Aequilībrium harmoniae

(The Harmony of Balance, or The Balance of Harmony)

a mathematical analysis of scales, modes, & the circle of fifths

by Aaron Freed ⟨aaronfreed.github.io⟩, composer and programmer

  1. Contents
  2. Introduction
    1. Aequilibriwhat?
    2. Scales 101
    3. Modes 101
      1. C major’s relative modes
      2. Mode transformations
    4. These are just examples; it could also be something much better
    5. The modes in circle of fifths order
      1. Modes descending from Lydian
      2. Mode transformations revisited
      3. Mode transformations re-revisited
    6. The principles of inverse operations
      1. Mode transformations inverted
    7. An audio demonstration
    8. Further notes
      1. Diabolus in mūsicā
  3. The major scale’s modes & the circle of fifths
    1. 12 major scales × 7 modes = 84 permutations
      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 Ionian scale’s seven modes in twelve-tone equal temperament
    3. Why is this happening?
      1. Traversing the circle of fifths
        1. C++
        2. You were expecting modes, but it was me, Dio the circle of fifths!
      2. Scale generators: A brief introduction
        1. All scale generators in twelve-tone equal temperament
      3. Interval distribution analysis
        1. Ionian interval spacing
    4. Chord analysis by mode
      1. Chord tonalities by scale position & mode (linear order)
      2. Chord tonalities by scale position & mode (circle of fifths order)
      3. Chords for C major’s parallel modes
      4. Chords for C major’s relative modes
  4. Beyond the Ionian scale
    1. Other heptatonic scales & tonalities
      1. Other heptatonic scales
      2. Harmonic minor & melodic minor’s modes at a glance
    2. Beyond pentatonic and heptatonic scales
      1. Ἐπίθετᾰ πρός ᾰ̓ρῐθμούς τόνων   (Adjectives for numbers of notes)
  5. The pentatonic scale
    1. As complement of Ionian
    2. As truncation of Ionian
      1. The incorrect “upshift” hypothesis
      2. The incorrect “downshift” hypothesis
    3. We have to go deeper (Scaleception)
      1. The correct “tritone deletion” explanation
    4. Analysis of modes in linear order
      1. The correct “root note” explanation
    5. Comparison of interval spacing
      1. Interval analysis (circle order)
      2. Interval analysis (linear order)
    6. A brief explanation of scale rotation
    7. The pentatonic circle of fourths, or, contrary motion explained
      1. Pentatonic interval spacing (root order)
      2. Ionian interval spacing (root order)
      3. Ionian interval spacing (circle of fifths order)
      4. Pentatonic interval spacing (circle of fourths order)
      5. Land of Confusion
      6. Ionian tetrachords & pentatonic trichords & synaphai, oh my
    8. An analysis of five-semitone scale rotation
      1. Ionian & pentatonic building blocks
      2. Outcome of swapping n-chord with synaphe
    9. Other single-note scale rotations
      1. The hendecatonic scale
        1. Transforming the hendecatonic scale
      2. Heptatonic chromatic
        1. The heptatonic chromatic scale
        2. The heptatonic chromatic scale (complete transformation)
      3. Other truncations of the chromatic scale
        1. Note movements and the 12-TET chromatic scale
      4. Alternating heptamode & alternating heptamode inverse
        1. Alternating heptamode & alternating heptamode inverse
        2. Alternating heptamode & alternating heptamode inverse (revisited)
        3. Alternating heptamode: A closer examination
      5. Melodic Phrygian (or Neapolitan “major”, as it’s misleadingly known)
        1. Analyzing the melodic Phrygian scale
        2. Transforming the melodic Phrygian scale
      6. Apathetic minor & Pacific
        1. Apathetic minor and Pacific on C
        2. Apathetic minor and Pacific on C (simplified)
        3. Single-note transformations of apathetic minor
        4. Single-note transformations of Pacific
      7. A brief analysis of note distributions
        1. Note distributions across parallel modes
      8. Why Ionian’s scale mutation requires a coprime scale length with 12
        1. Tetratonic truncation of Ionian
        2. Hexatonic truncation of Ionian
        3. Octatonic expansions of Ionian
  6. Transformations of the Ionian scale
    1. Harmonic minor
      1. Harmonic minor vs. modes from Aeolian (rooted on C, linear order)
      2. Harmonic minor vs. modes from Aeolian (rooted on scale, linear 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)
      5. Chord tonalities by scale position & mode (harmonic minor, linear order)
      6. Chord tonalities by scale position & mode (harmonic minor, “circle of fifths” order)
      7. Chords for C harmonic minor’s parallel modes
      8. Chords for C harmonic minor’s relative modes
    2. Melodic minor
      1. Melodic minor vs. modes from Ionian (rooted on C, linear order)
      2. Melodic minor vs. modes from Ionian (rooted on scale, linear order)
      3. Melodic minor vs. modes from Ionian (rooted on C, in “circle of fifths” order)
      4. Melodic minor vs. modes from Ionian (rooted on scale, in “circle of fifths” order)
      5. Melodic minor vs. modes from Dorian (rooted on scale, linear order)
      6. Melodic minor vs. modes from Dorian (rooted on C, in “circle of fifths” order)
      7. Melodic minor vs. modes from Dorian (rooted on scale, in “circle of fifths” order)
      8. Chord tonalities by scale position & mode (melodic minor, linear order)
      9. Chord tonalities by scale position & mode (melodic minor, “circle of fifths” order)
      10. Chords for C melodic minor’s parallel modes
      11. Chords for C melodic minor’s relative modes
    3. The Ionian scale’s stability
      1. Melodic minor vs. Ionian & Dorian (rooted on C±½, “circle of fifths” order)
      2. Melodic minor vs. modes from Mixolydian (rooted on C±½, linear order)
      3. Melodic minor vs. modes from Mixolydian (rooted on C±½, “circle of fifths” order)
      4. Harmonic minor’s modes revisited, or, please throw some snacks down this rabbit hole
      5. Harmonic minor & melodic minor’s “circle of fifths” progressions
      6. One weird trick to transform harmonic minor to melodic minor
    4. Mathematical proof of even spacing
    5. Other single-note transformations of Ionian
      1. Threshold of Transformation
      2. Expand, expand, expand. Clear forest, make land, fresh blood on hands
      3. Why just shells? Why limit yourself? She sells seashells; sell oil as well
      4. Guns, sell stocks, sell diamonds, sell rocks, sell water to a fish, sell the time to a clock
    6. Scale transformations and symmetry
      1. Step on the gas, take your foot off the brakes; run to be the president of the United States
      2. Big smile, mate; big wave, that’s great; now the truth is overrated; tell lies out the gate
      3. Polarise the people; controversy is the game; it don’t matter if they hate you if they all say your name
  7. “What immortal hand or eye / Could frame thy fearful symmetry?”
    1. Rotational symmetry: Modes of limited transposition
      1. Definition
        1. Ionian’s pitch sets across the chromatic scale
      2. Mode 1: The whole-tone scale
        1. Whole-tone note sets
        2. Transposing the whole-tone scale
      3. Mode 2: The octatonic scale
        1. Mode 2’s modes
        2. Mode 2’s notes
        3. Transposing the second mode of limited transposition
      4. Mode 3: The triple chromatic scale
        1. The third mode of limited transposition
      5. Mode 4: The double chromatic scale
        1. The fourth mode of limited transposition
      6. Mode 5: The tritone chromatic scale
        1. The fifth mode of limited transposition
      7. Mode 6: The whole-tone chromatic scale
        1. The sixth mode of limited transposition
      8. Mode 7: Duplex genus secundum inverse
        1. The seventh mode of limited transposition
      9. Truncations & implications
        1. Transposing the diminished seventh chord
        2. Transposing the augmented chord
      10. Mode 0: The chromatic scale
        1. Permuatations of the chromatic scale
      11. All modes of limited transposition in 12-TET
        1. MoLTs at a glance
        2. Note distributions of parallel modes of limited transposition
        3. Note distributions of parallel modes of limited transposition (adjusted for inflation)
      12. Single-note transformations of modes of limited transposition
        1. Note distributions across parallel modes (…again‽ But that trick never works!)
      13. Microtonal corollaries
        1. The 12-tone chromatic scale as a mode of limited microtonal transposition
    2. Achiral scales: Reflective & translational symmetry
      1. Reflective symmetry
        1. One-note reflective symmetry: the octave
        2. Two-note reflective symmetry: the tritone
        3. Three-note reflective symmetry (five scales, five modes)
        4. Tetratonic reflective symmetry (three scales, five modes)
        5. Pentatonic reflective symmetry (ten scales, ten modes)
        6. Hexatonic reflective symmetry (six scales, ten modes)
        7. Heptatonic reflective symmetry (ten scales, ten modes)
        8. Octatonic reflective symmetry (five scales, ten modes)
        9. Enneatonic reflective symmetry (five scales, five modes)
        10. Decatonic reflective symmetry (three scales, five modes)
        11. Hendecatonic reflective symmetry (one scales, one modes)
        12. Dodecatonic reflective symmetry (the chromatic scale)
        13. Symmetrical scale complements (5 & 7)
        14. Symmetrical scale complements (3 & 9)
      2. Translational symmetry
        1. Tetratonic translational symmetry (twelve additional scales)
        2. Hexatonic translational symmetry (fourteen additional scales)
        3. Octatonic translational symmetry (ten additional scales)
      3. Counting achiral scales
        1. Achiral scales by size
      4. Self-complementary scales, symmetry, and scale generators
        1. Self-complementing scales
  8. A crash course in Ancient Greek harmony
    1. Etymology
      1. Αἱ ἐτῠμολογῐ́αι τῶν ἑπτᾰ́ τόνων   (The seven modes’ etymologies)
    2. Ancient Greek harmony: The Cliffs Notes
      1. Interval key
      2. Interval ratios of a diatonic tetrachord
      3. Interval Genera: A Feed from Cloud Mountain
    3. Pythagoras with the looking glass: Comparing interval ratios
      1. A comparison of Pythagorean tuning & twelve-tone equal temperament
      2. Pythagorean interval division (or is it subtraction?)
      3. The Fibonacci sequence in Pythagorean tuning
    4. Ancient Greek tonoi & modern modes
      1. Approximate intervals of Ancient Greek tonoi & modern diatonic modes
      2. Greek enharmonic tonoi (C roots, linear order)
      3. Greek chromatic tonoi & their inversions (C roots, linear order)
      4. Greek chromatic tonoi & their inversions (mode-based roots, linear order)
      5. Greek chromatic tonoi & their variants (mode-based roots, “circle of fifths” order)
      6. Greek diatonic tonoi (C roots, circle of fifths order)
    5. Chromatic tonos analysis: The circle of fifths
      1. Greek chromatic tonoi & their variants (C roots, linear order)
      2. Greek chromatic tonoi & their variants (C roots, “circle of fifths” order)
      3. Greek chromatic tonoi & their variants (C roots, “aligned synaphai” order)
    6. Chromatic tonos analysis: Tetrachord swap
      1. Greek chromatic tonoi & their variants (C roots, “cyclical tetrachord swap” order)
      2. Greek chromatic tonoi & their variants (C roots, “linear tetrachord swap” order)
      3. Greek chromatic tonoi & their variants (C roots, “aligned tetrachord swap” order)
    7. Chromatic tonos analysis: Minor third position
      1. Greek chromatic tonoi & their variants (C roots, “cyclical aligned minor thirds” order)
      2. Greek chromatic tonoi & their variants (C roots, “linear aligned minor thirds” order)
      3. Greek chromatic tonoi & their variants (C roots, “doubly aligned minor thirds” order)
    8. Why our modes have historically inaccurate names
      1. Medieval names for the Greek diatonic tonoi
      2. Inverting the Greek diatonic tetrachord
      3. Mode transformations re-re-revisited
      4. Modes and the notes they redshift
    9. Applied Greek harmony: Tetrachords in modern scales
    10. Acknowledgements & sources
    11. Appendix 1: Greek musical terminology
      1. Λεξικογραφία Ἑλληνῐκῶν μουσῐκῶν ῐ̓́δῐογλῶσσῐῶν
    12. Appendix 2: Interval ratios of 12- and 24-tone equal temperament
      1. 24-tone equal temperament’s interval ratios
  9. Embryonic surveys of other musical traditions
    1. Back to Babylon, or, the Mesopotamian major scale
      1. Mesopotamian modes
    2. The carnatic numbered mēḷakartā
      1. The carnatic numbered mēḷakartā
      2. Mēḷakartā note ranges
  10. Scale counts in 12-TET by scale size
    1. 12-TET mode counts
    2. All mModes of limited transposition
    3. 12-TET discrete scales
    4. Appendix: All 2,048 modes of all 351 scales in 12-TET
  11. Yes, but, why?
  12. 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

Aequilibriwhat?

