Musical Modes and the Circle of Fifths

Table of Contents

  1. Contents
  2. Introduction
    1. The Seven Modes of the C Major Scale
    2. These are just examples; it could also be something much better
    3. The Modes in “Circle of Fifths” Order
      1. Modes Descending from Lydian
    4. The Principles of Inverse Operations
    5. An Audio Demonstration
    6. Further Notes
      1. Diabolus in mūsicā
  3. The Major Scale’s Modes & the Circle of Fifths
    1. 12 Major Scales × 7 Modes = 84 Combinations
      1. C   (B♯)
      2. B   (C♭)
      3. A♯ / B♭
      4. A
      5. G♯ / A♭
      6. G
      7. F♯ / G♭
      8. F   (E♯)
      9. E   (F♭)
      10. D♯ / E♭
      11. D
      12. C♯ / D♭
    2. Key Signature Cheat Sheet
      1. Key Signatures of the Seven Modes for the Twelve-Note Chromatic Scale
    3. Why is this happening?
      1. C++
      2. You were expecting modes, but it was me, Dio the circle of fifths!
    4. Chord Analysis by Mode
      1. Chord Tonalities by Scale Position and Mode
  4. Beyond the Major Scale
    1. Other Scales & Tonalities
      1. Other Seven-Note Scales
      2. Harmonic Minor & Ascending Melodic Minor’s Modes at a Glance
    2. Harmonic Minor’s Modes in Detail
      1. Harmonic Minor vs. Modes from Aeolian (rooted on C, in ascending order)
      2. Harmonic Minor vs. Modes from Aeolian (rooted on scale, in ascending order)
      3. Harmonic Minor vs. Modes from Aeolian (rooted on C, in “circle of fifths” order)
      4. Harmonic Minor vs. Modes from Aeolian (rooted on scale, in “circle of fifths” order)
    3. Ascending Melodic Minor’s Modes in Detail
      1. Ascending Melodic Minor vs. Modes from Ionian (rooted on C, in ascending order)
      2. Ascending Melodic Minor vs. Modes from Ionian (rooted on scale, in ascending order)
      3. Ascending Melodic Minor vs. Modes from Ionian (rooted on C, in “circle of fifths” order)
      4. Ascending Melodic Minor vs. Modes from Ionian (rooted on scale, in “circle of fifths” order)
  5. A Crash Course in Ancient Greek Harmony
    1. Etymology
      1. Αἱ ἐτῠμολογῐ́αι τῶν ἑπτᾰ́ τόνων
    2. Ancient Greek Tonoi & Modern Modes
      1. Interval Key
      2. Interval Ratios of a Diatonic Tetrachord
      3. Interval Genera: A Feed from Cloud Mountain
    3. Ancient Greek Tonoi & Modern Modes
      1. Approximate Intervals of Ancient Greek Tonoi & Modern Diatonic Modes
      2. Greek Chromatic Tonoi & Their Inversions (C roots, linear order)
      3. Greek Chromatic Tonoi & Their Inversions (mode-based roots, linear order)
      4. Greek Chromatic Tonoi & Their Inversions (mode-based roots, “circle of fifths” order)
      5. Greek Diatonic Tonoi (C roots, circle of fifths order)
    4. Why Our Modes Have Historically Inaccurate Names
      1. A Great Mode Discombobulation
      2. Inverting the Diatonic Major Scale
      3. A Medieval Off-by-One Error
    5. Applied Greek Harmony: Tetrachords in Modern Scales
    6. Acknowledgements & Sources
    7. Appendix: Interval Ratios of 12- and 24-Tone Equal Temperament
      1. 24-Tone Equal Temperament’s Interval Ratios
  6. Yes, but, why?
  7. Endnotes
    1. Greek Chromatic Tonoi & Their Inversions (mode-based roots, OCD order 1)
    2. Greek Chromatic Tonoi & Their Inversions (mode-based roots, OCD order 2)

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 ⟨aaronfreed.github.io/aboutme.html⟩ 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.

The Seven Modes of the C Major Scale
# Mode 12 3 4 5 6 7 Comment
1Ionian C D E F G A BThe traditional major scale.
2Dorian D Ef G A Bc
3Phrygian Efg A Bcd
4Lydian F G AB C D EThe only mode that raises a note above Ionian.
5MixolydianG A B C D Ef
6Aeolian A Bc D EfgThe natural minor scale.
7Locrian Bcd EfgaRarely 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’.

