned a discussion of what musical modes are, which in turn resulted in additional music theory discussion that sent me down a rabbit hole of mathematical patterns. This is probably the HTML equivalent of the bulletin board with string connecting pieces of a conspiracy, except that the conspiracy is ultimately just that music and mathematics are low-key the same thing. If you don’t believe me, spend enough time studying calculus to understand Euler’s identity and you’ll see the music in math.)

Endnotes

Endnotes
#	Note
1.	I spent an unreasonable amount of time trying to formulate a pun on Marcel Duchamp’s classic Modernist painting Nude Descending a Staircase, something along the lines of ‘Lydian Descending a Scale’, but every phrasing I formulated ultimately felt too awkward, shoehorned, and/or unhelpful… so I just created this not-at-all awkward, shoehorned, or unhelpful endnote instead.
2.	Sanskrit: राग
3.	Sanskrit: स्वर; also sometimes romanized as swara
4.	Sanskrit: श्रुति; also sometimes romanized as shruti
5.	Thus, if you’ve ever misspelled chord as cord (or vice versa⁽¹⁰⁾), you weren’t entirely wrong, especially since chord and cord both come from ἡ χορδή (hē khordḗ) in ancient Greek. We can blame the French for the two senses being romanized differently, since they changed the Latin chorda to corde, which later reverted to chord for the musical sense.
6.	It is worth recalling here that typical definitions of “Ancient Greece” use timespans of no fewer than 750 years. The earliest possible end point is the Roman conquest in 31-30 BCE; the second Arab siege of Constantinople in 717-718 CE is the latest typical end date. Typical start dates are Ischia’s settlement in 785 BCE or Euboea’s rise ca. 1200 BCE – already a possible span of nearly two millennia. However, it would be technically correct (the best kind of correct!) to start it with the rise of Aegean Greece ca. 3300 BCE, which passes two crucial tests: it was over five millennia ago (i.e., it was ancient), and its name includes “Greece” (i.e., it was Greek). Thus, the broadest possible view of the term encompasses over four millennia.
7.	⁽ᵃ⁾ ⁽ᵇ⁾ For instance, here’s the Stanford Encyclopedia of Philosophy on Archytas: Archytas’ final contribution to music theory has to do with the structure of the scale (for a more detailed account than what follows, see Huffman 2005: 402–25 and Barker 2007: 292–302). The Greeks used a number of different scales, which were distinguished by the way in which the fourth, or tetrachord, was constructed. These scales were grouped into three main types, or genera. One genus was called the diatonic; one example of this is the Pythagorean diatonic described above, which is built on the tetrachord with the intervals 9:8, 9:8, and 256:243, and was used by Philolaus and Plato. There is no doubt that Archytas knew of this diatonic scale, but his own diatonic tetrachord was somewhat different, being composed of the intervals 9:8, 8:7, and 28:27. Archytas also defined scales in the two other major genera, the enharmonic and chromatic. Archytas’ enharmonic tetrachord is composed of the intervals 5:4, 36:35, and 28:27, and his chromatic tetrachord of the intervals 32:27, 243:224, and 28:27. There are several puzzles about the tetrachords which Archytas adopts in each of the genera. First, why does Archytas reject the Pythagorean diatonic used by Philolaus and Plato? Second, Ptolemy, who is our major source for Archytas’ tetrachords (A16), argues that Archytas adopted as a principle that all concordant intervals should correspond to superparticular ratios. The ratios in Archytas’ diatonic and enharmonic tetrachords are indeed superparticular, but two of the ratios in his chromatic tetrachord are not superparticular (32:27 and 243:224). Why are these ratios not superparticular as well? Finally, Plato criticizes Pythagorean harmonics in the Republic for seeking numbers in heard harmonies rather than ascending to generalized problems (531c). Can any sense be made of this criticism in light of Archytas’ tetrachords? The basis for an answer to all of these questions is contained in the work of Winnington-Ingram (1932) and Barker (1989, 46–52). The crucial point is that Archytas’ account of the tetrachords in each of the three genera can be shown to correspond to the musical practice of his day; Ptolemy’s criticisms miss the mark because of his ignorance of musical practice in Archytas’ day, some 500 years before Ptolemy (Winnington-Ingram 1932, 207). Archytas is giving mathematical descriptions of scales actually in use; although mathematical considerations did play a role (Barker 2007: 295–302), he arrived at his numbers in part by observation of the way in which musicians tuned their instruments (Barker 1989, 50–51). He did not follow the Pythagorean diatonic scale because it did not correspond to any scale actually in use, although it does correspond to a method of tuning. The unusual numbers in Archytas’ chromatic tetrachord do correspond to a chromatic scale in use in Archytas’ day. Ptolemy was wrong to suggest that Archytas adhered to the principle that all concordant intervals should have superparticular ratios (Huffman 2005: 422–3), although Barker suggests that he may have been following a different but related principle (2007: 301). Archytas thus provides a brilliant analysis of the music of his day, but it is precisely his focus on actual musical practice that draws Plato’s ire. Plato does not want him to focus on the music he hears about him (“heard harmonies”), but rather to ascend to consider quite abstract questions about which numbers are harmonious with which. Plato might well have welcomed a principle of concordance based solely on mathematical considerations, such as the principle that only superparticular ratios are concordant, but Archytas wanted to explain the numbers of the music he actually heard played. There is an important metaphysical issue at stake here. Plato is calling for the study of number in itself, apart from the sensible world, while Archytas, like Pythagoreans before him, envisages no split between a sensible and an intelligible world and is looking for the numbers which govern sensible things. For discussions of Archytas as the target of Plato’s complaints in the Republic, see Huffman 2005: 423–5 and Barker 2014: 192–3. (I’ve lightly edited the above text by removing spaces within ratios and adding commas and line breaks for legibility’s sake [the entire quote was originally a single paragraph!], but it is otherwise unchanged.) I feel justified in concluding from this that Plato and Ptolemy, vastly more famous though they are, are very likely less reliable than Archytas where ancient Greek musical practice is concerned (even if much of our knowledge of Archytas’ musical writings comes secondhand from Ptolemy).
8.	We hates it, precious. We hates it forever.
9.	I’ve tentatively titled it «Λοκρῶν θρῆνος» [romanized: Lokrôn thrênos], Ancient Greek for “Locrian Lament”, but this may change. It’s still unfinished, but I’ve added it to Compositions 2023-2024, my collection of recent original compositions. In addition to Locrian mode, it also uses a different time signature every measure, from 1/8 in the first measure to 21/8 in the twenty-first, before the pattern repeats, which also makes it incredibly disorienting. Naturally, this is entirely intentional.
10.	“He who fights monsters should see to it that he himself does not become a monster; and if you gaze for long into an abyss, the abyss gazes also into you.” –Friedrich Nietzsche, Beyond Good and Evil Aphorism 146
11.	While I’m at it, the ancient Romans would’ve pronounced vice versa as something like wee-keh wer-sah.