Be warned: This is a dense technical analysis of scales and modes. If you don’t have a solid grasp of music theory, the charts may look pretty, but much of my analysis may fly straight over your head. If you want to learn music theory, I have introductions to rhythm ⟨aaronfreed.github.io/whatistime.html⟩ and harmony ⟨aaronfreed.github.io/basicmusicalharmony.html⟩ that provide far better starting points.

I’m sure my decision to title this work in Latin will come as a surprise to absolutely no one who knows me. I had a reason to do so beyond purely being pretentious, though. Latin’s genitive case conveys a two-way possessive relationship. That is, it means both “The X of Y” and “The Y of X”. I’d briefly retitled this work The Balance of Harmony, and my repeated inability to remember whether I’d retitled it The Harmony of Balance or The Balance of Harmony clarified to me that I wanted my title to convey both. As pretentious as Aequilībrium harmoniae might be, The Balance of Harmony and the Harmony of Balance seems even more so. The idea of a possessive relationship extending in both directions is entirely foreign to English grammar, which I’m sure must baffle anyone who’s ever cared for a cat or a dog. And in any case, the Latin genitive case’s reciprocality seems to underscore the entire idea of balance. That is, balance is harmony, and harmony is balance. One isn’t more important than the other; they’re equally important, since they’re ultimately the same thing.

(Before we proceed: As always, please contact me ⟨aaronfreed.github.io/aboutme.html⟩ if you notice any errors or omissions.)

This book started as an analysis of the familiar Ionian (diatonic/heptatonic major) scale, its seven modes, and their interrelationships. You know the one. Whole step, whole step, half-step, whole step, whole step, whole step, half-step (WWHWWWH for short). Play the white keys on the piano from C to C. Doe, a deer, a female deer. There’s your Ionian scale.

What began this analysis was discovering that raising or lowering specific pat­terns of notes in the scale, one by one, produces a cycle that encompasses not merely all seven modes of the scale, but all eighty-four possible sets of its modes and root notes, in a manner inextricably linked to the circle of fifths. That’s where it began, at least; I may never manage to climb back out of this rabbit hole again. But at least I’ll have plenty of company down here: while writing this book, I learned that some four millennia ago, the core of Mesopotamian tuning practice centered around this exact pattern of mutations to this exact scale. You read that right: the diatonic scale is almost 4,000 years old – assuming that it doesn’t predate history entirely.

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Scales 101

A brief (lol, lmao) explanation is perhaps in order. First, it may help to define what I mean by scale:

  1. Scales repeat every octave. If A0 appears in a scale, so do A1, A2, A3, A4, A5, A6, A7, and so on.
  2. Scales cannot duplicate any pitches. A scale may not include both G and A in standard Western tuning, since they are the same pitch. (Note that this is not true in all tunings!)
  3. Scales must include their roots. That is, a scale defined to start on C must actually include C. (The sole exception I ever allow to this rule is for the null set, which contains no notes; this is occasionally necessary as a contrast to the chromatic scale, which contains all notes.)
  4. Scales are defined in either ascending or descending order. Modern Western harmony almost always defines scales in ascending order. Late in this book, I address ancient Greek and Mesopotamian harmony. Both the Ancient Greeks and the Mesopotamians defined scales in descending order, but to avoid confusing readers, I’ll convert them to ascending order.
  5. Scales have identical ascents and descents. The most common definition of the melodic minor scale uses a different descent than ascent. For our purposes, this is not a scale. I will therefore use “melodic minor” throughout this book as a synonym for “ascending melodic minor”.

Those are the only restrictions I’m placing on my definition of “scale”, because otherwise, certain patterns will be impossible to analyze. This has a few additional implications, including:

  1. Scales’ intervals add up to an octave. (Do you understand why?)

And, specifically in our tuning system, twelve-tone equal temperament (or 12-TET):

  1. Scales contain at least one note and no more than twelve notes.
  2. Each interval of an n-note scale must be no smaller than one semitone and no larger than (13 − n) semitones. The largest possible interval size is 12 semitones, for the one-note scale.

12-TET contains 2¹¹ = 2,048 patterns of notes that meet this definition. (It’s not 2¹² because combinations that exclude the root do not meet our definition.) However, not all of these qualify as discrete scales; as it turns out, our tuning system only has 351 scales, which collectively possess 2,048 modes. I explain the oddly specific number of 351 in §7’s discussion of reflective symmetry.

Note that, in music theory, “the diatonic scale”, preceded by the definite article, almost always refers specifically to the diatonic major scale. In this context, “diatonic” means “of two interval sizes”. Now, technically, even among heptatonic (seven-note) scales, four scales contain only two interval sizes. (All other seven-note scales with two half-steps and five whole-steps are modes of the first three, as we shall see. Also, “T” stands for “tritone”.)

  1. Melodic Phrygian [HWWWWWH], often misleadingly called Neapolitan “major”
  2. Melodic minor [WHWWWWH]
  3. Major [WWHWWWH]
  4. Heptatonic chromatic [HHHHHHT]

However, only the major scale is the diatonic scale. Likewise, “the pentatonic scale” means C-D-E-G-A, even though our tuning system actually contains dozens of pentatonic (five-note) scales.

The diatonic and pentatonic scales likely earned the definite article in part for the same reason: each has the most evenly distributed notes scales of their sizes can have in our tuning system. The diatonic scale derives its name from an effectively identical Ancient Greek genus (§8 contains detailed comparisons, notably §8.3-4). The Greeks didn’t invent it, though: the Hurrian songs from ca. 1400 BCE employ it, and Mesopotamian tablets from ca. 1800 BCE describe tuning practices mathematically guaranteed to produce it.

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Modes 101

Modes are a bit trickier to define. I shall attempt to do so by example of the Ionian scale itself.

Of the seven modes that are §2 and §3’s central focus, six move varying numbers of Ionian’s intervals either from the end to the beginning, or vice versa. (Ionian itself is the first mode.) These modes are usually numbered by their roots - i.e., which degree (note of the parent scale) they start on. Following this numbering, I’ll list the C Ionian scale’s seven modes.

C major’s relative modes
# Mode 12 3 4 5 6 7 Comment
1Ionian C D E F G A B The traditional major scale.
2Dorian D Ef G A Bc
3Phrygian Efg A Bcd
4Lydian F G AB C D E The only mode that raises a note from Ionian.
5Mixolydian G A B C D Ef
6Aeolian A Bc D Efg The natural minor scale.
7Locrian Bcd Efga Rare due to its diminished root chord; many pieces that use it modulate out of it at times, creating a sense that we’re rarely truly ‘home’. It’s more eerie or mysterious than unsettling, though; the Ionian scale is too melodic for the latter, diminished root or not.
C major’s relative modes
# Mode 12 3 4 5 6 7
1 Ionian C D E F G A B
The traditional major scale.
2 Dorian D E f G A B c
3 Phrygian E f g A B c d
4 Lydian F G A B C D E
The only mode that raises a note from Ionian.
5 Mixolydian G A B C D E f
6 Aeolian A B c D E f g
The natural minor scale.
7 Locrian B c d E f g a
Rare due to its diminished root chord; many pieces that use it modulate out of it at times, creating a sense that we’re rarely truly ‘home’. It’s more eerie or mysterious than unsettling, though; the Ionian scale is too melodic for the latter, diminished root or not.

The other way to think of it is as a scale transformation. This produces parallel modes, such as a major scale’s parallel minor (e.g., C major to C minor). Here, we start the modes on the same note and make the following changes to the major scale. Note that, in this context, “” means “lower the respective scale degree of the major scale by a semitone” (i.e. F becomes F), and “” means “raise the respective scale degree of the major scale by a semitone” (i.e., G becomes G).

Mode transformations
# ModeMode 12 3 4 5 6 7
1Ionian Ionian
2Dorian Dorian 3 7
3Phrygian Phrygian 23 67
4Lydian Lydian 4
5Mixolydian Mixolydian 7
6Aeolian Aeolian 3 67
7Locrian Locrian 23 567

There are a few mnemonics for these. Three of my favorites are:

Of course, the last of these requires you to recall that we pronounce ph as f. This is because Phrygian is an anglicized version of Φρῠ́γῐος (romanization: Phrŭ́gĭos). This has undergone several pronunciation shifts (remember, Greek is an ancient language).⁽⁰⁾ Approximate pronunciations (note that ü is pronounced as in German, g is a hard g, and Modern Greek omits many of the diacritics found in earlier dialects of Greek):

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

Defining modes is not an exact science: some songs start on chords that aren’t the root, and determining the root in such cases is essentially a case of what feels like home, which is inherently subjective. (If a song uses any mode’s IV-V-I progression often enough, there’s a case for that mode being the song’s intended mode, but this isn’t a universal rule, and many songs never use any such progression.) Often, we can only be certain of a song’s intended mode if its songwriter explicitly specifies a mode besides major or minor. (The latter might be oversimplifications: they might just say G major because they figure people won’t know C Lydian.)

Those caveats aside, some immediately recognizable examples of each mode include:

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The modes in circle of fifths order

I mostly won’t be analyzing the modes in their traditional order: I’m analyzing how lowering a regular pattern of notes by a semitone each walks us through every mode on every key. I call this the circle of fifths order for reasons that will soon become clear, and I refer to scales with single-note transformations between all their modes as mutable or mutant scales, and the process as scale mutation, to distinguish it from scale transformations more generally. A few notes (pun intended):

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 4C D E F G A B C
5C Mixolydian 7C D E F G A B C
2C Dorian 3C D E F G A B C
6C Aeolian 6C D E F G A B C
3C Phrygian 2C D E F G A B C
7C Locrian 5C D E F G A B C
4C Lydian 1C D E F G A B C
Mode transformations revisited
# ModeMode 12 3 4 5 6 7
4Lydian Lydian 4
1Ionian Ionian
5Mixolydian Mixolydian 7
2Dorian Dorian 3 7
6Aeolian Aeolian 3 6 7
3Phrygian Phrygian 2 3 6 7
7Locrian Locrian 2 3 5 6 7

The significance of that note sequence may not be immediately obvious, but if we reshuffle the above table back into linear order, it becomes easier to understand its inverse relationship to the modes themselves:

Mode transformations re-revisited
# ModeMode 12 3 4 5 6 7
1Ionian Ionian
2Dorian Dorian 3 7
3Phrygian Phrygian 2 3 6 7
4Lydian Lydian 4
5Mixolydian Mixolydian 7
6Aeolian Aeolian 3 6 7
7Locrian Locrian 2 3 5 6 7

The Russian author Anton Chekhov (1860-1904) might have a few words to say about this table. (This is my roundabout way of advising you to remember it.)

