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 can at least tolerate the double-flat symbol, but the double-sharp symbol is a monstrosity that doesn’t even bear any resemblance to the thing it’s supposed to be doubling.) 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 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
# | 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 ⟨aaronfreed
.github .io /c_ lydian_ to_ b_ lydian .flac⟩ in Logic Pro. For the following modes, in order: - C Lydian
- C Ionian
- C Mixolydian
- C Dorian
- C Aeolian
- C Phrygian
- C Locrian
- C♭/B Lydian
- the scale in ascending order
- arpeggiated and block versions of:
- the root (I) chord
- the subdominant (IV) chord
- the dominant (V) chord
- the root (I) chord
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
- Although this is only somewhat relevant here, one pair of intervals always constitutes an exception to the “perfect fourths” and “perfect fifths” rules: one fourth is always augmented, and one fifth is always diminished. These are actually the same interval, a tritone. The following steps on each scale are tritones in either direction:
The Major Scale’s Modes & the Circle of Fifths
12 Major Scales × 7 Modes = 84 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 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, ascending melodic minor, harmonic minor, and chromatic Hypolydian; possibly others as well).
- “RM” is an initialism for “Relative Major”, and “KS” is for “Key Signature”. A cheat sheet for what modes use what key signatures can be found in the section immediately following this one.
- I’ve used zero-based indexing for these tables, so they’re indexed from zero to eighty-three. This is partly because I prefer zero-based indexing, but to be honest, it was mostly to give C Ionian an index of one.
C (B♯) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
# | Root | Mode | RM | KS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
0 | C♮ | 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♯ | G♯ | D♯ | A♯ | E♯ | B♯ | 7♯ | ♯ | ♯ | ♯ | ♯ | ♯ | ♯ | ♯ |
B | F♯ | C♯ | G♯ | D♯ | A♯ | E♯ | 6♯ | ♯ | ♯ | ♯ | ♯ | ♯ | ♯ | |
E | B | F♯ | C♯ | G♯ | D♯ | A♯ | 5♯ | ♯ | ♯ | ♯ | ♯ | ♯ | ||
A | E | B | F♯ | C♯ | G♯ | D♯ | 4♯ | ♯ | ♯ | ♯ | ♯ | |||
D | A | E | B | F♯ | C♯ | G♯ | 3♯ | ♯ | ♯ | ♯ | ||||
G | D | A | E | B | F♯ | C♯ | 2♯ | ♯ | ♯ | |||||
C | G | D | A | E | B | F♯ | 1♯ | ♯ | ||||||
F | C | G | D | A | E | B | ♮ | |||||||
B♭ | F | C | G | D | A | E | 1♭ | ♭ | ||||||
E♭ | B♭ | F | C | G | D | A | 2♭ | ♭ | ♭ | |||||
A♭ | E♭ | B♭ | F | C | G | D | 3♭ | ♭ | ♭ | ♭ | ||||
D♭ | A♭ | E♭ | B♭ | F | C | G | 4♭ | ♭ | ♭ | ♭ | ♭ | |||
G♭ | D♭ | A♭ | E♭ | B♭ | F | C | 5♭ | ♭ | ♭ | ♭ | ♭ | ♭ | ||
C♭ | G♭ | D♭ | A♭ | E♭ | B♭ | F | 6♭ | ♭ | ♭ | ♭ | ♭ | ♭ | ♭ | |
F♭ | C♭ | G♭ | D♭ | A♭ | E♭ | B♭ | 7♭ | ♭ | ♭ | ♭ | ♭ | ♭ | ♭ | ♭ |
- Legend for the above:
- Lyd = Lydian
- Maj = major (Ionian)
- Mix = Mixolydian
- Dor = Dorian (if you’re not sure why it’s grey, ask your English teacher about the picture)
- Min = natural minor (Aeolian)
- Phr = Phrygian
- Loc = Locrian
- KS = key signature (highlighted because it’s the key to the table – pun coincidental, though I definitely didn’t even try to avoid it)
- Additional notes:
- 5♯ 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, | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
# | 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 to make it clear how they change the diatonic major scale:
Other Seven-Note Scales Scale 1 2 3 4 5 6 7 Intervals Chromatic Dorian Kanakangi C♮ D♭ E𝄫 F♮ G♮ A♭ B𝄫 ½ ½ 1½ 1 ½ ½ 1½ Altered diminished Super-Locrian 𝄫2 C♮ D♭ E♭ F♭ G♭ A♭ B𝄫 ½ 1 ½ 1 1 ½ 1½ Altered dominant 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 Jazz minor inverse C♮ D♭ E♭ F♮ G♮ A♮ B♭ ½ 1 1 1 1 ½ 1 Phrygian dominant C♮ D♭ E♮ F♮ G♮ A♭ B♭ ½ 1½ ½ 1 ½ 1 1 Flamenco mode Major Phrygian C♮ D♭ E♮ F♮ G♮ A♭ B♮ ½ 1½ ½ 1 ½ 1½ ½ 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 Harmonic minor C♮ D♮ E♭ F♮ G♮ A♭ B♮ 1 ½ 1 1 ½ 1½ ½ Ascending melodic minor C♮ D♮ E♭ F♮ G♮ A♮ B♮ 1 ½ 1 1 1 1 ½ Hungarian minor C♮ D♮ E♭ F♯ G♮ A♭ B♮ 1 ½ 1½ ½ ½ 1½ ½ Ukrainian Dorian C♮ D♮ E♭ F♯ G♮ A♮ B♭ 1 ½ 1½ ½ 1 ½ 1 Aeolian dominant Mixolydian ♭6 C♮ D♮ E♮ F♮ G♮ A♭ B♭ 1 1 ½ 1 ½ 1 1 Augmented major Ionian ♯5 C♮ D♮ E♮ F♮ G♯ A♮ B♮ 1 1 ½ 1½ ½ 1 ½ Lydian augmented C♮ D♮ E♮ F♯ G♯ A♮ B♮ 1 1 1 1 ½ 1 ½ Lydian dominant Mixolydian ♯4 C♮ D♮ E♮ F♯ G♮ A♮ B♭ 1 1 1 ½ 1 ½ 1 Chromatic Dorian inverse C♮ D♯ E♮ F♮ G♮ A♯ B♮ 1½ ½ ½ 1 1½ ½ ½ Chromatic Hypophrygian C♮ D♯ E♯ F♯ G♮ A♯ B♮ 1½ 1 ½ ½ 1½ ½ ½ Aeolian harmonic Lydian ♯2 C♮ D♯ E♮ F♯ G♮ A♮ B♮ 1½ ½ 1 ½ 1 1 ½ Superlydian augmented C♮ D♯ E♯ F♯ G♯ A♯ B♮ 1½ 1 ½ 1 1 ½ ½ Since it would be impossible to alphabetize scales with more than one name (note that many scales have more names than I’ve printed here), I’ve applied the principles of alphabetization to the intervals, with the first interval counted as the most significant. Bold, orange text denotes raised notes and minor thirds; thin, blue text denotes lowered notes and minor seconds (doubly lowered notes are also fainter).