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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

Immediately recognizable examples of each mode include:

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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):

Modes Descending from Lydian⁽¹⁾
# Root & mode Pitch lowered 1 2 3 4 5 6 7 1
4C Lydian C D E F♯G A B C
1C Ionian 4 C D E F G A B C
5C Mixolydian7 C D E F G A B♭C
2C Dorian 3 C D E♭F G A B♭C
6C Aeolian 6 C D E♭F G A♭B♭C
3C Phrygian 2 C D♭E♭F G A♭B♭C
7C Locrian 5 C D♭E♭F G♭A♭B♭C
4C♭Lydian 1 C♭D♭E♭F G♭A♭B♭C♭

Back to top · My discography · Marathon soundtracks · Contact me · Website index

The Principles of Inverse Operations

Back to top · My discography · Marathon soundtracks · Contact me · Website index

An Audio Demonstration

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Further Notes

Back to top · My discography · Marathon soundtracks · Contact me · Website index

The Major Scale’s Modes & the Circle of Fifths

12 Major Scales × 7 Modes = 84 Combinations

C   (B♯)
# Root Mode RMKS1 2 3 4 5 6 7 1
 0C4 – Lydian G 1♯ CDEF♯ GABC
 1C1 – Ionian C CDEFGABC
 2C5 – MixolydianF 1♭ CDEFGAB♭ C
 3C2 – Dorian B♭ 2♭ CDE♭ FGAB♭ C
 4C6 – Aeolian E♭ 3♭ CDE♭ FGA♭ B♭ C
 5C3 – Phrygian A♭ 4♭ CD♭ E♭ FGA♭ B♭ C
 6C7 – Locrian D♭ 5♭ CD♭ E♭ FG♭ A♭ B♭ C

Back to top · My discography · Marathon soundtracks · Contact me · Website index

B   (C♭)
# Root Mode RMKS1 2 3 4 5 6 7 1
 7C♭ 4 – Lydian G♭ 6♭ C♭ D♭ E♭ FG♭ A♭ B♭ C♭
 7B4 – Lydian F♯ 6♯ BC♯ D♯ E♯ F♯ G♯ A♯ B
 8B1 – Ionian B 5♯ BC♯ D♯ EF♯ G♯ A♯ B
 9B5 – MixolydianE 4♯ BC♯ D♯ EF♯ G♯ AB
10 B2 – Dorian A 3♯ BC♯ DEF♯ G♯ AB
11 B6 – Aeolian D 2♯ BC♯ DEF♯ GAB
12 B3 – Phrygian G 1♯ BCDEF♯ GAB
13 B7 – Locrian C BCDEFGAB

Back to top · My discography · Marathon soundtracks · Contact me · Website index

A♯ / B♭
# RootMode RMKS12 3 4 5 6 7 1
14 B♭ 4 – Lydian F 1♭ B♭ CDEFGAB♭
15 B♭ 1 – Ionian B♭ 2♭ B♭ CDE♭ FGAB♭
16 B♭ 5 – MixolydianE♭ 3♭ B♭ CDE♭ FGA♭ B♭
17 B♭ 2 – Dorian A♭ 4♭ B♭ CD♭ E♭ FGA♭ B♭
18 B♭ 6 – Aeolian E♭ 5♭ B♭ CD♭ E♭ FG♭ A♭ B♭
19 B♭ 3 – Phrygian G♭ 6♭ B♭ C♭ D♭ E♭ FG♭ A♭ B♭
19 A♯ 3 – Phrygian F♯ 6♯ A♯ BC♯ D♯ E♯ F♯ G♯ A♯
20 A♯ 7 – Locrian B 5♯ A♯ BC♯ D♯ EF♯ G♯ A♯

Back to top · My discography · Marathon soundtracks · Contact me · Website index

A
#Root Mode RMKS1 2 3 4 5 6 7 1
21 A4 – Lydian E 4♯ ABC♯ D♯ EF♯ G♯ A
22 A1 – Ionian A 3♯ ABC♯ DEF♯ G♯ A
23 A5 – MixolydianD 2♯ ABC♯ DEF♯ GA
24 A2 – Dorian G 1♯ ABCDEF♯ GA
25 A6 – Aeolian C ABCDEFGA
26 A3 – Phrygian F 1♭ AB♭ CDEFGA
27 A7 – Locrian B♭ 2♭ AB♭ CDE♭ FGA