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The principles of inverse operations

Mode transformations invertedMode transformations inverted
circle of fifths ordercircle of fifths orderlinear order
# ModeMode 12 3 4 5 6 7 # Mode 12 3 4 5 6 7
7Locrian Locrian 2 3 5 6 7 7Locrian 23 567
3Phrygian Phrygian 2 3 6 7 6Aeolian 3 67
6Aeolian Aeolian 3 6 7 5Mixolydian 7
2Dorian Dorian 3 7 4Lydian 4
5Mixolydian Mixolydian 7 3Phrygian 23 67
1Ionian Ionian 2Dorian 3 7
4Lydian Lydian 4 1Ionian
linear order
Mode 12 3 4 5 6 7
Locrian 23 5 6 7
Aeolian 3 6 7
Mixolydian 7
Lydian 4
Phrygian 2 3 67
Dorian 3 7
Ionian

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An audio demonstration

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Further notes

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The major scale’s modes & the circle of fifths

12 major scales × 7 modes = 84 permutations

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
 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

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

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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
20 A 7 – Locrian B 5 A B C D E F G A

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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
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

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G / A
#RootMode RMKS12 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 – MixolydianD 5 A B C D E F G A
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

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G
#Root Mode RMKS1 2 3 4 5 6 7 1
35 G4 – Lydian D 2 G A B C D E F G
36 G1 – Ionian G 1 G A B C D E F G
38 G2 – Dorian F 1 G A B C D E F G
39 G6 – Aeolian B 2 G A B C D E F G
40 G3 – Phrygian E 3 G A B C D E F G
41 G7 – Locrian A 4 G A B C D E F G

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F / G
#RootMode RMKS12 3 4 5 6 7 1
42 G 4 – Lydian D 5 G A B CD E FG
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

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F   (E)
#Root Mode RMKS1 2 3 4 5 6 7 1
50 F 1 – Ionian F 1 FGAB CDEF
51 F 5 – MixolydianB 2 FGAB CDE F
52 F 2 – Dorian E 3 FGA B CDE F
53 F 6 – Aeolian A 4 FGA B CD E F
54 F 3 – Phrygian D 5 FG A B CD E F

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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
62 E7 – Locrian F 1 EFGAB CDE

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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
68 D 3 – Phrygian B 5 D EF G A BC D
69 D 7 – Locrian E 4 D EF G ABC D

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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
74 D6 – Aeolian F 1 DEFGAB CD
75 D3 – Phrygian B 2 DE FGAB CD
76 D7 – Locrian E 3 DE FGA B CD

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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
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

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Key signature cheat sheet

Key signatures of the Ionian scale’s seven modes in twelve-tone equal temperament
LydMajMixDorMin PhrLoc KS A B C D E F G
F C G D A E B 7               
BF C G D A E 6                
EBF C G D A 5                 
AEBF C G D 4                  
DAEBF C G 3                   
GDAEBF C 2                    
CGDAEBF 1                     
FCGDAEB                       
B FCGDAE1                     
E B FCGDA2                    
A E B FCGD3                   
D A E B FCG4                  
G D A E B FC5                 
C G D A E B F6                
F C G D A E B 7               

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Why is this happening?

Traversing the circle of fifths

Whether we realized it or not, we’ve been traversing the circle of fifths this entire time. My introduction notes that traveling from C Lydian to C Ionian is, in a sense, traveling from G major to C major. Here’s the C table again. Note Relative Major’s traversal down the circle of fifths:

C++
# Root Mode RMKS1 2 3 4 5 6 7 1
 0C4 – Lydian G 1 C D E F G A B C
 1C1 – Ionian C    C D E F G A B C
 2C5 – MixolydianF 1 C D E F G A B C
 3C2 – Dorian B 2 C D E F G A B C
 4C6 – Aeolian E 3 C D E F G A B C
 5C3 – Phrygian A 4 C D E F G A B C
 6C7 – Locrian D 5 C D E F G A B C

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

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 G A B C D E F G
 11 – Ionian C    C D E F G A B C
 25 – MixolydianF 1 F G A B C D E F
 32 – Dorian B 2 B C D E F G A B
 46 – Aeolian E 3 E F G A B C D E
 53 – Phrygian A 4 A B C D E F G A
 67 – 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. That’s the first part of the explanation.

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Scale generators: A brief introduction

The second part of the explanation explains why the modes’ circle of fifths order is what it is. Reminder: octave numbering starts at C, so the note above B0 is C1, which in turn is eleven semitones below B1. (I’m sure this must’ve been done purely to annoy people on the obsessive-compulsive spectrum. Other obsessive-compulsive programmers will understand what an atrocity this is without my having to explain it, and for everyone else, I don’t believe the English language possesses adequate expressiveness to explain.)

Anyhow, let’s go to F1 near the bottom of the piano. (The grand piano spans A0 to C8.) What’s a perfect fifth above that? C2. A perfect fifth above C2? G2. Move up another perfect fifth. We’re at D3. Up another perfect fifth. A3. Another perfect fifth. E4. Another. B4. So, to recap, we have the notes:

  1. F1
  2. C2
  3. G2
  4. D3
  5. A3
  6. E4
  7. B4

F, C, G, D, A, E, B: the fourth, first, fifth, second, sixth, third, and seventh degrees of the C major scale. There’s our circle of fifths order. The entire scale is literally just 7/12 of the circle of fifths, rearranged into linear order. The mode depends merely on which note in the sequence you use as the base:

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

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

The notes are deterministic: the lowest note in the sequence of perfect fifths determines the other six. The note selected as scale root, meanwhile, determines the mode. Not coincidentally, these correspond exactly to the key signature table: if we start our perfect fifths on C, then a root of C yields Lydian, a root of G yields Ionian, a root of D yields Mixolydian, and so on. This is also why flats are ordered B, E, A, D, G, C, F, and sharps are ordered F, C, G, D, A, E, B.

It’s worth noting that we can also construct the major scale by starting on B0 and moving up a perfect fourth six times. In scale theory, a perfect fifth is the same pitch class as a perfect fourth, because a perfect fourth up and a perfect fifth down are the same note when we disregard octaves. (However, swapping the intervals will reverse the order of the modes’ correspondence to the interval stack, thus putting Locrian’s root on the bottom and Lydian’s on top. This holds true for the same reason that making the notes into a scale rearranges them to occur within the same octave: a perfect fourth up equates to a perfect fifth down.)

In short, the diatonic major scale consists of seven stacked and flattened perfect fifths. This is known as a scale generator. Only a small fraction of scales have these, and only two mutate in exactly the same way as diatonic major. One is the pentatonic scale (as we will see in §5.1-8), which consists of four stacked perfect fifths or perfect fourths; the other is the hendecatonic scale (§5.9.1), which can be expressed as ten stacked perfect fourths or perfect fiths (or minor seconds or major sevenths, for that matter).

No other scale sizes can mutate exactly like this: the one-note scale has no modes to mutate into, and no other numbers are coprime with 12 (the parent tonality): that is, they share no prime factors with 12. We’ll examine why the length must be coprime with 12 below (§5.9.8). That said, the heptatonic chromatic scale, consisting of six stacked minor seconds (or major sevenths), exhibits similar behavior to diatonic major in several important ways, which we’ll also explore below (§5.9.2).

Only minor seconds, perfect fourths, perfect fifths, and major sevenths work as heptatonic scale generators, for two reasons: the scale’s length, and (again) only 1, 5, 7, and 11 (those intervals’ sizes in semitones) being coprime with 12. Generators have upper size limits for intervals that aren’t coprime with 12. For instance, stacking six major thirds (a four-semitone interval) duplicates the first note twice and the others once, giving us not a heptatonic scale but a three-note scale consisting solely of an augmented chord.

Scale generators can produce the following scale sizes:

All scale generators in twelve-tone equal temperament
Interval(s) # Notes
Minor second Major seventh 2 3 4 5 6 7 8 9 10 11 12
Perfect fourth Perfect fifth 2 3 4 5 6 7 8 9 10 11 12
Major second Minor seventh 2 3 4 5 6
Minor third Major sixth 2 3 4
Major third Minor sixth 2 3
Tritone 2
Octave 1

A few additional notes:

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Interval distribution analysis

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

Ionian interval spacing
#Mode1 234567Intervals
4Lydian C D E F G A B 1 1 1 ½ 1 1 ½
1Ionian C D E F G A B 1 1 ½ 1 1 1 ½
5Mixolydian C D E F G A B 1 1 ½ 1 1 ½ 1
2Dorian C D E F G A B 1 ½ 1 1 1 ½ 1
6Aeolian C D E F G A B 1 ½ 1 1 ½ 1 1
3Phrygian C D E F G A B ½ 1 1 1 ½ 1 1
7Locrian C D E F G A B ½ 1 1 ½ 1 1 1

A brief explanation of the above table format is in order, since you’ll be seeing it a lot. The first interval is the number of whole steps between the first and second notes. The second interval is the number of whole steps between the second and third notes. And so on, until the final interval, which, for an l-note scale, is the number of whole steps between notes l and l + 1. “How can an l-note scale have a note l + 1?”, I hear you object. Simple: A scale is a pattern that repeats every octave. Thus, note l + 1 is an octave above note 1. Note (2 × l) + 1 is two octaves above note 1. And so on. Such a scale’s interval i(l), expressed in whole steps, should always equal 6 minus the sum of intervals i(1) through i(l − 1); also, for n = the number of whole steps between notes 1 and l, it should equal 6 − n. This is a mathematical property of how scales work; if any scale’s intervals ever sum up to anything but six whole steps, I made a mistake.

The Ionian scale is virtually unique among 12-TET’s seven-note scales in that, for every mode of the scale, it is possible to swap two notes (or two consecutive intervals) and produce a different mode of the same scale, and it is possible to cycle through the entire chromatic scale by doing these transformations. The other seven-note scale that most unambiguously displays this trait is actually, for various reasons, its polar opposite in virtually every important way. In §5, particularly §5.9, I will go over more about why, precisely, this is. For now, the important point is that each step down the circle of fifths order swaps only one pair of consecutive intervals, and therefore moves only one note. If more intervals changed, the pattern would break.

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Chord analysis by mode

Chord tonalities by scale position & mode (linear order)
Mode1234567
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
Chord tonalities by scale position & mode (circle of fifths order)
Mode1234567
4Lydian Maj Majmindim Majminmin
1Ionian Majminmin Maj Majmindim
5Mixolydian Majmindim Majminmin Maj
2Dorian minmin Maj Majmindim Maj
6Aeolian mindim Majminmin Maj Maj
3Phrygian min Maj Majmindim Majmin
7Locrian dim Majminmin Maj Majmin

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Beyond the Ionian scale

Other heptatonic scales & tonalities

While the above analysis focuses exclusively on the Ionian scale’s modes, numerous possible scales (and modes thereof) don’t fit its pattern, many of which I’ll now analyze. Some examples:

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Beyond pentatonic and heptatonic scales

Of course, scales needn’t contain seven notes. Pentatonic (five-note) scales are so complex to unpack that they’ll need their own section (mostly because both five and seven are coprime with twelve: neither share any prime factors with it). Here’s a brief overview of other scale sizes in ascending order:

So far, we’ve exclusively been considering 12-TET, but of course, plenty of other tunings have been and still are used; nothing even constrains octaves to twelve notes. For instance:

For the record:

Ἐπίθετᾰ πρός ᾰ̓ρῐθμούς τόνων
Epíthetă prós ărĭthmoús tónōn
Adjectives for numbers of notes
#Ἐπίθετον
Epítheton
Adjective
Ἑλληνική «–τονος»
Hellēnĭke «–tonos»
Greek “–toned”
Ῥωμᾰῐ̈σμένη
Rhṓmăĭ̈sméni
Romanized
Ἑλληνική «–τονικός»
Hellēnĭke «–tonikós»
Greek “–tonic”
Ῥωμᾰῐ̈σμένη
Rhṓmăĭ̈sméni
Romanized
1monotonicμονότονοςmonótonosμονότονικόςmonótonikós
2diatonicδιατονοςdiatonosδιατονικόςdiatonikós
3tritonicτρίτονοςtrítonosτρίτονικόςtrítonikos
4tetratonicτετράτονοςtetrátonosτετράτονικόςtetrátonikós
5pentatonicπέντατονοςpéntatonosπέντατονικόςpéntatonikós
6hexatonicἑξατονοςhexatonosἑξατονικόςhexatonikós
7heptatonicἑπτάτονοςheptátonosἑπτάτονικόςheptátonikós
8octatonicὀκτάτονοςoktátonosὀκτάτονικόςoktátonikós
9enneatonicἐννεάτονοςenneátonosἐννεάτονικόςenneátonikós
10decatonicδέκατονοςdékatonosδέκατονικόςdékatonikós
11hendecatonicἕνδεκάτονοςhendekátonosἕνδεκάτονικόςhendekátonikós
12dodecatonicδωδεκάτονοςdōdekátonosδωδεκάτονικόςdōdekátonikós

The suffix -tonic is Greek. Friends don’t let friends mix Latin prefixes and Greek suffixes. (Unless Latin already did so, that is. It did borrow τόνος as tonus, but only as a noun, never an adjective; it did not borrow τονικός.)

I only managed to find attestations of some of the Greek forms in this list, but it seems likely they all must have existed at some time. The ones I found are in bold; the ones I was unable to find are in fainter text.