Note that this list is far from exhaustive; it doesn’t even delve into scales with different ascents and descents. Also, none of these scales use B♯ because, while it is semantically different in music theory 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 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.
One additional advantage of printing Chromatic Dorian first: It’s directly relevant to the section below on ancient Greek harmony.
-
Note that several sets of the above scales are modes of each other. In particular:
Harmonic Minor & Ascending Melodic Minor’s Modes at a Glance Scale 1 2 3 4 5 6 7 Intervals 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 Augmented major Ionian ♯5 C♮ D♮ E♮ F♮ G♯ A♮ B♮ 1 1 ½ 1½ ½ 1 ½ Ukrainian Dorian C♮ D♮ E♭ F♯ G♮ A♮ B♭ 1 ½ 1½ ½ 1 ½ 1 Phrygian dominant C♮ D♭ E♮ F♮ G♮ A♭ B♭ ½ 1½ ½ 1 ½ 1 1 Aeolian harmonic Lydian ♯2 C♮ D♯ E♮ F♯ G♮ A♮ B♮ 1½ ½ 1 ½ 1 1 ½ Altered diminished Super-Locrian 𝄫2 C♮ D♭ E♭ F♭ G♭ A♭ B𝄫 ½ 1 ½ 1 1 ½ 1½ Ascending melodic minor C♮ D♮ E♭ F♮ G♮ A♮ B♮ 1 ½ 1 1 1 1 ½ Jazz minor inverse C♮ D♭ E♭ F♮ G♮ A♮ B♭ ½ 1 1 1 1 ½ 1 Lydian augmented C♮ D♮ E♮ F♯ G♯ A♮ B♮ 1 1 1 1 ½ 1 ½ Lydian dominant Mixolydian ♯4 C♮ D♮ E♮ F♯ G♮ A♮ B♭ 1 1 1 ½ 1 ½ 1 Aeolian dominant Mixolydian ♭6 C♮ D♮ E♮ F♮ G♮ A♭ B♭ 1 1 ½ 1 ½ 1 1 Half-diminished C♮ D♮ E♭ F♮ G♭ A♭ B♭ 1 ½ 1 ½ 1 1 1 Altered dominant Super-Locrian C♮ D♭ E♭ F♭ G♭ A♭ B♭ ½ 1 ½ 1 1 1 1 We can apply similar principles to them as we applied to the diatonic major 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.
- 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; blues and jazz often use eight- and nine-note scales.
- And of course, all of the above is based on the ⟨en
.wikipedia .org /wiki /12_ 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 ⟨en
.wikipedia .org /wiki /Quarter_ tone⟩ is common in Arabic and Turkic music. - Indian ⟨en
.wikipedia .org /wiki /Raga⟩ typically use anywhere from four to seven svaras (roughly comparable to the notes of a Western diatonic scale) selected from the ⟨en .wikipedia .org /wiki /Shruti_ (music)⟩ (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:
- ⟨kinggizzard
.bandcamp .com /album /flying-microtonal-banana⟩ by King Gizzard & the Lizard Wizard (24-TET) - ⟨votsband
.bandcamp .com /album /nowherer⟩ by Victory over the Sun (17-TET) - any metal album by ⟨jutegyte
.bandcamp .com /music⟩ since Discontinuities (24-TET)
- ⟨kinggizzard
- The Ancient Greek enharmonic genus, as we’ll see after our examinations of harmonic and melodic minor, also used microtonality.
- A ⟨en
Harmonic Minor’s Modes in Detail
Since learning the modes of harmonic minor and ascending melodic minor is absolutely essential for anyone who wants to play jazz, I’ve created several sets of tables to help people visualize them better.
Harmonic Minor vs. Modes from Aeolian (rooted on C, in ascending order) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Scale | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
Harmonic minor | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | 1 | ½ | 1 | 1 | ½ | 1½ | ½ | |
Aeolian | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |
Locrian ♮6 | Maqam Tarznauyn | C♮ | D♭ | E♭ | F♮ | G♭ | A♮ | B♭ | ½ | 1 | 1 | ½ | 1½ | ½ | 1 |
Locrian | C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | |
Augmented major | Ionian ♯5 | C♮ | D♮ | E♮ | F♮ | G♯ | A♮ | B♮ | 1 | 1 | ½ | 1½ | ½ | 1 | ½ |
Ionian | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |
Ukrainian Dorian | C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♭ | 1 | ½ | 1½ | ½ | 1 | ½ | 1 | |
Dorian | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |
Phrygian dominant | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♭ | ½ | 1½ | ½ | 1 | ½ | 1 | 1 | |
Phrygian | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |
Aeolian harmonic | Lydian ♯2 | C♮ | D♯ | E♮ | F♯ | G♮ | A♮ | B♮ | 1½ | ½ | 1 | ½ | 1 | 1 | ½ |
Lydian | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |
Altered diminished | Super-Locrian 𝄫2 | C♮ | D♭ | E♭ | F♭ | G♭ | A♭ | B𝄫 | ½ | 1 | ½ | 1 | 1 | ½ | 1½ |
Mixolydian | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 |
Thus, harmonic minor’s modes respectively raise by a half-step:
- Aeolian’s seventh degree,
- Locrian’s sixth degree,
- Ionian’s fifth degree,
- Dorian’s fourth degree,
- Phrygian’s third degree,
- Lydian’s second degree,
- C-C-C-C-COMBO BREAKER!