Back to top · My discography · Marathon soundtracks · Contact me · Website index

G♯ / A♭
#RootMode RMKS12 3 4 5 6 7 1
28 A♭ 4 – Lydian E♭ 3♭ A♭ B♭ CDE♭ FGA♭
29 A♭ 1 – Ionian A♭ 4♭ A♭ B♭ CD♭ E♭ FGA♭
30 A♭ 5 – MixolydianD♭ 5♭ A♭ B♭ CD♭ E♭ FG♭ A♭
31 A♭ 2 – Dorian G♭ 6♭ A♭ B♭ C♭ D♭ E♭ FG♭ A♭
31 G♯ 2 – Dorian F♯ 6♯ G♯ A♯ BC♯ D♯ E♯ F♯ G♯
32 G♯ 6 – Aeolian B 5♯ G♯ A♯ BC♯ D♯ EF♯ G♯
33 G♯ 3 – Phrygian E 4♯ G♯ ABC♯ D♯ EF♯ G♯
34 G♯ 7 – Locrian A 3♯ G♯ ABC♯ DEF♯ G♯

Back to top · My discography · Marathon soundtracks · Contact me · Website index

G
#Root Mode RMKS1 2 3 4 5 6 7 1
35 G4 – Lydian D 2♯ GABC♯ DEF♯ G
36 G1 – Ionian G 1♯ GABCDEF♯ G
37 G5 – MixolydianC GABCDEFG
38 G2 – Dorian F 1♭ GAB♭ CDEFG
39 G6 – Aeolian B♭ 2♭ GAB♭ CDE♭ FG
40 G3 – Phrygian E♭ 3♭ GA♭ B♭ CDE♭ FG
41 G7 – Locrian A♭ 4♭ GA♭ B♭ CD♭ E♭ FG

Back to top · My discography · Marathon soundtracks · Contact me · Website index

F♯ / G♭
#RootMode RMKS12 3 4 5 6 7 1
42 G♭ 4 – Lydian D♭ 5♭ G♭ A♭ B♭ CD♭ E♭ FG♭
43 G♭ 1 – Ionian G♭ 6♭ G♭ A♭ B♭ C♭ D♭ E♭ FG♭
43 F♯ 1 – Ionian F♯ 6♯ F♯ G♯ A♯ BC♯ D♯ E♯ F♯
44 F♯ 5 – MixolydianB 5♯ F♯ G♯ A♯ BC♯ D♯ EF♯
45 F♯ 2 – Dorian E 4♯ F♯ G♯ ABC♯ D♯ EF♯
46 F♯ 6 – Aeolian A 3♯ F♯ G♯ ABC♯ DEF♯
47 F♯ 3 – Phrygian D 2♯ F♯ GABC♯ DEF♯
48 F♯ 7 – Locrian G 1♯ F♯ GABCDEF♯

Back to top · My discography · Marathon soundtracks · Contact me · Website index

F   (E♯)
#Root Mode RMKS1 2 3 4 5 6 7 1
49 F4 – Lydian C FGABCDEF
50 F1 – Ionian F 1♭ FGAB♭ CDEF
51 F5 – MixolydianB♭ 2♭ FGAB♭ CDE♭ F
52 F2 – Dorian E♭ 3♭ FGA♭ B♭ CDE♭ F
53 F6 – Aeolian A♭ 4♭ FGA♭ B♭ CD♭ E♭ F
54 F3 – Phrygian D♭ 5♭ FG♭ A♭ B♭ CD♭ E♭ F
55 F7 – Locrian G♭ 6♭ FG♭ A♭ B♭ C♭ D♭ E♭ F
55 E♯ 7 – Locrian F♯ 6♯ E♯ F♯ G♯ A♯ BC♯ D♯ E♯

Back to top · My discography · Marathon soundtracks · Contact me · Website index

E   (F♭)
#Root Mode RMKS1 2 3 4 5 6 7 1
56 E4 – Lydian B 5♯ EF♯ G♯ A♯ BC♯ D♯ E
57 E1 – Ionian E 4♯ EF♯ G♯ ABC♯ D♯ E
58 E5 – MixolydianA 3♯ EF♯ G♯ ABC♯ DE
59 E2 – Dorian D 2♯ EF♯ GABC♯ DE
60 E6 – Aeolian G 1♯ EF♯ GABCDE
61 E3 – Phrygian C EFGABCDE
62 E7 – Locrian F 1♭ EFGAB♭ CDE