Bolded English words, meanwhile, have attested usages for scale size in music theory contexts. Monotonic, diatonic, and tritonic are printed more faintly because they have completely different meanings that have nothing to do with the number of pitches in a scale, so using them to mean that will likely just confuse readers. The latter two are also struck through because their alternate meanings are ubiquitous in music theory contexts. You might be technically correct to use them to refer to scale size, but is that really the hill you want to die on?

  1. Monotonic mostly refers to the modern Greek accent system (cf. the old polytonic system with markers for word pitch and breathing). It has no widely established meaning in music theory, so this case is less clear-cut than the others.
  2. Diatonic means of two interval sizes. In English and Ancient Greek alike, it most often means what became our diatonic major scale: it’s at least 2,500 years old.
  3. Tritonic means spanning an interval of three whole tones, i.e., a tritone.

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The pentatonic scale

As complement of Ionian

Pentatonic (“5t” for short) scales are ubiquitous in rock, blues, and jazz, though they’re much older than that and exist in many cultures. The most common pentatonic scale is literally Ionian’s scale complement.

What exactly is a scale complement? It’s the equivalent of a binary XOR. Say we represent a scale as a set of twelve 1s (“this tone is part of the scale”) or 0s (“this tone is not part of the scale”). Now, flip all the bits. Tones that had notes are no longer part of the scale; tones that didn’t now are. Since scales must start with a note, we now must rotate the scale so that the first bit is a 1. Once we’ve done so, we have a mode of the complement.

This means we can play the pentatonic scale using all the piano keys we didn’t use to play Ionian. Whenever I write “the pentatonic scale”, preceded by the definite article, I mean this pentatonic scale. To wit:

To a huge extent, the pentatonic and Ionian scales’ relationship even extends to their modes. For instance:

I must clarify, however, that scale complements apply on a scale-wide basis, not on a modal basis. That is, because the complement of any non-hexatonic scale in 12-TET will have a different number of notes, its complement will also have a different number of modes (barring a few exceptions, known as modes of limited transposition, which are covered in §7’s discussion of symmetry). As a result, it’s not possible to make 1:1 comparisons between scale complements’ modes. I feel the need to emphasize this because we’re to compare the pentatonic and Ionian scales through a second lens, and 1:1 comparisons do apply through the second one.

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As truncation of Ionian

Conveniently, though, it’s not just a complement, though: it’s also a truncation. We can get the pentatonic scale simply by deleting two notes of Ionian. As a result, the two scales’ modes correspond in countless ways.

Since the pentatonic scale has two fewer notes than Ionian, our analysis must delete two modes. But which two? We can derive the pentatonic scale from Ionian in at least three different ways.

Quick warning before we proceed further: we’re taking a quick detour into “right for the wrong reasons” land. After the third table, I’ll explain how, why, and where the first two tables go wrong.

Let’s try disregarding Phrygian and Locrian, the lowest modes in the circle of fifths progression. In this analysis:

  1. Major pentatonic deletes Lydian’s fourth and seventh notes: F G A C D (1, 1, , 1, ).
  2. Scottish pentatonic deletes Ionian’s third and seventh notes: C D F G A (1, , 1, 1, ).
  3. Neutral pentatonic deletes Mixolydian’s third and sixth notes: G A C D F (1, , 1, , 1). Neutral pentatonic is comparable to Dorian mode in two ways: it is a symmetrical scale, and it’s the midpoint of the pentatonic circle of fifths order (which is complex enough to merit its own section below).
  4. Minor pentatonic deletes Dorian’s second and sixth notes: D F G A C (, 1, 1, , 1).
  5. Blues minor deletes Aeolian’s second and fifth notes: A C D F G (, 1, , 1, 1).

Got all that? Let’s recap. (Note: “H” = half-tone, ”W” = whole tone, “M” = minor third)

The incorrect “upshift” hypothesis
[7t]   Mode   [5t][7t]   Piano keys   [5t][7t]   Intervals   [5t]Del
Lydian Major 5t. FGABCDE F G A C D W W W H W W H W W M W M 4 7
Ionian Scottish 5t. CDEFGAB C D F G A W W H W W W H W M W W M 3 7
Mixolydian Neutral 5t. GABCDEF G A C D F W W H W W H W W M W M W 3 6
Dorian Minor 5t. DEFGABC D F G A C W H W W W H W M W W M W 2 6
Aeolian Blues Minor ABCDEFG A C D F G W H W W H W W M W M W W 2 5

I probably don’t even need to point out how many patterns recur in both scales.

We just analyzed the pentatonic modes based on notes a half-step above them, but we could just as easily have used the notes a half-step above. This means instead disregarding Lydian and Ionian. Oddly enough, we delete the same scale degrees either way:

The incorrect “downshift” hypothesis
[7t]   Mode   [5t][7t]   Piano keys   [5t][7t]   Intervals   [5t]Del
Mixolydian Major 5t. GABCDEF G A B D E W W H W W H W W W M W M 4 7
Dorian Scottish 5t. DEFGABC D E G A B W H W W W H W W M W W M 3 7
Aeolian Neutral 5t. ABCDEFG A B D E G W H W W H W W W M W M W 3 6
Phrygian Minor 5t. EFGABCD E G A B D H W W W H W W M W W M W 2 6
Locrian Blues Minor BCDEFGA B D E G A H W W H W W W M W M W W 2 5

(End warning.)

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We have to go deeper (Scaleception)

But wait, there’s more! Consulting the heptatonic circle of fifths progression enables us to “average” the above two tables. Eerily, averaging them doesn’t invalidate their results in any way except a trivial one: this is the first analysis that’s actually correct. But, as the famed philosopher Nigel Tufnel put it, that’s nitpicking, innit?

How pentatonic transforms Ionian: The correct “tritone deletion” explanation
[7t]   Mode   [5t][7t]   Piano keys   [5t][7t]   Intervals   [5t]Del
Ionian Major 5t. CDEFGAB CDEGA W W H W W W H W W M W M 4 7
Mixolydian Scottish 5t. GABCDEF GACDE W W H W W H W W M W W M 3 7
Dorian Neutral 5t. DEFGABC DEGAC W H W W W H W W M W M W 3 6
Aeolian Minor 5t. ABCDEFG ACDEG W H W W H W W M W W M W 2 6
Phrygian Blues Minor EFGABCD EGACD H W W W H W W M W M W W 2 5

So, to reiterate: Only the tritone deletion table is completely accurate; its two predecessors are filthy half-truths, owing to a mathematical pattern they don’t account for. Can you figure out why?

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

    1. We should compare the symmetrical modes to each other.
    2. Symmetrical modes should be their circle-order comparisons’ center data rows.
    3. Rows whose intervals are mirrors in the base should remain mirrors in the transformation.
    4. Both circle-order interval comparisons should possess 180° rotational symmetry.

    All of these are false in the first two tables and true in the third.

  2. Since we didn’t delete the root, we must compare the base scales. Only table three does so. Not comparing the base scales is a surefire recipe for confusion.
  3. If we delete a note, we must delete its mode. We deleted notes four and seven. Only the third table deletes both notes’ modes (Lydian and Locrian).
  4. Our analysis must compare the same notes within each scale. We didn’t move notes, only remove them, so our analysis can’t either. Examining the first table closely reveals why this is a problem: Mixolydian four, Dorian seven, Aeolian three, Phrygian six, and Locrian two are Ionian’s root!

    The problem is less obvious in the second table, but it tells us to remove Ionian’s third note. Major pentatonic, like Ionian, opens with two whole steps, so it hasn’t removed Ionian’s third note! This is why we should only shift notes in our analysis if we delete the root. (And if so, good luck – you’ll need it.)

“Tritone Deletion”, “Tritone Substitution”, or “Tritone Shift” could all fit for the third table. The notes it removes correspond exactly to the Ionian scale’s sole tritone; it also lists the pentatonic scales exactly a tritone from where its two predecessors had them. I chose “Tritone Deletion” in the end because it’s a more accurate description of what we’re actually doing, and the difference in pitch is a direct consequence of removing the tritone instead of removing other notes, then shifting the scale.

Since pentatonic’s mode nomenclature isn’t as well established as Ionian’s, my brain’s cutesy part wants to rename them Nianoi, Niadyloxim, Niarod, Nialoea, and Niagyrhp. I’m afraid that even after the above explanation, that might confuse people, but that won’t stop me from using them as alternate names.

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Analysis of modes in linear order

Viewing the modes in linear order, with the Ionian scale’s missing modes included, may help further clarify why the “upshift” and “downshift” tables are wrong:

How pentatonic transforms Ionian: The correct “root note” explanation
[7t]   Mode   [5t][7t]   Piano keys   [5t][7t]   Intervals   [5t]Del
Ionian Major 5t. CDEFGAB CDEGA W W H W W W H W W M W M 7 4
Dorian Neutral 5t. DEFGABC DEGAC W H W W W H W W M W M W 6 3
Phrygian Blues Minor EFGABCD EGACD H W W W H W W M W M W W 5 2
Lydian FGABCDE W W W H W W H 4 1
Mixolydian Scottish 5t. GABCDEF GACDE W W H W W H W W M W W M 3 7
Aeolian Minor 5t. ABCDEFG ACDEG W H W W H W W M W W M W 2 6
Locrian BCDEFGA H W W H W W W 1 5

(Note: I reordered a few “deleted notes” entries in this table to clarify its pattern.)

Modes and roots are inextricably linked. We must compare the same notes in each scale:

  1. We didn’t delete this note; its mode, Ionian, becomes major pentatonic5t..
  2. We didn’t delete this note; its mode, Dorian, becomes neutral pentatonic5t..
  3. We didn’t delete this note; its mode, Phrygian, becomes blues minor.
  4. Deleting this note deletes its mode; Lydian has no pentatonic equivalent.
  5. We didn’t delete this note; its mode, Mixolydian, becomes Scottish pentatonic5t..
  6. We didn’t delete this note; its mode, Aeolian, becomes minor pentatonic5t..
  7. Deleting this note deletes its mode; Locrian has no pentatonic equivalent.

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

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

So, applying the same principle to the pentatonic scale:

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

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Comparison of interval spacing

Why pentatonic and Ionian’s intervals are out of sync may not be obvious. Let’s revisit the above table, this time with deleted tones in red and combined intervals in purple:

How pentatonic transforms Ionian: Interval analysis (circle order)
[7t]   Mode   [5t][7t]   Piano keys   [5t][7t]   Intervals   [5t]Del
Lydian FGABCDE W W W H W W H 4 1
Ionian Major 5t. CDEFGAB CDEGA W W H W W W H W W M W M 4 7
Mixolydian Scottish 5t. GABCDEF GACDE W W H W W H W W M W W M 3 7
Dorian Neutral 5t. DEFGABC DEGAC W H W W W H W W M W M W 3 6
Aeolian Minor 5t. ABCDEFG ACDEG W H W W H W W M W W M W 2 6
Phrygian Blues Minor EFGABCD EGACD H W W W H W W M W M W W 2 5
Locrian BCDEFGA H W W H W W W 1 5

It may help to emphasize that we aren’t deleting intervals; we’re deleting notes and combining intervals. For instance, deleting a scale’s second note combines its first two intervals. Furthermore:

  1. Ionian has only semitones and whole tones
  2. We only delete notes that:

    1. follow semitones and precede whole tones
    2. follow whole tones and precede semitones

Thus, the two intervals around every deleted note turn become a single minor third aligning exactly to the pattern of deletions. Since this pattern is out of phase with the original one, it changes, but if the deleted notes were surrounded by different intervals, the new interval pattern wouldn’t map so precisely to the deletions.