We can’t raise Mixolydian’s first degree, because it’s the first degree! Instead, we lower every other degree by a half-step. Say wha?
Somehow, it’s actually both even weirder than that, and not weird at all: what we do in the above table is the equivalent of raising the first degree. Since we’re constraing ourselves to a root of C, raising the first note of a scale by a half-step requires us to lower every note of that scale by a half-step. This results in the first note being the only scale degree we don’t lower: ½ − ½ = 0.
In practice, though, harmonic minor has already raised its seventh degree for us. Let’s represent these modes again, but this time, we’ll root each mode in its respective note within its parent C minor scale:
Harmonic Minor vs. Modes from Aeolian (rooted on scale, in ascending order) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Scale | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
Harmonic minor | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | 1 | ½ | 1 | 1 | ½ | 1½ | ½ | |
Aeolian | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |
Locrian ♮6 | Maqam Tarznauyn | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | C♮ | ½ | 1 | 1 | ½ | 1½ | ½ | 1 |
Locrian | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | |
Augmented major | Ionian ♯5 | E♭ | F♮ | G♮ | A♭ | B♮ | C♮ | D♮ | 1 | 1 | ½ | 1½ | ½ | 1 | ½ |
Ionian | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | D♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |
Ukrainian Dorian | F♮ | G♮ | A♭ | B♮ | C♮ | D♮ | E♭ | 1 | ½ | 1½ | ½ | 1 | ½ | 1 | |
Dorian | F♮ | G♮ | A♭ | B♭ | C♮ | D♮ | E♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |
Phrygian dominant | G♮ | A♭ | B♮ | C♮ | D♮ | E♭ | F♮ | ½ | 1½ | ½ | 1 | ½ | 1 | 1 | |
Phrygian | G♮ | A♭ | B♭ | C♮ | D♮ | E♭ | F♮ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |
Aeolian harmonic | Lydian ♯2 | A♭ | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | 1½ | ½ | 1 | ½ | 1 | 1 | ½ |
Lydian | A♭ | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |
Altered diminished | Super-Locrian 𝄫2 | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | ½ | 1 | ½ | 1 | 1 | ½ | 1½ |
Mixolydian | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 |
Thus, B♭ Mixolydian is to C Aeolian as B altered diminished is to C harmonic minor: each starts from its parent scale’s seventh note. There’s actually nothing odd going on here at all; it’s exactly how modes are supposed to behave. C altered diminished equates to lowering every note of C Mixolydian except C by a half-step – and to raising only the B in B Mixolydian by a half-step.
Observant readers may have noticed that the “rooted on C” table above actually appears to contain several shifts. The missing puzzle piece is that it lists the modes in ascending order rather than “circle of fifths” order, which I did to make the scales’ intervals easier to relate to each other. So let’s return to “circle of fiths” order.
Harmonic Minor vs. Modes from Aeolian (rooted on C, in “circle of fifths” order) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Scale | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
Aeolian harmonic | Lydian ♯2 | C♮ | D♯ | E♮ | F♯ | G♮ | A♮ | B♮ | 1½ | ½ | 1 | ½ | 1 | 1 | ½ |
Lydian | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |
Augmented major | Ionian ♯5 | C♮ | D♮ | E♮ | F♮ | G♯ | A♮ | B♮ | 1 | 1 | ½ | 1½ | ½ | 1 | ½ |
Ionian | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |
Altered diminished | Super-Locrian 𝄫2 | C♮ | D♭ | E♭ | F♭ | G♭ | A♭ | B𝄫 | ½ | 1 | ½ | 1 | 1 | ½ | 1½ |
Mixolydian | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | |
Ukrainian Dorian | C♮ | D♮ | E♭ | F♯ | G♮ | A♮ | B♭ | 1 | ½ | 1½ | ½ | 1 | ½ | 1 | |
Dorian | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |
Harmonic minor | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | 1 | ½ | 1 | 1 | ½ | 1½ | ½ | |
Aeolian | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |
Phrygian dominant | C♮ | D♭ | E♮ | F♮ | G♮ | A♭ | B♭ | ½ | 1½ | ½ | 1 | ½ | 1 | 1 | |
Phrygian | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |
Locrian ♮6 | Maqam Tarznauyn | C♮ | D♭ | E♭ | F♮ | G♭ | A♮ | B♭ | ½ | 1 | 1 | ½ | 1½ | ½ | 1 |
Locrian | C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 |
We now see that, broadly speaking, in all except one case, each scale has one fewer sharp or one more flat than its predecessor two entries above. The clear outlier is altered diminished, and this table may further clarify why the mode corresponding to Mixolydian is the one thus affected. Mixolydian corresponds to Ionian’s fifth scale degree, Dorian’s fourth scale degree, Phrygian’s third scale degree… and that’s the degree that harmonic minor’s modes raise. For completeness, here are the modes in “circle of fifths” order, rooted to their respective notes within their parent C minor scales:
Harmonic Minor vs. Modes from Aeolian (rooted on scale, in “circle of fifths” order) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Scale | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
Aeolian harmonic | Lydian ♯2 | A♭ | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | 1½ | ½ | 1 | ½ | 1 | 1 | ½ |
Lydian | A♭ | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |
Augmented major | Ionian ♯5 | E♭ | F♮ | G♮ | A♭ | B♮ | C♮ | D♮ | 1 | 1 | ½ | 1½ | ½ | 1 | ½ |
Ionian | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | D♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |
Altered diminished | Super-Locrian 𝄫2 | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | ½ | 1 | ½ | 1 | 1 | ½ | 1½ |
Mixolydian | B♭ | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | |
Ukrainian Dorian | F♮ | G♮ | A♭ | B♮ | C♮ | D♮ | E♭ | 1 | ½ | 1½ | ½ | 1 | ½ | 1 | |
Dorian | F♮ | G♮ | A♭ | B♭ | C♮ | D♮ | E♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |
Harmonic minor | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | 1 | ½ | 1 | 1 | ½ | 1½ | ½ | |
Aeolian | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |
Phrygian dominant | G♮ | A♭ | B♮ | C♮ | D♮ | E♭ | F♮ | ½ | 1½ | ½ | 1 | ½ | 1 | 1 | |
Phrygian | G♮ | A♭ | B♭ | C♮ | D♮ | E♭ | F♮ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |
Locrian ♮6 | Maqam Tarznauyn | D♮ | E♭ | F♮ | G♮ | A♭ | B♮ | C♮ | ½ | 1 | 1 | ½ | 1½ | ½ | 1 |
Locrian | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | C♮ | ½ | 1 | 1 | ½ | 1 | 1 | 1 |
Finally, this table also clarifies why these charts have to use scare quotes for “circle of fifths”: once we abandon the diatonic major scale’s regularity, we’re not actually moving every note up or down by a fifth. A perfect fifth up from E♭ isn’t B; it’s B♭. A perfect fifth below F isn’t B; it’s B♭. This is very likely why the diatonic major scale uses the intervals it does: it possesses a borderline fractal level of regularity. Manifestations of the circle of fifths occur throughout it. Even bumping one of its notes a half-step up or down breaks the pattern in places. We can still approximate “circle of fifths” order, of course, but our approximations will have to carry asterisks.