Back to top · My discography · Marathon soundtracks · Contact me · Website index

D♯ / E♭
#RootMode RMKS12 3 4 5 6 7 1
63 E♭ 4 – Lydian B♭ 2♭ E♭ FGAB♭ CDE♭
64 E♭ 1 – Ionian E♭ 3♭ E♭ FGA♭ B♭ CDE♭
65 E♭ 5 – MixolydianA♭ 4♭ E♭ FGA♭ B♭ CD♭ E♭
66 E♭ 2 – Dorian D♭ 5♭ E♭ FG♭ A♭ B♭ CD♭ E♭
67 E♭ 6 – Aeolian G♭ 6♭ E♭ FG♭ A♭ B♭ C♭ D♭ E♭
67 D♯ 6 – Aeolian F♯ 6♯ D♯ E♯ F♯ G♯ A♯ BC♯ D♯
68 D♯ 3 – Phrygian B 5♯ D♯ EF♯ G♯ A♯ BC♯ D♯
69 D♯ 7 – Locrian E 4♯ D♯ EF♯ G♯ ABC♯ D♯

Back to top · My discography · Marathon soundtracks · Contact me · Website index

D
#Root Mode RMKS1 2 3 4 5 6 7 1
70 D4 – Lydian A 3♯ DEF♯ G♯ ABC♯ D
71 D1 – Ionian D 2♯ DEF♯ GABC♯ D
72 D5 – MixolydianG 1♯ DEF♯ GABCD
73 D2 – Dorian C DEFGABCD
74 D6 – Aeolian F 1♭ DEFGAB♭ CD
75 D3 – Phrygian B♭ 2♭ DE♭ FGAB♭ CD
76 D7 – Locrian E♭ 3♭ DE♭ FGA♭ B♭ CD

Back to top · My discography · Marathon soundtracks · Contact me · Website index

C♯ / D♭
#RootMode RMKS12 3 4 5 6 7 1
77 D♭ 4 – Lydian A♭ 4♭ D♭ E♭ FGA♭ B♭ CD♭
78 D♭ 1 – Ionian D♭ 5♭ D♭ E♭ FG♭ A♭ B♭ CD♭
79 D♭ 5 – MixolydianG♭ 6♭ D♭ E♭ FG♭ A♭ B♭ C♭ D♭
79 C♯ 5 – MixolydianF♯ 6♯ C♯ D♯ E♯ F♯ G♯ A♯ BC♯
80 C♯ 2 – Dorian B 5♯ C♯ D♯ EF♯ G♯ A♯ BC♯
81 C♯ 6 – Aeolian E 4♯ C♯ D♯ EF♯ G♯ ABC♯
82 C♯ 3 – Phrygian A 3♯ C♯ DEF♯ G♯ ABC♯
83 C♯ 7 – Locrian D 2♯ C♯ DEF♯ GABC♯

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Key Signature Cheat Sheet

Key Signatures of the Seven Modes for the Twelve-Note Chromatic Scale
LydMajMixDorMin PhrLoc 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♭

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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 RMKS1 2 3 4 5 6 7 1
 0C4 – Lydian G 1♯ CDEF♯ GABC
 1C1 – Ionian C CDEFGABC
 2C5 – MixolydianF 1♭ CDEFGAB♭ C
 3C2 – Dorian B♭ 2♭ CDE♭ FGAB♭ C
 4C6 – Aeolian E♭ 3♭ CDE♭ FGA♭ B♭ C
 5C3 – Phrygian A♭ 4♭ CD♭ E♭ FGA♭ B♭ C
 6C7 – Locrian D♭ 5♭ CD♭ E♭ FG♭ 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 KS1 2 3 4 5 6 7 1
 04 – Lydian G 1♯ GABCDEF♯ G
 11 – Ionian C CDEFGABC
 25 – MixolydianF 1♭ FGAB♭ CDEF
 32 – Dorian B♭2♭ B♭ CDE♭ FGAB♭
 46 – Aeolian E♭3♭ E♭ FGA♭ B♭ CDE♭
 53 – Phrygian A♭4♭ A♭ B♭ CD♭ E♭ FGA♭
 67 – Locrian D♭5♭ D♭ E♭ FG♭ A♭ B♭ CD♭

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.