Interestingly, that relationship may be less obvious in linear order, even when highlighted:

How pentatonic transforms Ionian: Interval analysis (linear order)
[7t]   Mode   [5t][7t]   Piano keys   [5t][7t]   Intervals   [5t]Del
Ionian Major 5t. CDEFGAB CDEGA W W H W W W H W W M W M 7 4
Dorian Neutral 5t. DEFGABC DEGAC W H W W W H W W M W M W 6 3
Phrygian Blues Minor EFGABCD EGACD H W W W H W W M W M W W 5 2
Lydian FGABCDE W W W H W W H 4 1
Mixolydian Scottish 5t. GABCDEF GACDE W W H W W H W W M W W M 3 7
Aeolian Minor 5t. ABCDEFG ACDEG W H W W H W W M W W M W 2 6
Locrian BCDEFGA H W W H W W W 1 5

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

  1. Major pentatonic 5t. deletes Ionian’s fourth and seventh notes: C D E G A (1, 1, , 1, ). It’s the root form of the scale and the pentatonic circle of fourths’ lowest mode.
  2. Scottish pentatonic 5t. deletes Mixolydian’s third and seventh notes: G A C D E (1, , 1, 1, ). It’s the pentatonic circle of fourths’ second-lowest mode.
  3. Neutral pentatonic 5t. deletes Dorian’s third and sixth notes: D E G A C (1, , 1, , 1). Like Dorian, it’s symmetrical and the midpoint of its own circle of fourths.
  4. Minor pentatonic 5t. deletes Aeolian’s second and sixth notes: A C D E G (, 1, 1, , 1). It’s the pentatonic circle of fourths’ second-highest mode.
  5. Blues minor deletes Phrygian’s second and fifth notes: E G A C D (, 1, , 1, 1). It’s the pentatonic circle of fourths’ highest mode.

Oh, right. I haven’t explained why I call it a circle of fourths, which in turn means I need to explain scale rotation.

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A brief explanation of scale rotation

Scale rotation is the practice of forming a mode by moving intervals of a scale from its start to its end, or vice versa. When I refer to moving a scale’s intervals “left”, I’m referring both to the piano keyboard and to the scale interval tables I use. Rotating a scale left means moving most of its intervals to a lower point in the scale. Since scales repeat every octave, the rest move to the top.

“It might not always be ‘most’,” I hear you object. True, you could rotate a seven-note scale six degrees to the right, but why would you, when that’s the same as rotating it one degree to the left?

…Which means I need to explain rotation by degrees, too. No, we’re not talking angles here. Rotating a scale n degrees left moves its first n intervals to the end; rotating it n degrees right moves its last n intervals to the start. Dorian is one degree left of Ionian; Lydian is three degrees left of Ionian. And so on.

I may also refer to rotation by semitones. Rotating a scale by five semitones means the intervals moved sum up to five semitones. Thus, rotating Ionian five semitones to the left also takes you to Lydian.

A scale rotation’s size, measured either by degrees or by interval sum, has nothing to do with how far it moves the scale’s notes. Lydian is a five-semitone leftward rotation from Ionian, but it only moves one note (the fourth degree of the scale) by a semitone (F to F, when rooted on C).

I may also refer to scale rotations by how many notes they move. A single-note rotation only moves one note. This does not signify anything about how many intervals it moves forward or backward, the size of those intervals, or even about the interval by which the note is moved.

So, to summarize: Ionian to Lydian is a single-note rotation; it moves the note by one semitone, but it rotates the scale by five semitones (and three degrees).

One final note: A parallel rotation preserves the original mode’s root while changing its interval order and note composition; a relative rotation preserves the original mode’s key signature while changing its interval order and root. With rare exceptions (modes of limited transposition, discussed in §7.1), rotation by anything other than exact multiples of an octave cannot preseve both the root and the note composition.

I’ll try to keep this terminology from being ambiguous, but words are an imperfect medium for discussing music at the best of times, and when we throw mathematics, geometry, and set theory into the mix, forget it. If anything feels confusingly worded, please let me know, and I’ll try to clarify.

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The pentatonic circle of fourths, or, contrary motion explained

The pentatonic and Ionian circle of fifths orders move their parent scales in exact opposite directions. In fact, to emphasize this, I’m not even gonna call it the pentatonic circle of fifths anymore. I’ma call it monkeydude Josh the pentatonic circle of fourths. I find this fitting for at least two reasons:

  1. In scale analysis, octaves don’t matter: we get to the same note whether we go a perfect fifth down or a perfect fourth up. Altering the nomenclature helps call attention to their contrary directions.
  2. I’ve analyzed heptatonic scales through a seven-semitone lens. Analyzing pentatonic scales through anything but a five-semitone lens would feel wrong.

But why do they move in opposite directions? It’s probably easiest to analyze in terms of interval spacing.

Pentatonic changes two of Ionian’s interval pairs from “tone, semitone” to “minor third”. It so happens that one of Ionian’s two semitones closes out the scale. Thus, compare what happens when we shift pentatonic major’s intervals to the left to what happens when we shift Ionian’s.

Pentatonic interval spacing (root order)
#Mode1 2345Intervals
1Nainoi Major 5t CDEGA111
2Nairod Neutral 5t CDF GA11 1
3Naigyrhp Blues Minor CD F GA 1 11
4NaidyloximScottish 5tCDFGA111
5Nailoea Minor 5t CDF GA 11 1
Ionian interval spacing (root order)
#Mode1 234567Intervals
1Ionian CDEFGAB 1 1 ½ 1 1 1½
2Dorian CDE FGAB 1 ½ 1 1 1½ 1
3Phrygian CD E FGA B ½ 1 1 1½ 1 1
4Lydian CDEF GAB 1 1 1½ 1 1 ½
5MixolydianCDEFGAB 1 1½ 1 1 ½ 1
6Aeolian CDE FGA B 1½ 1 1 ½ 1 1
7Locrian CD E FG A B ½ 1 1 ½ 1 11

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That’s a lot to unpack. Some analysis:

Both scales contain a repeated interval set (called a tetrachord in Ionian and a trichord in pentatonic) and a spare whole step, or synaphe. I’ve marked one possible reading within the interval list. (Two such readings exist of pentatonic, and three of Ionian; in this example, I avoided splitting tetrachords or trichords in the base scale.)

Synaphe (plural: synaphai or synaphes) comes from the Attic Greek word σῠνᾰφή (sŭnăphḗ, literally connection, union, junction; point or line of junction; conjunction of two tetrachords). Its Attic pronunciation was roughly suh-nup-HEY pre-φ shift and suh-nuh-FAY (so, basically how a drunk person would say Santa Fe) afterward, but I think English speakers, mistakenly assuming it to be French, might say sy-NAFF.

(Pro tip: If a word contains ph and doesn’t split it across two syllables, it’s almost always transliterated Greek. Also, pronouncing foreign words using the wrong orthography is a great way to make a linguistics nerd’s blood boil. Speaking of which, orthography descends from ορθο- (ortho-, correct) and -γραφίᾱ (-graphíā, writing).)

Ionian and pentatonic share similar structures and five notes… but for this analysis’ purposes, that’s almost where their similarities stop.

Ionian interval spacing (circle of fifths order)
#Mode1 234567Intervals
4Lydian CDEF GAB 1 1 1½ 1 1 ½
1Ionian CDEFGAB 1 1 ½ 1 1 1½
5MixolydianCDEFGAB 1 1½ 1 1 ½ 1
2Dorian CDE FGAB 1 ½ 1 1 1½ 1
6Aeolian CDE FGA B 1½ 1 1 ½ 1 1
3Phrygian CD E FGA B ½ 1 1 1½ 1 1
7Locrian CD E FG A B ½ 1 1 ½ 1 11

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I’m analyzing Ionian’s tetrachord as whole tone, whole tone, semitone, placing the synaphe mid-scale. Swapping the synaphe with the tetrachord above shifts the scale down, note by note. (Remember, a scale is a repeating note pattern, so in Mixolydian, the tetrachord above is intervals 1-3; in Aeolian and Locrian, it’s split across the start and end of the scale.) Only Lydian starts with larger intervals than Ionian – it swaps Ionian’s first semitone and third whole tone, with the following results:

Pentatonic interval spacing (circle of fourths order)
#Mode1 2345Intervals
3Naigyrhp Blues Minor CD F GA 1 11
5Nailoea Minor 5t CDF GA 11 1
2Nairod Neutral 5t CDF GA11 1
4NaidyloximScottish 5tCDFGA111
1Nainoi Major 5t CDEGA111

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I’m analyzing pentatonic with the synaphe up front, making its trichord whole tone, minor third. Since major pentatonic’s intervals are as back-loaded as possible, its other C-rooted modes move at least one note up a semitone, and its rotations must lower the root before any other notes.

Thus, while Ionian is second in its circle of fifths, major pentatonic is last in its own circle of fourths. The next transposition in the above sequence yields C blues minor (or B blues minor, whichever you please).

A few additional observations about both scales:

A final note: Don’t read too much into my decision to highlight the synaphai. Their positions are only part of why the scales move in different directions. In fact, since both scales are, apart from two outliers, made entirely of whole steps, multiple intervals can be read as their synaphai; the choice depends entirely on the arbitrary choice of trichord or tetrachord pattern. Two such patterns can fit for the pentatonic scale and three for Ionian; each result in different synaphai and n-chord divisions. I settled on divisions that wouldn’t split the base scale’s n-chords. (I’ve highlighted my approaches below.)

Ionian tetrachords & pentatonic trichords & synaphai, oh my
Scale Pattern n-chord 1 n-chord 2 Synaphe
Major 5t 1 23 45 1
Major 5t 1 34 51 2
Ionian 11½ 567 123 4
Ionian 1½1 671 234 5
Ionian ½11 712 345 6

Note that the Ancient Greeks used the final analysis for their diatonic genus, which was nearly identical to our Ionian scale in all but name. When in Greece, I shall do as the Greeks did, but in this section, I figured it was better to defer the added complexity until this part of my explanation.

In short, the extra whole step’s position per se doesn’t affect the scale’s direction; their different directions are mostly due to Ionian ending with a semitone and major pentatonic with a minor third. But the circle orders are direct results of the pentatonic and Ionian scales’ atypically even note distribution. Transforming one mode of most other scales into another requires far more work. Let’s explore why.

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An analysis of five-semitone scale rotation

The pentatonic and Ionian scales are, respectively, the only pentatonic and heptatonic scales in 12-TET for which moving a single note by a semitone amounts to a scale rotation. There’s a relatively simple mathematical explanation for why, too:

In short, for moving a single note of any five- or seven-note scale in 12-TET by a semitone to rotate the scale, its intervals must be almost completely uniform, with only two identical outliers that:

  1. Differ in size from the remainder by only a semitone
  2. Are separated by five semitones

Readers may still have one final question: why is the number of outliers so important? Actually, it isn’t; it’s just important that the outliers be identical. If a seven-note scale could be completely uniform apart from one outlier, moving that interval would also rotate the scale. And, as it turns out, it can, but not, ironically, by making its note distribution more uniform.

  1. Take the temperament modulo the scale size to get the number of extra semitones to distribute: 12 modulo 7 is 5. We have five extra semitones to distribute.
  2. Take the note count modulo the extra semitones to figure out the most uniform note distribution possible: 7 modulo 5 is 2.

In short, two intervals must be outliers in the most uniform heptatonic note distribution possible. The way to get a single outlier, therefore, is to go in the exact opposite direction and make the outlier as big as possible. Which brings us to our next point of analysis.

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Other single-note scale rotations

The hendecatonic scale

I’ve focused most of my analysis so far on pentatonic and heptatonic scales, but now that I better understand the mathematical principles explaining why this happens, I’ve expanded my scope somewhat to see if I can uncover other examples of similar patterns with scale rotations of various interval sizes. I’ve uncovered a few, which I’ll explain in this section.

Other equal temperaments certainly have similar examples (for instance, in 24-TET, rotation by 11 or 13 quarter-tones should produce similar results for similar 11- and 13-note scales), but I haven’t finished developing tools for scale analysis outside 12-TET, so they’ll have to wait.

Do other scales exist in 12-TET that don’t contain the above composition for which moving a single note by a semitone will qualify as a scale rotation? As it happens, yes: I can say with complete confidence, without even having to think about it, that the hendecatonic scale must demonstrate the same principle. And I say the hendecatonic scale for a simple reason: 12-TET contains only a single hendecatonic scale. The reason may be self-explanatory, but if it isn’t, I’ll give you a hint: It’s the same reason there’s only one dodecatonic scale.

In 12-TET, hendecatonic scales must contain all but one note of the chromatic scale. Thus, it must contain ten semitones and one whole tone, and swapping its whole tone with any of its semitones qualifies as a scale rotation by default. There are only eleven ways to remove notes that aren’t the root; thus, a single scale with eleven modes, which displays similar patterns not just for the circle of fifths but for every possible interval in 12-TET.