Ascending Melodic Minor’s Modes in Detail
Ascending melodic minor is perhaps better related to the modes starting with Ionian. I haven’t drawn borders this time, because… well, it’s probably better to just show the table first.
Ascending Melodic Minor vs. Modes from Ionian (rooted on C, in ascending order) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Scale | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
Ionian | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |
Ascending melodic minor | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | |
Dorian | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |
Jazz minor inverse | C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | |
Phrygian | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |
Lydian augmented | C♮ | D♮ | E♮ | F♯ | G♯ | A♮ | B♮ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | |
Lydian | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |
Lydian dominant | Mixolydian ♯4 | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 |
Mixolydian | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | |
Aeolian dominant | Mixolydian ♭6 | C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | ½ | 1 | ½ | 1 | 1 |
Aeolian | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |
Half-diminished | C♮ | D♮ | E♭ | F♮ | G♭ | A♭ | B♭ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | |
Locrian | C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | |
Altered dominant | Super-Locrian | C♮ | D♭ | E♭ | F♭ | G♭ | A♭ | B♭ | ½ | 1 | ½ | 1 | 1 | 1 | 1 |
In short, it may be helpful to think of ascending melodic minor as Ionian and Dorian’s midpoint; of jazz minor inverse as Dorian and Phrygian’s midpoint; and so on. (This is, of course, an oversimplification, since it’s completely ignoring “circle of fiths” order.)
Transposed versions of the above scales:
Ascending Melodic Minor vs. Modes from Ionian (rooted on scale, in ascending order) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Scale | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
Ionian | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |
Ascending melodic minor | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | |
Dorian | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | C♮ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |
Jazz minor inverse | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | C♮ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | |
Phrygian | E♮ | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |
Lydian augmented | E♭ | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | |
Lydian | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |
Lydian dominant | Mixolydian ♯4 | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | E♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 |
Mixolydian | G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | F♮ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | |
Aeolian dominant | Mixolydian ♭6 | G♮ | A♮ | B♮ | C♮ | D♮ | E♭ | F♮ | 1 | 1 | ½ | 1 | ½ | 1 | 1 |
Aeolian | A♮ | B♮ | C♮ | D♮ | E♮ | F♮ | G♮ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |
Half-diminished | A♮ | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | |
Locrian | B♮ | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | |
Altered dominant | Super-Locrian | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | ½ | 1 | ½ | 1 | 1 | 1 | 1 |
“Circle of fifths” order makes it clear that the big note shift from Phrygian to Lydian augmented occurs in the “rooted on C” chart for the same reason the note shift between Mixolydian and altered diminished occurs with the harmonic minor scale: rooting everything to C means we can’t lower the first note and must instead raise the other notes by however much we’d have lowered the first note.
Ascending Melodic Minor vs. Modes from Ionian (rooted on C, in “circle of fifths” order) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Scale | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
Lydian | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |
Lydian dominant | Mixolydian ♯4 | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 |
Ionian | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |
Ascending melodic minor | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | |
Mixolydian | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | |
Aeolian dominant | Mixolydian ♭6 | C♮ | D♮ | E♮ | F♮ | G♮ | A♭ | B♭ | 1 | 1 | ½ | 1 | ½ | 1 | 1 |
Dorian | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |
Jazz minor inverse | C♮ | D♭ | E♭ | F♮ | G♮ | A♮ | B♭ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | |
Aeolian | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |
Half-diminished | C♮ | D♮ | E♭ | F♮ | G♭ | A♭ | B♭ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | |
Phrygian | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |
Lydian augmented | C♮ | D♮ | E♮ | F♯ | G♯ | A♮ | B♮ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | |
Locrian | C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | |
Altered dominant | Super-Locrian | C♮ | D♭ | E♭ | F♭ | G♭ | A♭ | B♭ | ½ | 1 | ½ | 1 | 1 | 1 | 1 |
Phrygian mode corresponds to Ionian’s third degree; that’s the note ascending melodic minor lowers from Ionian. Thus, Phrygian is the mode that undergoes the note shift in the above table. Moreover, C Lydian augmented raises every note of C Phrygian except its root because its parent scale lowers its corresponding note.
For completeness, here’s “circle of fifths” order without transposition.