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Chord Analysis by Mode

Chord Tonalities by Scale Position & Mode
ModeIIIIIIIVVVIVII
1Ionian Majminmin Maj Majmindim
2Dorian minmin Maj Majmindim Maj
3Phrygian min Maj Majmindim Majmin
4Lydian Maj Majmindim Majminmin
5Mixolydian Majmindim Majminmin Maj
6Aeolian mindim Majminmin Maj Maj
7Locrian dim Majminmin Maj Majmin

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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:

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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 234567Intervals
Harmonic minorCDE♭FGA♭B1½11½½
AeolianCDE♭FGA♭B♭1½11½11
Locrian ♮6Maqam TarznauynCD♭E♭FG♭AB♭½11½½1
LocrianCD♭E♭FG♭A♭B♭½11½111
Augmented majorIonian ♯5CDEFG♯AB11½½1½
IonianCDEFGAB11½111½
Ukrainian DorianCDE♭F♯GAB♭1½½1½1
DorianCDE♭FGAB♭1½111½1
Phrygian dominantCD♭EFGA♭B♭½½1½11
PhrygianCD♭E♭FGA♭B♭½111½11
Aeolian harmonicLydian ♯2CD♯EF♯GAB½1½11½
LydianCDEF♯GAB111½11½
Altered diminishedSuper-Locrian 𝄫2CD♭E♭F♭G♭A♭B𝄫½1½11½
MixolydianCDEFGAB♭11½11½1

Thus, harmonic minor’s modes respectively raise by a half-step:

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 234567Intervals
Harmonic minor CDE♭FGA♭B 1½11½½
Aeolian CDE♭FGA♭B♭ 1½11½11
Locrian ♮6Maqam Tarznauyn DE♭FGA♭BC ½11½½1
Locrian DE♭FGA♭B♭C ½11½111
Augmented majorIonian ♯5 E♭FGA♭BCD 11½½1½
Ionian E♭FGA♭B♭CD 11½111½
Ukrainian Dorian FGA♭BCDE♭ 1½½1½1
Dorian FGA♭B♭CDE♭ 1½111½1
Phrygian dominant GA♭BCDE♭F ½½1½11
Phrygian GA♭B♭CDE♭F ½111½11
Aeolian harmonicLydian ♯2 A♭BCDE♭FG ½1½11½
Lydian A♭B♭CDE♭FG 111½11½
Altered diminishedSuper-Locrian 𝄫2 BCDE♭FGA♭ ½1½11½
Mixolydian B♭CDE♭FGA♭ 11½11½1

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 234567Intervals
Aeolian harmonicLydian ♯2CD♯EF♯GAB½1½11½
LydianCDEF♯GAB111½11½
Augmented majorIonian ♯5CDEFG♯AB11½½1½
IonianCDEFGAB11½111½
Altered diminishedSuper-Locrian 𝄫2CD♭E♭F♭G♭A♭B𝄫½1½11½
MixolydianCDEFGAB♭11½11½1
Ukrainian DorianCDE♭F♯GAB♭1½½1½1
DorianCDE♭FGAB♭1½111½1
Harmonic minorCDE♭FGA♭B1½11½½
AeolianCDE♭FGA♭B♭1½11½11
Phrygian dominantCD♭EFGA♭B♭½½1½11
PhrygianCD♭E♭FGA♭B♭½111½11
Locrian ♮6Maqam TarznauynCD♭E♭FG♭AB♭½11½½1
LocrianCD♭E♭FG♭A♭B♭½11½111

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 234567Intervals
Aeolian harmonicLydian ♯2 A♭BCDE♭FG ½1½11½
Lydian A♭B♭CDE♭FG 111½11½
Augmented majorIonian ♯5 E♭FGA♭BCD 11½½1½
Ionian E♭FGA♭B♭CD 11½111½
Altered diminishedSuper-Locrian 𝄫2 BCDE♭FGA♭ ½1½11½
Mixolydian B♭CDE♭FGA♭ 11½11½1
Ukrainian Dorian FGA♭BCDE♭ 1½½1½1
Dorian FGA♭B♭CDE♭ 1½111½1
Harmonic minor CDE♭FGA♭B 1½11½½
Aeolian CDE♭FGA♭B♭ 1½11½11
Phrygian dominant GA♭BCDE♭F ½½1½11
Phrygian GA♭B♭CDE♭F ½111½11
Locrian ♮6Maqam Tarznauyn DE♭FGA♭BC ½11½½1
Locrian DE♭FGA♭B♭C ½11½111

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.