Transforming the hendecatonic scale
#1234567891011Intervals
C 1CC DD EFF GG AA ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ 1
C 2CC DD EFF GG AB½ ½ ½ ½ ½ ½ ½ ½ ½ 1 ½
C 3CC DD EFF GG A B½ ½ ½ ½ ½ ½ ½ ½ 1 ½ ½
C 4CC DD EFF GAA B½ ½ ½ ½ ½ ½ ½ 1 ½ ½ ½
C 5CC DD EFF G AA B½ ½ ½ ½ ½ ½ 1 ½ ½ ½ ½
C 6CC DD EFGG AA B½ ½ ½ ½ ½ 1 ½ ½ ½ ½ ½
C 7CC DD EF GG AA B½ ½ ½ ½ 1 ½ ½ ½ ½ ½ ½
C 8CC DD FF GG AA B½ ½ ½ 1 ½ ½ ½ ½ ½ ½ ½
C 9CC DEFF GG AA B½ ½ 1 ½ ½ ½ ½ ½ ½ ½ ½
C 10CC D EFF GG AA B½ 1 ½ ½ ½ ½ ½ ½ ½ ½ ½
C 11CDD EFF GG AA B1 ½ ½ ½ ½ ½ ½ ½ ½ ½ ½
C 1C DD EFF GG AA B½ ½ ½ ½ ½ ½ ½ ½ ½ ½ 1

Of course, the very fact that only one hendecatonic scale exists in 12-TET somehow makes this fact feel vastly less impressive, even though it has exactly the same cause as Ionian’s circle of fifths pattern. Funny how that works. (In fact, the hendecatonic scale can be generated using the exact same generator as Ionian; it just runs for four more notes.)

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The heptatonic chromatic scale

Applying this same principle, we can determine that one other heptatonic scale exists in 12-TET for which moving a series of single notes by a constant interval size will rotate the scale. However, you don’t move its notes by a semitone; you move them by a perfect fourth. And, ironically, this doesn’t rotate it by five semitones; it rotates it by one. And its root won’t progress through the chromatic scale by semitones: it’ll progress through it by perfect fourths. (This will still take it all the way around the chromatic scale, just in a different order.) It’s the heptatonic chromatic scale, which has the following interval spacing:

semitone, semitone, semitone, semitone, semitone, semitone, tritone

Trying to restrict ourselves to using every letter of the scale gives us an absolutely cursed set of notations. (For this set of tables and only this set of tables, I’ve used chromatic coloring rather than Doppler-shift coloring.)

Modes of the heptatonic chromatic scale
Mode 1 234567Intervals
Heptatonic Chromatic ICDE𝄫F𝄫G𝄫♭A𝄫𝄫B𝄫𝄫♭½½½½½½3
Heptatonic Chromatic IICDE𝄫F𝄫G𝄫♭A𝄫𝄫B½½½½½3½
Heptatonic Chromatic IIICDE𝄫F𝄫G𝄫♭AB½½½½3½½
Heptatonic Chromatic IVCDE𝄫F𝄫G𝄪AB½½½3½½½
Heptatonic Chromatic VCDE𝄫F𝄪♯G𝄪AB½½3½½½½
Heptatonic Chromatic VICDE𝄪♯F𝄪♯G𝄪AB½3½½½½½
Heptatonic Chromatic VIICD𝄪𝄪E𝄪♯F𝄪♯G𝄪AB3½½½½½½

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“D𝄪𝄪” and “B𝄫𝄫♭” use the same color because they represent the same note. You probably know it as F or G.

Sorry. Like I said, it’s an absolutely cursed set of notations. Perhaps it’ll be more legible if we abandon any pretense of using every note name once. In fact, let’s observe the entire set of transformations:

The heptatonic chromatic scale (complete transformation)
# 1 234567Intervals
C 1CC DD EFF ½½½½½½3
C 2CC DD EFB½½½½½3½
C 3CC DD EA B½½½½3½½
C 4CC DD AA B½½½3½½½
C 5CC DG AA B½½3½½½½
C 6CC GG AA B½3½½½½½
C 7CF GG AA B3½½½½½½
F 1FF GG AA B½½½½½½3
F 2FF GG AA E½½½½½3½
F 3FF GG AD E½½½½3½½
F 4FF GG DD E½½½3½½½
F 5FF GC DD E½½3½½½½
F 6FF CC DD E½3½½½½½
F 7FBCC DD E3½½½½½½
A1A BCC DD E½½½½½½3
A2A BCC DD A½½½½½3½
A3A BCC DG A½½½½3½½
A4A BCC GG A½½½3½½½
A5A BCF GG A½½3½½½½
A6A BFF GG A½3½½½½½
A7A EFF GG A3½½½½½½
D1D EFF GG A½½½½½½3
D2D EFF GG D½½½½½3½
D3D EFF GC D½½½½3½½
D4D EFF CC D½½½3½½½
D5D EFBCC D½½3½½½½
D6D EA BCC D½3½½½½½
D7D AA BCC D3½½½½½½
G1G AA BCC D½½½½½½3
G2G AA BCC G½½½½½3½
G3G AA BCF G½½½½3½½
G4G AA BFF G½½½3½½½
G5G AA EFF G½½3½½½½
G6G AD EFF G½3½½½½½
G7G DD EFF G3½½½½½½
C1C DD EFF G½½½½½½3
C2C DD EFF C½½½½½3½
C3C DD EFBC½½½½3½½
C4C DD EA BC½½½3½½½
C5C DD AA BC½½3½½½½
C6C DG AA BC½3½½½½½
C7C GG AA BC3½½½½½½
F1F GG AA BC½½½½½½3
F2F GG AA BF½½½½½3½
F3F GG AA EF½½½½3½½
F4F GG AD EF½½½3½½½
F5F GG DD EF½½3½½½½
F6F GC DD EF½3½½½½½
F7F CC DD EF3½½½½½½
B 1BCC DD EF½½½½½½3
B 2BCC DD EA ½½½½½3½
B 3BCC DD AA ½½½½3½½
B 4BCC DG AA ½½½3½½½
B 5BCC GG AA ½½3½½½½
B 6BCF GG AA ½3½½½½½
B 7BFF GG AA 3½½½½½½
E 1EFF GG AA ½½½½½½3
E 2EFF GG AD ½½½½½3½
E 3EFF GG DD ½½½½3½½
E 4EFF GC DD ½½½3½½½
E 5EFF CC DD ½½3½½½½
E 6EFBCC DD ½3½½½½½
E 7EA BCC DD 3½½½½½½
A 1AA BCC DD ½½½½½½3
A 2AA BCC DG ½½½½½3½
A 3AA BCC GG ½½½½3½½
A 4AA BCF GG ½½½3½½½
A 5AA BFF GG ½½3½½½½
A 6AA EFF GG ½3½½½½½
A 7AD EFF GG 3½½½½½½
D 1DD EFF GG ½½½½½½3
D 2DD EFF GC ½½½½½3½
D 3DD EFF CC ½½½½3½½
D 4DD EFBCC ½½½3½½½
D 5DD EA BCC ½½3½½½½
D 6DD AA BCC ½3½½½½½
D 7DG AA BCC 3½½½½½½
G 1GG AA BCC ½½½½½½3
G 2GG AA BCF ½½½½½3½
G 3GG AA BFF ½½½½3½½
G 4GG AA EFF ½½½3½½½
G 5GG AD EFF ½½3½½½½
G 6GG DD EFF ½3½½½½½
G 7GC DD EFF 3½½½½½½

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Other truncations of the chromatic scale

Similar corollaries apply to certain other scale sizes: moving successive notes of the pentatonic chromatic scale (½ ½ ½ ½ 4, or semitone, semitone, semitone, semitone, minor sixth) by seven semitones will inevitably take the scale through all its modes, moving the scale root up by a perfect fifth each time. (The pentatonic chromatic scale is heptatonic chromatic’s scale complement.)

However, this only works for a few scale sizes. Why? Rotating the hexatonic chromatic scale (½ ½ ½ ½ ½ , or semitone, semitone, semitone, semitone, semitone, perfect fifth) moves each note by a tritone. Two tritones add up to an octave, so we skip five-sixths of the chromatic scale.

The only non-semitone interval of any such truncation of the chromatic scale is (13 - n) semitones, where n is the scale’s note count. Thus, scale rotation moves such scales’ notes (12 - n) semitones. For such a rotation to take us through every mode across the entire chromatic scale, this interval must be coprime with 12. In fact, in any n-TET, for scale rotations that move single notes by more than a chromatic step to cover the entire chromatic scale, the interval size (in units of 1/n octave) must be coprime with n. Thus, in 12-TET:

Note movements and the 12-TET chromatic scale
Semitones Notes used Pattern
1 11 all notes linear order
2 10 six notes major second
3 9 four notes minor third
4 8 three notes major third
5 7 all notes circle of fourths
6 two notes tritone

Why do most rows list two interval sizes? Moving a note up by seven semitones equates to moving it down by five. Thus, in 12-TET, the only truncations of the chromatic scale for which this method of scale rotation will work contain 1, 5, 7, and 11 notes, and their rotations will respectively move single notes by a semitone, a perfect fourth, a perfect fourth, and a semitone.

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Alternating heptamode and alternating heptamode inverse

Thus, the Ionian and heptatonic chromatic scales are the only heptatonic scales in 12-TET with single-note rotations that cycle through the entire chromatic scale. I’ve only found five other heptatonic scales in 12-TET with single-note rotations of any sort, and they’re complicated. In fact, they have an asterisk: all five require moving notes from the start of the scale to the end, or vice versa, which means that in some respects, they aren’t even single-note transformations. Two of these scales respectively have the following interval spacing:

Alternating heptamode: semitone, whole tone, semitone, whole tone, semitone, whole tone, minor third
Jhankāradhvani 5: whole tone, semitone, whole tone, semitone, whole tone, semitone, minor third

Now, if we try to use every letter of the scale, it might not even be obvious that this even works as a single-note transformation. (I haven’t given these proper names yet; sorry.)

Alternating heptamode & alternating heptamode inverse
Mode 1 234567Intervals
1212132CDEFGA𝄫B½1½1½1
1213212CDEFGAB½1½1½1
1321212CDEFGAB½1½1½1
2121213CDEFGAB𝄫1½1½1½
2121321CDEFGAB1½1½1½
2132121CDEFGAB1½1½1½
3212121CDEFGAB1½1½1½
1212123CDEFGA𝄫B𝄫½1½1½1
1212312CDEFGAB½1½1½1
1231212CDEFGAB½1½1½1
3121212CDEFGAB½1½1½1
2121231CDEFGAB1½1½1½
2123121CDEFGAB1½1½1½
2312121CDEFGAB1½1½1½

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Let’s try just using sharps instead of trying to use every letter of the scale – which is still confusing, since the notes don’t stay in the same positions between scales. For instance, D appears in all scales except 1321212 (the third scale) and 2312121 (the last), but in 3212121 (the seventh scale) and 3121212 (the fourth from the last), it’s the second note of the scale, and in all others it appears in, it’s the third note. Likewise, F appears in all scales except 1213212 (the second scale) and 2123121 (the second from the last), and it switches between positions four and five.

Alternating heptamode & alternating heptamode inverse (revisited)
Mode 1 234567Intervals
1212132CCDEFGA½1½1½1
1213212CCDEGAA½1½1½1
1321212CCEFGAA½1½1½1
2121213CDDFFGA1½1½1½
2121321CDDFFAB1½1½1½
2132121CDDFGAB1½1½1½
3212121CDFFGAB1½1½1½
1212123CCDEFGA½1½1½1
1212312CCDEFAA½1½1½1
1231212CCDFGAA½1½1½1
3121212CDEFGAA½1½1½1
2121231CDDFFGB1½1½1½
2123121CDDFGAB1½1½1½
2312121CDFFGAB1½1½1½

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In case it isn’t obvious why these don’t cycle through the entire chromatic scale: in both cases, the single-note rotation moves a note by a minor third. Four minor thirds make up an octave, so we will only cycle through four notes of the chromatic scale.

That said, we’re somewhat examining this scale through the wrong lens. The issue is that the Ionian scale’s root only changes once in each of its transformation cycles: between Locrian and Lydian. As I hinted at, that’s not actually true of this scale. If we examine the pattern of notes being moved, we notice something interesting:

  1. 1212132 to 1213212: F is raised to A
  2. 1213212 to 1321212: D is raised to F
  3. 1321212 to 2121213: C is raised to D

…and this is where things get confusing.