Ascending Melodic Minor vs. Modes from Ionian (rooted on scale, in “circle of fifths” order) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Scale | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
Lydian | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ | |
Lydian dominant | Mixolydian ♯4 | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | E♭ | 1 | 1 | 1 | ½ | 1 | ½ | 1 |
Ionian | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ | |
Ascending melodic minor | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | 1 | ½ | 1 | 1 | 1 | 1 | ½ | |
Mixolydian | G♮ | A♮ | B♮ | C♮ | D♮ | E♮ | F♮ | 1 | 1 | ½ | 1 | 1 | ½ | 1 | |
Aeolian dominant | Mixolydian ♭6 | G♮ | A♮ | B♮ | C♮ | D♮ | E♭ | F♮ | 1 | 1 | ½ | 1 | ½ | 1 | 1 |
Dorian | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | C♮ | 1 | ½ | 1 | 1 | 1 | ½ | 1 | |
Jazz minor inverse | D♮ | E♭ | F♮ | G♮ | A♮ | B♮ | C♮ | ½ | 1 | 1 | 1 | 1 | ½ | 1 | |
Aeolian | A♮ | B♮ | C♮ | D♮ | E♮ | F♮ | G♮ | 1 | ½ | 1 | 1 | ½ | 1 | 1 | |
Half-diminished | A♮ | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | 1 | ½ | 1 | ½ | 1 | 1 | 1 | |
Phrygian | E♮ | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | ½ | 1 | 1 | 1 | ½ | 1 | 1 | |
Lydian augmented | E♭ | F♮ | G♮ | A♮ | B♮ | C♮ | D♮ | 1 | 1 | 1 | 1 | ½ | 1 | ½ | |
Locrian | B♮ | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | ½ | 1 | 1 | ½ | 1 | 1 | 1 | |
Altered dominant | Super-Locrian | B♮ | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | ½ | 1 | ½ | 1 | 1 | 1 | 1 |
(At some point, I plan to make equivalents of §3’s charts for at least ascending melodic minor and harmonic minor, and perhaps for some of the Greek scales I discuss below as well… but not until I’ve written programs to automate their generation, which could take anywhere from a few days to months.)
A Crash Course in Ancient Greek Harmony
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 | ⟨en |
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, note, tone, or tension; its dual nominative form was τὼ τόνω, tṑ tónō, and its plural nominative form was οἱ τόνοι, hoi tónoi.)
As for the word diatonic, that comes from the Ancient Greek διατονικός (diatonikós), literally meaning two tones, in reference to the diatonic tetrachord, which… well, keep reading.
Ancient Greek Harmony: The Cliffs Notes
This is an oversimplification by necessity, as ancient Greek music theory wasn’t unified; (Φιλόλαος, Philólaos), (Ἀρχύτας), (Ἀριστόξενος ὁ Ταραντῖνος, Aristóxenos ho Tarantínos), (Πτολεμαῖος, Ptolemaios), and others had quite different conceptions of it. I will list modern sources in an acknowledgement section below.
I should first clarify that very little music of ancient Greece actually survives to this day; the earliest known piece that can reasonably be described as complete is ⟨en
I’ll be using numbers to represent the intervals of ancient Greek harmony within ⟨en
Interval Key | ||||||||
---|---|---|---|---|---|---|---|---|
# | Interval | Tone | Exact | Approximate | ||||
¼ | Infra second | Quarter-tone | ²⁴√2 | : | 1 | ≈ | 1.02930223664 | |
½ | Minor second | Semitone | ¹²√2 | : | 1 | ≈ | 1.05946309436 | |
1 | Major second | Whole tone | ⁶√2 | : | 1 | ≈ | 1.12246204831 | |
1½ | Minor third | Three semitones | ⁴√2 | : | 1 | ≈ | 1.18920711500 | |
2 | Major third | Two whole tones | ³√2 | : | 1 | ≈ | 1.25992104989 |
Note that in scales with only whole-steps and half-steps, I’ll use H (i.e., Half) interchangeably with ½, and W (i.e., Whole) interchangeably with 1. In all other scales, I’ll only use the numbers.
24-TET is by necessity an oversimplification as well, not least since different authors defined the same interval differently. For instance, here are how Philolaus, Archytas, and 24-TET would define the intervals within a diatonic tetrachord. (A tetrachord is a major element of Greek harmony consisting of four notes. Also, I’ve taken the liberty of reversing what they called the first and third intervals – ancient Greek harmony defined scales in 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 | = | 1.0370370370… | 8 | : | 7 | = | 1.142857142857… | 9 | : | 8 | = | 1.125 |
24-TET | ¹²√2 | : | 1 | ≈ | 1.05946309436 | ⁶√2 | : | 1 | ≈ | 1.12246204831 | ⁶√2 | : | 1 | ≈ | 1.12246204831 |
Thus, representing ancient Greek harmony using 24-TET is a substantial oversimplification, and in point of fact, we’d get closer to Archytas’ definitions in 24-TET by using a quarter tone for the low interval and 1¼-tone (that is, the interval between a major second and a minor third, known as an infra third or ultra second) for the middle interval. However, the intervals I’ve selected (a minor second and two major seconds) are closer to Philolaus’ definitions and, crucially, can be represented in modern Western music’s 12-TET.
As one further example, both Philolaus and Archytas define a diatonic tetrachord’s high and low notes as having 4:3 (1.33333…) ratios, which 24-TET would represent using a perfect fourth (≈1.33483985417). Most people’s ears are insufficiently trained to tell the difference – even trained singers with perfect pitch frequently wander off-key by larger differences than that (at least when they’re not subjected to pitch correction).
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.
Ancient Greek Tonoi & Modern Modes
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 interval notation will be slightly easier to understand for the enharmonic tonoi. Below this table, I’ll present examples of the chromatic tonoi based on C.
- A thicker font denotes a larger interval size: 2, 1½, 1, ½, ¼.
- I’ve placed borders between tetrachords (again, four tonoi split one of their tetrachords) and around the extra whole step (whose background I’ve also highlighted).
- I’ve printed the modern modes with blue text and brighter borders, with a thicker border below since the next tonos will not be related.
- I’ve printed the tonoi in the order ancient Greek harmony assigned them: Mixolydian (modern Locrian’s equivalent) first, Hypodorian (modern Aeolian’s equivalent) last.
- Our Dorian, Phrygian, Lydian, and Mixolydian modes don’t match their eponymous diatonic tonoi because medieval Europeans erroneously believed the ancient Greeks described tonoi in ascending order. (Swiss poet Heinrich Glarean appears to have popularized the misconception.) I’ll explain this further below.