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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 234567Intervals
IonianCDEFGAB11½111½
Ascending melodic minorCDE♭FGAB1½1111½
DorianCDE♭FGAB♭1½111½1
Jazz minor inverseCD♭E♭FGAB♭½1111½1
PhrygianCD♭E♭FGA♭B♭½111½11
Lydian augmentedCDEF♯G♯AB1111½1½
LydianCDEF♯GAB111½11½
Lydian dominantMixolydian ♯4CDEF♯GAB♭111½1½1
MixolydianCDEFGAB♭11½11½1
Aeolian dominantMixolydian ♭6CDEFGA♭B♭11½1½11
AeolianCDE♭FGA♭B♭1½11½11
Half-diminishedCDE♭FG♭A♭B♭1½1½111
LocrianCD♭E♭FG♭A♭B♭½11½111
Altered dominantSuper-LocrianCD♭E♭F♭G♭A♭B♭½1½1111

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 234567Intervals
Ionian CDEFGAB 11½111½
Ascending melodic minor CDE♭FGAB 1½1111½
Dorian DEFGABC 1½111½1
Jazz minor inverse DE♭FGABC ½1111½1
Phrygian EFGABCD ½111½11
Lydian augmented E♭FGABCD 1111½1½
Lydian FGABCDE 111½11½
Lydian dominantMixolydian ♯4 FGABCDE♭ 111½1½1
Mixolydian GABCDEF 11½11½1
Aeolian dominantMixolydian ♭6 GABCDE♭F 11½1½11
Aeolian ABCDEFG 1½11½11
Half-diminished ABCDE♭FG 1½1½111
Locrian BCDEFGA ½11½111
Altered dominantSuper-Locrian BCDE♭FGA ½1½1111

“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 234567Intervals
LydianCDEF♯GAB111½11½
Lydian dominantMixolydian ♯4CDEF♯GAB♭111½1½1
IonianCDEFGAB11½111½
Ascending melodic minorCDE♭FGAB1½1111½
MixolydianCDEFGAB♭11½11½1
Aeolian dominantMixolydian ♭6CDEFGA♭B♭11½1½11
DorianCDE♭FGAB♭1½111½1
Jazz minor inverseCD♭E♭FGAB♭½1111½1
AeolianCDE♭FGA♭B♭1½11½11
Half-diminishedCDE♭FG♭A♭B♭1½1½111
PhrygianCD♭E♭FGA♭B♭½111½11
Lydian augmentedCDEF♯G♯AB1111½1½
LocrianCD♭E♭FG♭A♭B♭½11½111
Altered dominantSuper-LocrianCD♭E♭F♭G♭A♭B♭½1½1111

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 234567Intervals
Lydian FGABCDE 111½11½
Lydian dominantMixolydian ♯4 FGABCDE♭ 111½1½1
Ionian CDEFGAB 11½111½
Ascending melodic minor CDE♭FGAB 1½1111½
Mixolydian GABCDEF 11½11½1
Aeolian dominantMixolydian ♭6 GABCDE♭F 11½1½11
Dorian DEFGABC 1½111½1
Jazz minor inverse DE♭FGABC ½1111½1
Aeolian ABCDEFG 1½11½11
Half-diminished ABCDE♭FG 1½1½111
Phrygian EFGABCD ½111½11
Lydian augmented E♭FGABCD 1111½1½
Locrian BCDEFGA ½11½111
Altered dominantSuper-Locrian BCDE♭FGA ½1½1111

(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.)

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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
1Ionian Ἰωνία Iōnía region on the western coast of Anatolia (modern Turkey)
2Dorian Δωρῐεύς Dōrieús one of the four major Hellenic tribes
3Phrygian Φρῠγῐ́ᾱ Phrugíā kingdom in west-central Anatolia
4Lydian Λῡδῐ́ᾱ Lūdíā Anatolian kingdom most famously ruled by Croesus
5Mixolydianμιξο-Λῡ́δῐοςmixo-Lū́diosliterally “mixed Lydian”
6Aeolian Αἰολῐ́ᾱ Aiolíā region of northwestern Anatolia
7Locrian Λοκρῐ́ς Lokrís Three separate regions ⟨en.wikipedia.org/wiki/Locris⟩ 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.