With Ionian’s scale transformations, we only have to mess with the root once: between Locrian and Lydian. Alternating heptamode isn’t so simple: we have to transpose the root on multiple occasions. In fact, only eight notes can ever be part of this set of transformations, and going forward, I’ll represent them all with flats except F, since doing so will make E the only repeated letter. Thus, starting on C with 1212132 gives us:

Alternating heptamode: A closer examination
ModeD¹E¹F¹B¹D²E²F²
1212132CDEEFG B
1213212CDEE GAB
1321212CD EFGAB
2121213 DEEFGAB
2121321 DEEFG BC
2132121 DEE GABC
3212121 D EFGABC
1212132 EEFGAB D
1213212 EEFG BCD
1321212 EE GABCD
2121213 EFGABCD
2121321 EFGAB DE
2132121 EFG BCDE
3212121 E GABCDE
1212132 FGABCD E
1213212 FGAB DEE
1321212 FG BCDEE
2121213 GABCDEE
2121321 GABCD EF
2132121 GAB DEEF
3212121 G BCDEEF
1212132 ABCDEE G
1213212 ABCD EFG
1321212 AB DEEFG
2121213 BCDEEFG
2121321 BCDEE GA
2132121 BCD EFGA
3212121 B DEEFGA

In each row, the orange note will move in the next scale, and the blue note has just moved. Note that these are always separated by a tritone within the same scale, and the note always moves up by a minor third from one scale to the next (except from 3212121 to 1212132, when it moves down by a major sixth). The note that moves is also always either a minor third below or a major sixth above the note that moved in the previous scale. The four notes that move by minor thirds and major sixths form C, E, F, and A diminished seventh chords; the four notes that only ever move by octaves form D, E, G, and A diminished seventh chords.

I’m going to confess that I don’t perfectly understand what’s going on here either, but I think what’s happening is that whenever we move the root, or whenever we move a note across the root, we have to move a note from the end of the scale to the start, or vice versa. But note that both this scale and its inverse contain three pairs of whole tones and semitones, and one additional minor third. A whole tone and a semitone, of course, add up to a minor third. Thus, although this is in some ways the least regular interval distribution we’ve examined, the fact that it can be divided into four three-semitone regions (three with two notes, one with one) necessitates a rotation that lines up exactly with the parent tonality. In short, while a scale needs to be nearly regular to be possible to rotate all its notes across the entire chromatic scale, the intervals themselves must be impossible to subdivide into regions of exactly equal size.

These scales’ complements experience exactly the same issue:

Major pentatonic 234: semitone, whole tone, minor third, minor third, minor third
Major pentatonic 23: whole tone, semitone, minor third, minor third, minor third

I will leave discerning why as an exercise for the reader.

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Melodic Phrygian (or Neapolitan “major”, as it’s misleadingly known)

Another scale with a similar transformation is just as confusing to track. It’s most frequently known by the misleading name “Neapolitan major”. This is a terrible name, because it’s objectively not a major scale – it doesn’t have a major third above the root! The reasoning behind the name is that a Neapolitan chord starts on a scale’s flat second degree, and it has a major sixth above the root, while its counterpart the Neapolitan minor has a minor sixth above the root. This is an awful justification: I can’t think of any scales for which “major” and “minor” mean the sixth scale degree rather than the third. A name that requires an explanation to make sense isn’t very helpful! I instead call them, respectively, melodic Phrygian and harmonic Phrygian; anyone who knows Phrygian, melodic minor, and harmonic minor can reasonably infer what those mean.

Melodic Phrygian transforms as follows:

Analyzing the melodic Phrygian scale
Mode 1 234567 Intervals
Locrian major CDEFGAB 11½½111
Super-Locrian 2 CDEFGAB 1½½1111
Leading whole-tone inverse CDEFGAB ½½11111
Melodic Phrygian CDEFGAB ½11111½
Leading whole-tone CDEFGAB 11111½½
Lydian dominant augmented CDEFGAB𝄫 1111½½1
Lydian dominant 6 CDEFGA𝄫B𝄫 111½½11

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(Lydian dominant 6 is also, equally inaccurately, called Lydian minor. I won’t dignify this with further discussion.)

Once again, trying to use all seven note names paints a misleading picture. It looks like more than one note is moving per scale transformation until we rename the notes. In the following table, I’ve renamed C to B, E to F, E to D, G to F, A𝄫 to G, A to G, B𝄫 to A, and B to A. I’ve also reshuffled the modes so that melodic Phrygian is at the bottom, intensified the border each time we move the root note, and highlighted the note we move in each scale transformation. Since we move the note by a whole step each time, we skip half the possible transpositions. This set gives us melodic Phrygian on A, G, F, E, D, and C; and its other modes on B, A, G, F, D, and C.

Transforming the melodic Phrygian scale
Mode 1 234567 Intervals
Leading whole-tone BCDFGAA 11111½½
Lydian dominant augmented BCDFGGA 1111½½1
Lydian dominant 6 BCDFFGA 111½½11
Locrian major BCDEFGA 11½½111
Super-Locrian 2 BCDDFGA 1½½1111
Leading whole-tone inverse BCCDFGA ½½11111
Melodic Phrygian ABCDFGA ½11111½
Leading whole-tone ABCDFGG 11111½½
Lydian dominant augmented ABCDFFG 1111½½1
Lydian dominant 6 ABCDEFG 111½½11
Locrian major ABCDDFG 11½½111
Super-Locrian 2 ABCCDFG 1½½1111
Leading whole-tone inverse AABCDFG ½½11111
Melodic Phrygian GABCDFG ½11111½
Leading whole-tone GABCDFF 11111½½
Lydian dominant augmented GABCDEF 1111½½1
Lydian dominant 6 GABCDDF 111½½11
Locrian major GABCCDF 11½½111
Super-Locrian 2 GAABCDF 1½½1111
Leading whole-tone inverse GGABCDF ½½11111
Melodic Phrygian FGABCDF ½11111½
Leading whole-tone FGABCDE 11111½½
Lydian dominant augmented FGABCDD 1111½½1
Lydian dominant 6 FGABCCD 111½½11
Locrian major FGAABCD 11½½111
Super-Locrian 2 FGGABCD 1½½1111
Leading whole-tone inverse FFGABCD ½½11111
Melodic Phrygian EFGABCD ½11111½
Leading whole-tone DFGABCD 11111½½
Lydian dominant augmented DFGABCC 1111½½1
Lydian dominant 6 DFGAABC 111½½11
Locrian major DFGGABC 11½½111
Super-Locrian 2 DFFGABC 1½½1111
Leading whole-tone inverse DEFGABC ½½11111
Melodic Phrygian DDFGABC ½11111½
Leading whole-tone CDFGABC 11111½½
Lydian dominant augmented CDFGAAB 1111½½1
Lydian dominant 6 CDFGGAB 111½½11
Locrian major CDFFGAB 11½½111
Super-Locrian 2 CDEFGAB 1½½1111
Leading whole-tone inverse CDDFGAB ½½11111
Melodic Phrygian CCDFGAB ½11111½

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Apathetic minor & Pacific

I’ve found one other pair of scales that exhibit this behavior. Unsurprisingly, they’re each other’s mirror images. I’ll call them “apathetic minor” and “Pacific”, but again, I don’t have good names for most of their modes yet. (For the record, “apathetic minor” is the mode labelled “1131141”, and Pacific is the mode labelled “1311141”. Again, I apologize for not baving better names for these.) Rooted on C, they look like this:

Apathetic minor and Pacific on C
Mode 1 234567 Intervals
3114111CDEFG𝄪AB½½2½½½
4111311CD𝄪EFGAB2½½½½½
1311411CDEFGAB½½½2½½
1411131CDEFGAB½2½½½½
1131141CDE𝄫FGA𝄫B½½½½2½
1141113CDE𝄫FGAB𝄫½½2½½½
1113114CDE𝄫F𝄫GA𝄫B𝄫♭½½½½½2
4113111CD𝄪EFG𝄪AB2½½½½½
3111411CDEFGAB½½½2½½
1411311CDEFGAB½2½½½½
1311141CDEFGA𝄫B½½½½2½
1141131CDE𝄫FGAB½½2½½½
1131114CDE𝄫FGA𝄫B𝄫♭½½½½½2
1114113CDE𝄫F𝄫GAB𝄫½½½2½½

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Which, again, is extremely difficult to parse. Let’s simplify.

Apathetic minor and Pacific on C (simplified)
Mode 1 234567 Intervals
3114111CDEFAAB½½2½½½
4111311CEFFGAB2½½½½½
1311411CCEFFAB½½½2½½
1411131CCFFGGB½2½½½½
1131141CCDFFGB½½½½2½
1141113CCDFGGA½½2½½½
1113114CCDDFGG½½½½½2
4113111CEFFAAB2½½½½½
3111411CDEFFAB½½½2½½
1411311CCFFGAB½2½½½½
1311141CCEFFGB½½½½2½
1141131CCDFGGB½½2½½½
1131114CCDFFGG½½½½½2
1114113CCDDGGA½½½2½½

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Of course, the transformations again require moving the root more often than that. And since we’re moving notes by six semitones, we again skip ⅚ of the chromatic scale for each mode.

Single-note transformations of apathetic minor
Mode 1 234567 Intervals
3114111CDEFAAB½½2½½½
4111311BDEFFAA2½½½½½
1311411BCDEFAA½½½2½½
1411131ABDEFFA½2½½½½
1131141ABCDEFA½½½½2½
1141113AABDEFF½½2½½½
1113114AABCDEF½½½½½2
3114111FAABDEF½½2½½½
4111311FAABCDE2½½½½½
1311411FFAABDE½½½2½½
1411131EFAABCD½2½½½½
1131141EFFAABD½½½½2½
1141113DEFAABC½½2½½½
1113114DEFFAAB½½½½½2
Single-note transformations of Pacific
Mode 1 234567 Intervals
1114113DDEFAAB½½½2½½
4113111BDEFGAA2½½½½½
3111411BDDEFAA½½½2½½
1411311ABDEFGA½2½½½½
1311411ABDDEFA½½½½2½
1141131AABDEFG½½2½½½
1131114AABDDEF½½½½½2
1114113GAABDEF½½½2½½
4113111FAABDDE2½½½½½
3111411FGAABDE½½½2½½
1411311EFAABDD½2½½½½
1311411EFGAABD½½½½2½
1141131DEFAABD½½2½½½
1131114DEFGAAB½½½½½2

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A brief analysis of note distributions

It may also be instructive to compare how many times each note appears on each mode of each scale on C.

Note distributions across parallel modes
ScaleCCDDEFFGGAAB
Ionian 7 2 5 4 3 6 2 6 3 4 5 2
Major pentatonic 5 0 3 2 1 4 0 4 1 2 3 0
Chromatic heptatonic 7 6 5 4 3 2 2 2 3 4 5 6
Chromatic pentatonic 5 4 3 2 1 0 0 0 1 2 3 4
Alternating heptamode 7 3 3 6 3 3 6 3 3 6 3 3
Major pentatonic 234 5 1 1 4 1 1 4 1 1 4 1 1
Alternating heptamode inverse 7 3 3 6 3 3 6 3 3 6 3 3
Major pentatonic 23 5 1 1 4 1 1 4 1 1 4 1 1
Melodic Phrygian 7 2 6 2 6 2 6 2 6 2 6 2
Augmented ninth 5 0 4 0 4 0 4 0 4 0 4 0
Apathetic minor 7 5 3 2 3 5 6 5 3 2 3 5
Rāga Saugandhini 5 3 1 0 1 3 4 3 1 0 1 3
Pacific 7 5 3 2 3 5 6 5 3 2 3 5
Rāga Saugandhini inverse 5 3 1 0 1 3 4 3 1 0 1 3

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  1. The reflective symmetry about F seen above holds uniformly for every scale.
  2. As a consequence of #1, a scale and its mirror image always have identical note distributions. This explains the identical note distributions seen above.
  3. After each heptatonic scale, I list its pentatonic complement. Each pentatonic scale uses every note of the chromatic scale in two fewer modes than its heptatonic complement does. This principle appears to hold uniformly for all scale complements, even counting the null set as the chromatic scale’s complement.