Here are all three genera of all seven tonoi, followed by the diatonic genus’ modern equivalent:
Approximate Intervals of Ancient Greek Tonoi & Modern Diatonic Modes | |||||||||
---|---|---|---|---|---|---|---|---|---|
Tonos | Genus | 1–2 | 2–3 | 3–4 | 4–5 | 5–6 | 6–7 | 7–8 | |
Mixolydian | Enharmonic | ¼ | ¼ | 2 | ¼ | ¼ | 2 | 1 | |
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 |
The above table is quite abstract, so to follow it up, here are the chromatic tonoi rooted in C and, for the sake of representing what medieval Europeans might have thought they were, their inversions. As in the previous table, I’ve separated the tetrachords in the interval listing to make it clear where they occur, and I’ve highlighted the extra whole-step (a bit more so, even, because it will become a bit less legible shortly).
Greek Chromatic Tonoi & Their Inversions (C roots, linear order) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B♭ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | ||
C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | ||
C♮ | D♯ | E♮ | F♮ | G♯ | A♯ | B♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | ||
C♮ | D♭ | E𝄫 | F♮ | G♮ | A♭ | B𝄫 | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | ||
C♮ | D♭ | E♮ | F♯ | G♮ | A♭ | B♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | ||
C♮ | D♯ | E♯ | F♯ | G♮ | A♯ | B♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | ||
C♮ | D♮ | E♭ | F♭ | G♮ | A♭ | B𝄫 | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | ||
C♮ | D♮ | E♯ | F♯ | G♮ | A♯ | B♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | ||
C♮ | D♭ | E♭ | F♯ | G♮ | A♭ | B♮ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | ||
C♮ | D♭ | E𝄫 | F♭ | G♮ | A♭ | B𝄫 | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | ||
C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | ||
C♮ | D♭ | E♮ | F♮ | G♭ | A♭ | B♮ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | ||
C♮ | D♭ | E𝄫 | F♮ | G♭ | A𝄫 | B𝄫 | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | ||
C♮ | D♯ | E♮ | F♮ | G♯ | A♮ | B♭ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 |
Remember how I said above that Chromatic Dorian was directly relevant to this section? Well, there you go.
Scale-based transposition now. My base scales are Chromatic Lydian and Chromatic Hypophrygian inverse; this is an admittedly arbitrary choice that I made purely because they use the fewest accidentals on C. This also creates a neat pattern in the table below:
Greek Chromatic Tonoi & Their Inversions (mode-based roots, linear order) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
B♮ | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | ||
C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | ||
D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | C♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | ||
E♮ | F♮ | G♭ | A♮ | B♮ | C♮ | D♭ | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | ||
F♮ | G♭ | A♮ | B♮ | C♮ | D♭ | E♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | ||
G♭ | A♮ | B♮ | C♮ | D♭ | E♮ | F♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | ||
A♮ | B♮ | C♮ | D♭ | E♮ | F♮ | G♭ | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | ||
F♮ | G♮ | A♯ | B♮ | C♮ | D♯ | E♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | ||
E♮ | F♮ | G♮ | A♯ | B♮ | C♮ | D♯ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | ||
D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | C♮ | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | ||
C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | ||
B♮ | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ | ||
A♯ | B♮ | C♮ | D♯ | E♮ | F♮ | G♮ | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | ||
G♮ | A♯ | B♮ | C♮ | D♯ | E♮ | F♮ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 |
My “circle of fifths” order for the ancient Greek tonoi is:
- Hypolydian
- Lydian
- Hypophrygian
- Phrygian
- Hypodorian
- Dorian
- Mixolydian
This may help explain how the Greeks got the names Hypolydian, Hypophrygian, and Hypodorian in the first place: ὑπό (hupó) is literally Ancient Greek for under, and remember, the ancient Greeks’ scales went in what we consider descending order. Note what’s going on with the intervals in the following table. (Also note that I inverted the order for the inverted scales this time around because, as the table above clearly demonstrates, they’re actually moving in the opposite direction from their namesakes.)
Greek Chromatic Tonoi & Their Inversions (mode-based roots, “circle of fifths” order) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Chromatic Tonos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | |||||||
F♮ | G♭ | A♮ | B♮ | C♮ | D♭ | E♮ | ½ | 1½ | 1 | ½ | ½ | 1½ | ½ | ||
C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | ½ | 1½ | ½ | ½ | 1½ | 1 | ½ | ||
G♭ | A♮ | B♮ | C♮ | D♭ | E♮ | F♮ | 1½ | 1 | ½ | ½ | 1½ | ½ | ½ | ||
D♭ | E♮ | F♮ | G♭ | A♮ | B♮ | C♮ | 1½ | ½ | ½ | 1½ | 1 | ½ | ½ | ||
A♮ | B♮ | C♮ | D♭ | E♮ | F♮ | G♭ | 1 | ½ | ½ | 1½ | ½ | ½ | 1½ | ||
E♮ | F♮ | G♭ | A♮ | B♮ | C♮ | D♭ | ½ | ½ | 1½ | 1 | ½ | ½ | 1½ | ||
B♮ | C♮ | D♭ | E♮ | F♮ | G♭ | A♮ | ½ | ½ | 1½ | ½ | ½ | 1½ | 1 | ||
F♮ | G♮ | A♯ | B♮ | C♮ | D♯ | E♮ | 1 | 1½ | ½ | ½ | 1½ | ½ | ½ | ||
C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | 1½ | ½ | ½ | 1 | 1½ | ½ | ½ | ||
G♮ | A♯ | B♮ | C♮ | D♯ | E♮ | F♮ | 1½ | ½ | ½ | 1½ | ½ | ½ | 1 | ||