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Ancient Greek Harmony: The Cliffs Notes

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. 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 the Seikilos epitaph ⟨en.wikipedia.org/wiki/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 ⟨en.wikipedia.org/wiki/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 how Greek music sounded; 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 ⟨en.wikipedia.org/wiki/Quarter_tone⟩ (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
 ¼Infra secondQuarter-tone ²⁴√2:11.02930223664
 ½Minor secondSemitone ¹²√2:11.05946309436
Major secondWhole tone ⁶√2:11.12246204831
Minor third Three semitones ⁴√2:11.18920711500
Major third Two whole tones ³√2:11.25992104989

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
Philolaus256 :2431.05349794239 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 1.05946309436⁶√2:11.12246204831 ⁶√2:11.12246204831

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⁽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 intervalMiddle intervalHigh interval
EnharmonicInfra secondInfra second Major third
Chromatic Minor secondMinor second Minor third
Diatonic Minor secondMajor second Major second

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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:

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 ½  ½ 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 1¼
Lydian Chromatic ½ ½  ½ 1½
Lydian Diatonic 1 1 ½ 1 1 1½
Ionian Modern 1 1 ½ 1 1 1½
Phrygian Enharmonic 2 ¼ ¼ 2 1¼ ¼
Phrygian Chromatic ½ ½ 1½ ½
Phrygian Diatonic 1 ½ 1 1 1½ 1
Dorian Modern 1 ½ 1 1 1½ 1
Dorian Enharmonic ¼ ¼ 2 1¼ ¼ 2
Dorian Chromatic ½  ½ 1½ ½
Dorian Diatonic ½ 1 1½ 1 1
Phrygian Modern ½ 1 1½ 1 1
Hypolydian Enharmonic ¼ 1¼ ¼ 2 ¼
Hypolydian Chromatic ½ 1½ ½ ½
Hypolydian Diatonic 1 1 1½ 1 1 ½
Lydian Modern 1 1 1½ 1 1 ½
HypophrygianEnharmonic 2 1¼ ¼ 2 ¼ ¼
HypophrygianChromatic 1½ ½ ½ ½
HypophrygianDiatonic 1 1½ 1 1 ½ 1
Mixolydian Modern 1 1½ 1 1 ½ 1
Hypodorian Enharmonic 1 ¼ ¼ 2 ¼ ¼ 2
Hypodorian Chromatic 1 ½ ½ ½ ½
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 234567Intervals
MixolydianCD♭E𝄫FG♭A𝄫B♭½½½½1
LydianCD♭EFG♭AB½½½1½
PhrygianCD♯EFG♯A♯B½½1½½
DorianCD♭E𝄫FGA♭B𝄫½½1½½
HypolydianCD♭EF♯GA♭B½1½½½
HypophrygianCD♯E♯F♯GA♯B1½½½½
HypodorianCDE♭F♭GA♭B𝄫1½½½½
Mixolydian inverseCDE♯F♯GA♯B1½½½½
Lydian inverseCD♭E♭F♯GA♭B½1½½½
Phrygian inverseCD♭E𝄫F♭GA♭B𝄫½½1½½
Dorian inverseCD♯EFGA♯B½½1½½
Hypolydian inverseCD♭EFG♭A♭B½½½1½
Hypophrygian inverseCD♭E𝄫FG♭A𝄫B𝄫½½½½1
Hypodorian inverseCD♯EFG♯AB♭½½½½1

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 234567Intervals
Mixolydian BCD♭EFG♭A ½½½½1
Lydian CD♭EFG♭AB ½½½1½
Phrygian D♭EFG♭ABC ½½1½½
Dorian EFG♭ABCD♭ ½½1½½
Hypolydian FG♭ABCD♭E ½1½½½
Hypophrygian G♭ABCD♭EF 1½½½½
Hypodorian ABCD♭EFG♭ 1½½½½
Mixolydian inverse FGA♯BCD♯E 1½½½½
Lydian inverse EFGA♯BCD♯ ½1½½½
Phrygian inverse D♯EFGA♯BC ½½1½½
Dorian inverse CD♯EFGA♯B ½½1½½
Hypolydian inverse BCD♯EFGA♯ ½½½1½
Hypophrygian inverse A♯BCD♯EFG ½½½½1
Hypodorian inverse GA♯BCD♯EF ½½½½1

My “circle of fifths” order for the ancient Greek tonoi is:

  1. Hypolydian
  2. Lydian
  3. Hypophrygian
  4. Phrygian
  5. Hypodorian
  6. Dorian
  7. 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 234567Intervals
Hypolydian FG♭ABCD♭E ½1½½½
Lydian CD♭EFG♭AB ½½½1½
Hypophrygian G♭ABCD♭EF 1½½½½
Phrygian D♭EFG♭ABC ½½1½½
Hypodorian ABCD♭EFG♭ 1½½½½
Dorian EFG♭ABCD♭ ½½1½½
Mixolydian BCD♭EFG♭A ½½½½1
Mixolydian inverse FGA♯BCD♯E 1½½½½
Dorian inverse CD♯EFGA♯B ½½1½½
Hypodorian inverse GA♯BCD♯EF ½½½½1
Phrygian inverse D♯EFGA♯BC ½½1½½
Hypophrygian inverse A♯BCD♯EFG ½½½½1
Lydian inverse EFGA♯BCD♯ ½1½½½
Hypolydian inverse BCD♯EFGA♯ ½½½1½