D♯ | E♮ | F♮ | G♮ | A♯ | B♮ | C♮ | ½ | ½ | 1 | 1½ | ½ | ½ | 1½ | ||
A♯ | B♮ | C♮ | D♯ | E♮ | F♮ | G♮ | ½ | ½ | 1½ | ½ | ½ | 1 | 1½ | ||
E♮ | F♮ | G♮ | A♯ | B♮ | C♮ | D♯ | ½ | 1 | 1½ | ½ | ½ | 1½ | ½ | ||
B♮ | C♮ | D♯ | E♮ | F♮ | G♮ | A♯ | ½ | 1½ | ½ | ½ | 1 | 1½ | ½ |
So effectively, Dorian and Hypodorian keep one tetrachord in the same place; the other tetrachord just swaps places with the extra whole-step. This actually continues to be true throughout the rest of the chromatic scales. Effectively, the extra whole-step either moves three places forward or four places back. Dividing the diatonic scale in this way shows us the same thing occurring with it:
Greek Diatonic Tonoi (C roots, circle of fifths order) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Modern | Ancient | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Intervals | ||||||
Lydian | Hypolydian | C♮ | D♮ | E♮ | F♯ | G♮ | A♮ | B♮ | 1 | 1 | 1 | ½ | 1 | 1 | ½ |
Ionian | Lydian | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♮ | 1 | 1 | ½ | 1 | 1 | 1 | ½ |
Mixolydian | Hypophrygian | C♮ | D♮ | E♮ | F♮ | G♮ | A♮ | B♭ | 1 | 1 | ½ | 1 | 1 | ½ | 1 |
Dorian | Phrygian | C♮ | D♮ | E♭ | F♮ | G♮ | A♮ | B♭ | 1 | ½ | 1 | 1 | 1 | ½ | 1 |
Aeolian | Hypodorian | C♮ | D♮ | E♭ | F♮ | G♮ | A♭ | B♭ | 1 | ½ | 1 | 1 | ½ | 1 | 1 |
Phrygian | Dorian | C♮ | D♭ | E♭ | F♮ | G♮ | A♭ | B♭ | ½ | 1 | 1 | 1 | ½ | 1 | 1 |
Locrian | Mixolydian | C♮ | D♭ | E♭ | F♮ | G♭ | A♭ | B♭ | ½ | 1 | 1 | ½ | 1 | 1 | 1 |
Every scale in this table lowers its predecessor’s extra whole-step to a half-step – and because the extra whole-step is always followed by a half-step, that half-step subsequently becomes a whole-step. This is, in fact, exactly the source of most of the patterns we’ve observed throughout our analysis of the diatonic major scale. I don’t know how much the ancient Greeks mapped this out and how much of it simply stemmed from intuition, but if it was by design, the designer was a genius, and I’m sad that their name has been lost to history.
I’ll represent the enharmonic tonoi with a similar table when I figure out how best to do so. As for the diatonic tonoi, well, just keep reading.
Why Our Modes Have Historically Inaccurate Names
Our Ionian mode’s chromatic counterpart is actually Chromatic Hypolydian, and our Aeolian mode’s counterpart is Chromatic Hypodorian. Why is that? Well, as I remarked above, medieval Europeans were confused. Four of our modern modes also had multiple names, three of which they borrowed from the Ancient Greek tonoi: their Hypodorian was our Aeolian, their Hypophrygian was our Locrian, and their Hypolydian was our Ionian. So, if we plug in those names and recenter the order around our Aeolian mode:
A Great Mode Discombobulation | ||
---|---|---|
Ancient | Medieval | Modern |
Dorian | Phrygian | Phrygian |
Hypolydian | Lydian | Lydian |
Hypophrygian | Mixolydian | Mixolydian |
Hypodorian | Hypodorian | Aeolian |
Mixolydian | Hypophrygian | Locrian |
Lydian | Hypolydian | Ionian |
Phrygian | Dorian | Dorian |
Visualizing the mistake they made becomes easier. In short, their misconception that Greek harmony went in ascending order led them to reverse the mode order. Since diatonic major has an odd number of notes, it also has an odd number of modes. Its inversion is also one of its own modes – which, to be clear, is not a given (for instance, it’s not true of the Greeks’ chromatic tonoi, which is why I listed their inversions separately above). For this to hold, one of the scale’s modes must be symmetrical – in this case, our Dorian mode:
Inverting the Diatonic Major Scale | |||||
---|---|---|---|---|---|
Modern | Ancient | Mode | Modern | Ancient | Inversion |
Aeolian | Hypodorian | W-H-W-W-H-W-W | Mixolydian | Hypophrygian | W-W-H-W-W-H-W |
Locrian | Mixolydian | H-W-W-H-W-W-W | Lydian | Hypolydian | W-W-W-H-W-W-H |
Ionian | Lydian | W-W-H-W-W-W-H | Phrygian | Dorian | H-W-W-W-H-W-W |
Dorian | Phrygian | W-H-W-W-W-H-W | Dorian | Phrygian | W-H-W-W-W-H-W |
Phrygian | Dorian | H-W-W-W-H-W-W | Ionian | Lydian | W-W-H-W-W-W-H |
Lydian | Hypolydian | W-W-W-H-W-W-H | Locrian | Mixolydian | H-W-W-H-W-W-W |
Mixolydian | Hypophrygian | W-W-H-W-W-H-W | Aeolian | Hypodorian | W-H-W-W-H-W-W |
Thus, by necessity, they were still going to get one right; it just happened to be our Aeolian mode. Why wasn’t it our Dorian mode? Apparently, they made an off-by-one error in assuming the Greeks listed the tonoi themselves in ascending order as well. Let’s move the tonoi on the right down by one and plug in their medieval names:
A Medieval Off-by-One Error | |||||
---|---|---|---|---|---|
Medieval | Ancient | Mode | Medieval | Ancient | Inversion − 1 |
Hypodorian | Hypodorian | W-H-W-W-H-W-W | Hypodorian | Hypodorian | W-H-W-W-H-W-W |
Hypophrygian | Mixolydian | H-W-W-H-W-W-W | Mixolydian | Hypophrygian | W-W-H-W-W-H-W |
Hypolydian | Lydian | W-W-H-W-W-W-H | Lydian | Hypolydian | W-W-W-H-W-W-H |
Dorian | Phrygian | W-H-W-W-W-H-W | Phrygian | Dorian | H-W-W-W-H-W-W |
Phrygian | Dorian | H-W-W-W-H-W-W | Dorian | Phrygian | W-H-W-W-W-H-W |
Lydian | Hypolydian | W-W-W-H-W-W-H | Hypolydian | Lydian | W-W-H-W-W-W-H |
Mixolydian | Hypophrygian | W-W-H-W-W-H-W | Hypophrygian | Mixolydian | H-W-W-H-W-W-W |
Now the ancient names on the left line up with the medieval names on the right, and vice versa.
Applied Greek Harmony: Tetrachords in Modern Scales
We’ve already seen how the diatonic major scale is a variant of a scale in which a tetrachord is repeated with a whole-tone separation (though in our Ionian mode, specifically, the second tetrachord is split midway through). To reiterate, let’s look at D Dorian, whose intervals are W-H-W-W-W-H-W:
- It opens with a W-H-W tetrachord (i.e., D, E, F, G).
- It features another whole step (i.e., G, A).
- It closes by repeating the W-H-W tetrachord (i.e., A, B, C, D).
And because the tetrachord itself is symmetrical, so is Dorian mode itself.
A few scales in Other Scales and Tonalities above are also built on two tetrachords separated by a whole step:
- Chromatic Dorian (½, ½, 1½, 1, ½, ½, 1½) and its modes literally are an ancient Greek genus.
- Chromatic Dorian inverse (1½, ½, ½, 1, 1½, ½, ½) and its modes also qualify.
- Flamenco mode (½, 1½, ½, 1, ½, 1½, ½) uses a ½–1½–½ tetrachord, a whole step, and another ½–1½–½ tetrachord. Flamenco mode also has the same reflective symmetry as our Dorian mode: since its repeated tetrachord is symmetrical and its extra whole-step occurs between them, it too is symmetrical.
Scales built on two tetrachords can be pleasing in their regularity, and they may be helpful starting places when you first write pieces that stray from the familiarity of our diatonic scale. Symmetrical scales built on two of the same tetrachord (e.g., flamenco mode or modes thereof) may be especially ideal starting places. I’d suggest inventing your own, but there aren’t any others.
But you can be creative in varying how the ancient Greeks constructed their harmony. One possibility: a nine-note scale featuring two of the same pentachord (five-note sequence) separated by a whole-step. To fit these criteria, your pentachord must span a perfect fourth (2½ steps), which unfortunately prevents it from being rotationally symmetrical - your options are W-H-H-H, H-W-H-H, H-H-W-H, or H-H-H-W.
If you want rotational symmetry, though, you could invert the second pentachord:
- W-H-H-H-W-H-H-H-W ( )
- H-W-H-H-W-H-H-W-H ( )
- H-H-W-H-W-H-W-H-H ( )
- H-H-H-W-W-W-H-H-H ( ; this one seems especially daring)
The second option, H-W-H-H-W-H-H-W-H, can also be constructed by repeating the same trichord (three-note sequence), H-W, with a half-step separation each time, which is another interesting variation on the ancient Greek idea. Within it, each trichord spans a minor third; the added half-step above it means that the same interval pattern repeats every major third. Above its root key, it also includes a minor third, a major third, a perfect fourth, and a perfect fifth above its root key. These make it potentially a very versatile scale. (Since its dominant chord is diminished and it excludes the major second above its root, it also shares some harmonic characteristics with Phrygian mode and Phrygian dominant.) But these aren’t the only possible variants – be creative!
Then again, you may prefer harmonic minor or ascending melodic minor, which respectively only lower one note of Aeolian mode and raise one note of Ionian mode; both are also so ubiquitous in Western music that they may be intuitive. Neither, however, possess the repeated tetrachord of the ancient Greek genera. (Ascending melodic minor does possess a symmetrical mode, Aeolian dominant [W-W-H-W-H-W-W]; harmonic minor does not.)
Acknowledgements & Sources
I first wish to thank Marty O’Donnell (yes, that Marty O’Donnell) for pointing out that it wasn’t sufficiently clear what parts of this section I was oversimplifying; his feedback resulted in a much clearer explanation. However, the opinions presented in this section are entirely my own. In particular, Marty believes modern scholars have extrapolated more from ancient Greek texts than is merited by their contents – and in the interest of fairness, I must point out that he has a degree in music theory, and I don’t. However, I must also be fair to myself: Marty got his degree decades ago, and a lot of music scholarship has been done since then.
But even then, I’ll be the first person to admit that I’m by no means an infallible source, so here are some starting places for readers wishing to learn more about this subject. ⟨en
I consulted several other resources researching this section; many were too technical to be of interest to non-specialists, but those seeking more detailed technical analysis of ancient Greek tuning systems may be interested in Robert Erickson’s ⟨ex-tempore
But I think it’s most helpful to quote the ancient Greeks in their own words (or as close to their words as English speakers without educations in Attic Greek will understand), so here, via Cris Forester’s ⟨chrysalis-foundation
The magnitude of harmonia is syllaba and di’oxeian. The di’oxeian is greater than the syllaba in epogdoic ratio. From hypate [E] to mese [A] is a syllaba, from mese [A] to neate [or nete, E¹] is a di’oxeian, from neate [E¹] to trite [later paramese, B] is a syllaba, and from trite [B] to hypate [E] is a di’oxeian. The interval between trite [B] and mese [A] is epogdoic [9:8], the syllaba is epitritic [4:3], the di’oxeian hemiolic [3:2], and the dia pason is duple [2:1]. Thus harmonia consists of five epogdoics and two dieses; di’oxeian is three epogdoics and a diesis, and syllaba is two epogdoics and a diesis.—Philolaus, translated by Andrew Barker, Greek Musical Writings, Vol. 2 (1989:
Cambridge University Press). [Text and ratios in brackets are Cris Forester’s.]
Difficult as this is to parse, a close reading reveals Philolaus to be describing the 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.
Philolaus’ description is so exact that I believe we can conclude from it that the ancient Greeks routinely used something very similar to our diatonic major scale. However, I must reiterate: “ancient Greek harmony” refers to over a millennium of musical practices that were by no means uniform; various authors described other tonoi and indeed entirely different tuning systems. I’ve focused on the tonoi described above for two reasons:
- They’re easy to equate to modern tuning systems.
- They clearly inspired (four of) our modern modes’ names.
(Keep an eye on this page – I still intend to add more information on the medieval church modes that served as the precursors to our modern modes.)