So effectively, Dorian and Hypodorian keep one tetrachord in the same place; the other tetrachord just swaps places with the extra whole-step.⁽⁹⁾ 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)
ModernAncient1 234567Intervals
LydianHypolydianCDEF♯GAB111½11½
IonianLydianCDEFGAB11½111½
MixolydianHypophrygianCDEFGAB♭11½11½1
DorianPhrygianCDE♭FGAB♭1½111½1
AeolianHypodorianCDE♭FGA♭B♭1½11½11
PhrygianDorianCD♭E♭FGA♭B♭½111½11
LocrianMixolydianCD♭E♭FG♭A♭B♭½11½111

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
AncientMedievalModern
DorianPhrygianPhrygian
HypolydianLydianLydian
HypophrygianMixolydianMixolydian
HypodorianHypodorianAeolian
MixolydianHypophrygianLocrian
LydianHypolydianIonian
PhrygianDorianDorian

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
ModernAncientModeModernAncientInversion
AeolianHypodorianW-H-W-W-H-W-WMixolydianHypophrygianW-W-H-W-W-H-W
LocrianMixolydianH-W-W-H-W-W-WLydianHypolydianW-W-W-H-W-W-H
IonianLydianW-W-H-W-W-W-HPhrygianDorianH-W-W-W-H-W-W
DorianPhrygianW-H-W-W-W-H-WDorianPhrygianW-H-W-W-W-H-W
PhrygianDorianH-W-W-W-H-W-WIonianLydianW-W-H-W-W-W-H
LydianHypolydianW-W-W-H-W-W-HLocrianMixolydianH-W-W-H-W-W-W
MixolydianHypophrygianW-W-H-W-W-H-WAeolianHypodorianW-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
MedievalAncientModeMedievalAncientInversion − 1
HypodorianHypodorianW-H-W-W-H-W-WHypodorianHypodorianW-H-W-W-H-W-W
HypophrygianMixolydianH-W-W-H-W-W-WMixolydianHypophrygianW-W-H-W-W-H-W
HypolydianLydianW-W-H-W-W-W-HLydianHypolydianW-W-W-H-W-W-H
DorianPhrygianW-H-W-W-W-H-WPhrygianDorianH-W-W-W-H-W-W
PhrygianDorianH-W-W-W-H-W-WDorianPhrygianW-H-W-W-W-H-W
LydianHypolydianW-W-W-H-W-W-HHypolydianLydianW-W-H-W-W-W-H
MixolydianHypophrygianW-W-H-W-W-H-WHypophrygianMixolydianH-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:

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:

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:

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. Wi­ki­pe­dia ⟨en.wikipedia.org/wiki/Musical_system_of_ancient_Greece⟩ has much, much more information ⟨en.wikipedia.org/wiki/Octave_species⟩; the Xenharmonic wiki ⟨en.xen.wiki/w24edo_scales⟩, Feel Your Sound ⟨feelyoursound.com/scale-chords⟩, Midicode ⟨web.archive.org/web/20120308164408/www.midicode.com/tunings/greek.shtml⟩, and Ian Ring’s Scale Finder were also helpful. (The latter is so incredibly helpful that I’ve begun linking to its entries for scales on this page.) Solra Bizna and I also wrote a Rust program to automate several aspects of scale analysis; some of its output is on this very page.

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 analysis of Archytas ⟨ex-tempore.org/ARCHYTAS/ARCHYTAS.html⟩ (who provided what modern scholars believe to be detailed, accurate descriptions of his era’s actual tuning practices). Our knowledge of Archytas’ musical writings evidently comes from Ptolemy’s Harmonics, whose author comments in depth on the former’s writings⁽⁷⁾; large fragments of AristoxenusElements of Harmony and smaller fragments of Philolaus’ musical writings survive to this day.

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 book on the subject ⟨chrysalis-foundation.org/musical-mathematics-pages/philolaus-and-euclid⟩, is a translated excerpt of 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.
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:

In other words:

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:

  1. They’re easy to equate to modern tuning systems.
  2. 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.)

Back to top · My discography · Marathon soundtracks · Contact me · Website index