    1. Hexatonic scales and their complements always have identical note distributions.
    2. Heptatonic distributions equal their pentatonic complements’ distributions plus two.
    3. Octatonic distributions equal their tetratonic complements’ distributions plus four.
    4. Enneatonic distributions equal their three-note complements’ distributions plus six.
    5. Decatonic distributions equal their two-note complements’ distributions plus eight.
    6. The hendecatonic scale’s distribution equals the octave’s distribution plus ten.
    7. The dodecatonic scale’s distribution equals the null set’s distribution plus twelve.

    This has a few further implications. At the start of this section, we noted that the hendecatonic scale must use each note of the chromatic scale except the root in all but one of its modes. For similar reasons, every note of the chromatic scale except the root appears in:

    1. At least eight modes of every decatonic scale.
    2. At least six modes of every enneatonic scale.
    3. At least four modes of every octatonic scale.
    4. At least two modes of every heptatonic scale.

    Meanwhile, all scales must use the root in every mode – this is a requirement of being a scale.

  4. Decrementing alternating heptamode and melodic Phrygian’s root counts by 1 would cause them to repeat the same pattern four and six times, respectively. This makes them very nearly rotationally as well as reflectively symmetrical. Explaining the full implications of this would require covering some principles we haven’t addressed yet, so we’ll return to these scales when we discuss modes of limited transposition (which, to be clear, these scales aren’t; however, they’re single-note transformations thereof).

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Why Ionian’s scale mutation requires a coprime scale length with 12

For the sake of Science, let’s try using a scale generator with four notes separated by perfect fifths. Starting on F, we’ll get a scale of C-D-F-G. What would that set of transformations look like?

Tetratonic truncation of Ionian
RIDMIDSIDIID1234Intervals
1065D093D38.β3252CDFA11
1185D138D38.δ5232CFGA11
165D062D38.α2325CDFG11
645D073D38.γ2523CDGA11
1065D093D38.β3252EGAD11

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(I’ll explain these names and/or replace them with better ones soon™.)

Each note moves by a major third. When we move the root, we also must move the scale’s second note to the end. That’s fair enough – we’ve seen it before – but because we’re moving notes by four semitones, we won’t cycle through the entire chromatic scale. Our next root will be G, and then we’ll be back on C, having skipped three quarters of the chromatic scale. (Egad. …Sorry.)

OK, so what about C-D-E-F-G-A?

Hexatonic truncation of Ionian (semitone transposition)
RIDMIDSIDIID123456Intervals
2709F296F71.β223221CDEGAB1 1 1 1 ½
693F273F71.ε221223CDEFGA1 1 ½ 1 1
1701F315F71.γ232212CDFGAA1 1 1 ½ 1
1197F239F71.ζ212232CDDFGA1 ½ 1 1 1
1449F383F71.δ322122CDFGGA1 1 ½ 1 1
1323F116F71.α122322CCDFGA½ 1 1 1 1
2709F296F71.β223221BCDFGA1 1 1 1 ½
693F273F71.ε221223BCDEFG1 1 ½ 1 1

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This will cycle us through the entire circle of fifths, but…

  1. Between F71.β and F71.ε, B moves down by a tritone to F.
  2. Between F71.ε and F71.γ, E moves up by a tritone to A.
  3. Between F71.γ and F71.ζ, A moves down by a tritone to D.
  4. Between F71.ζ and F71.δ, D moves up by a tritone to G.
  5. Between F71.δ and F71.α, G moves down by a tritone to C.

But then the pattern breaks. The transformation from F71.α on C to F71.β on B requires moving two notes, not one: F down to B, and C up to F – both by a tritone, but we’re no longer doing single-note transformations.

Another giveaway is that the interval pattern breaks – indeed, as we can see by examining F71.ε on B, we’d need even more transformations to continue it.

We can continue the pattern by transposing the scale by a tritone rather than a semitone:

Hexatonic truncation of Ionian (tritone transposition)
RIDMIDSIDIID123456Intervals
2709F296F71.β223221CDEGAB1 1 1 1 ½
693F273F71.ε221223CDEFGA1 1 ½ 1 1
1701F315F71.γ232212CDFGAA1 1 1 ½ 1
1197F239F71.ζ212232CDDFGA1 ½ 1 1 1
1449F383F71.δ322122CDFGGA1 1 ½ 1 1
1323F116F71.α122322CCDFGA½ 1 1 1 1
2709F296F71.β223221FGACDF1 1 1 1 ½
693F273F71.ε221223FGABCD1 1 ½ 1 1

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But of course, we skip ⅚ of the chromatic scale by doing this.

Well, what about octatonic, then? As far as I can work out, there is a single-note set of transformations, but again, it will not cycle us through the entire chromatic scale.

Octatonic expansions of Ionian
RIDMIDSIDIID12345678Intervals
3765H280H28.δ22122111CDEFGAAB11½11½½½
2805H271H28.η22111221CDEFFGAB11½½½11½
2775H095H28.β11221221CCDEFGAB½½11½11½
1965H256H28.ε21221112BCDEFGGA1½11½½½1
1725H224H28.θ21112212BCDDEFGA1½½½11½1
3435H170H28.γ12212211ABCDEFGA½11½11½½
1515H163H28.ζ12211122ABCDEFFG½11½½½11
1455H047H28.α11122122ABCCDFFG½½½11½11
3765H280H28.δ22122111GACCDFFG11½11½½½

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This set of mutations is difficult to parse, but as far as I can make out, this is what’s happening:

  1. H28.δ to H28.η: A moves down four semitones to F.
  2. H28.η to H28.β: One semitone below F, F moves down four semitones to C.
  3. H28.β to H28.ε: One semitone below C, C moves down four semitones to G. Because we moved the root below B, it now moves to the front of the scale.
  4. H28.ε to H28.θ: One semitone below G, G moves down four semitones to D.
  5. H28.θ to H28.γ: One semitone below D, D moves down four semitones to A. A becomes the new root, since it is below our previous root of B.
  6. H28.γ to H28.ζ: One semitone below A, A moves down four semitones to F.
  7. H28.ζ to H28.α: One semitone below F, E moves down four semitones to C.
  8. H28.α to H28.δ: One semitone below C, B moves down four semitones to G. Because we moved the root below G, it now moves to the front of the scale.

As we can see, this sequence of transformations will move the entire scale by major thirds each cycle, so this set of transformations will only transpose each mode to one third of the chromatic scale. This is a direct result of the scale length sharing a factor with 12: it’s surely no coincidence that their mutual factor is exactly equal to the number of semitones by which this set of transformations moves the scale each cycle. Effectively, for this form of mutation to work, the generator must use an interval size that is coprime with the parent temperament, and it must use a scale length that is also coprime with the parent temperament. So, the intervals that will work are one semitone, five semitones, seven semitones, and eleven semitones, and the scale lengths that will work are one note, five notes, seven notes, and eleven notes.

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Transformations of the Ionian scale

Harmonic minor

Since learning harmonic and melodic minor’s modes 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, linear order)
Mode 1 234567Intervals
Aeolian 7Harmonic minorCDEFGAB1½11½½
AeolianCDEFGAB 1½11½11
Locrian 6Maqam TarznauynCDEFGAB ½11½½1
LocrianCDEFGAB ½11½111
Ionian 5Ionian augmentedCDEFGAB11½½1½
IonianCDEFGAB11½111½
Dorian 4Lydian diminishedCDEFGAB 1½½1½1
DorianCDEFGAB 1½111½1
Phrygian 3Phrygian dominantCDEFGAB ½½1½11
PhrygianCDEFGAB ½111½11
Lydian 2Aeolian harmonicCDEFGAB½1½11½
LydianCDEFGAB111½11½
Mixolydian 1Super-Locrian 7CDEFGAB𝄫½1½11½
MixolydianCDEFGAB 11½11½1

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Thus, as the first set of mode names suggests, harmonic minor’s modes respectively raise by a half-step:

Since we’re transposing every mode to C, we can’t raise Mixolydian’s first degree, because it’s the first degree! Instead, we must 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, it’s usually already raised for us: it’s harmonic minor’s seventh degree! Let’s see what happens when we root these modes on the corresponding notes of their respective parent C minor scales:

Harmonic minor vs. modes from Aeolian (rooted on scale, linear order)
Mode 1 234567Intervals
Aeolian 7Harmonic minor CDEFGAB 1½11½½
Aeolian CDEFGAB 1½11½11
Locrian 6Maqam Tarznauyn DEFGABC ½11½½1
Locrian DEFGABC ½11½111
Ionian 5Ionian augmented EFGABCD 11½½1½
Ionian EFGABCD 11½111½
Dorian 4Lydian diminished FGABCDE 1½½1½1
Dorian FGABCDE 1½111½1
Phrygian 3Phrygian dominant GABCDEF ½½1½11
Phrygian GABCDEF ½111½11
Lydian 2Aeolian harmonic ABCDEFG ½1½11½
Lydian ABCDEFG 111½11½
Mixolydian 1Super-Locrian 7 BCDEFGA ½1½11½
Mixolydian BCDEFGA 11½11½1

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Thus, B Mixolydian is to C Aeolian as B Mixolydian 1 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 Mixolydian 1 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)
Mode 1 234567Intervals
Lydian 2Aeolian harmonicCDEFGAB½1½11½
LydianCDEFGAB111½11½
Ionian 5Ionian augmentedCDEFGAB11½½1½
IonianCDEFGAB11½111½
Mixolydian 1Super-Locrian 7CDEFGAB𝄫½1½11½
MixolydianCDEFGAB 11½11½1
Dorian 4Lydian diminishedCDEFGAB 1½½1½1
DorianCDEFGAB 1½111½1
Aeolian 7Harmonic minorCDEFGAB1½11½½
AeolianCDEFGAB 1½11½11
Phrygian 3Phrygian dominantCDEFGAB ½½1½11
PhrygianCDEFGAB ½111½11
Locrian 6Maqam TarznauynCDEFGAB ½11½½1
LocrianCDEFGAB ½11½111

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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 Mixolydian 1, 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)
Mode 1 234567Intervals
Lydian 2Aeolian harmonic ABCDEFG ½1½11½
Lydian ABCDEFG 111½11½
Ionian 5Ionian augmented EFGABCD 11½½1½
Ionian EFGABCD 11½111½
Mixolydian 1Super-Locrian 7 BCDEFGA ½1½11½
Mixolydian BCDEFGA 11½11½1
Dorian 4Lydian diminished FGABCDE 1½½1½1
Dorian FGABCDE 1½111½1
Aeolian 7Harmonic minor CDEFGAB 1½11½½
Aeolian CDEFGAB 1½11½11
Phrygian 3Phrygian dominant GABCDEF ½½1½11
Phrygian GABCDEF ½111½11
Locrian 6Maqam Tarznauyn DEFGABC ½11½½1
Locrian DEFGABC ½11½111

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Analysis of chord tonality by scale position:

Chord tonalities by scale position & mode (harmonic minor, linear order)
Mode1234567
1Aeolian 7 Harmonic minor mindim AUGmin Maj Majdim
2Locrian 6 Maqam Tarznauyn dim AUGmin Maj Majdimmin
3Ionian 5 Ionian augmented AUGmin Maj Majdimmindim
4Dorian 4 Lydian diminishedmin Maj Majdimmindim AUG
5Phrygian 3 Phrygian dominant Maj Majdimmindim AUGmin
6Lydian 2 Aeolian harmonic Majdimmindim AUGmin Maj
7Mixolydian 1Super-Locrian 7 dimmindim AUGmin Maj Maj
Chord tonalities by scale position & mode (harmonic minor, “circle of fifths” order)
Mode1234567
4Dorian 4 Lydian diminishedmin Maj Majdimmindim AUG
1Aeolian 7 Harmonic minor mindim AUGmin Maj Majdim
5Phrygian 3 Phrygian dominant Maj Majdimmindim AUGmin
2Locrian 6 Maqam Tarznauyn dim AUGmin Maj Majdimmin
6Lydian 2 Aeolian harmonic Majdimmindim AUGmin Maj
3Ionian 5 Ionian augmented AUGmin Maj Majdimmindim
7Mixolydian 1Super-Locrian 7 dimmindim AUGmin Maj Maj

The circle of fifths table here may clarify one reason harmonic minor requires so many more changes to rotate than Ionian does. Note how only one chord stays the same between any two successive rows of the harmonic minor “circle of fifths” table, while four stayed the same between any two successive rows of Ionian’s table (e.g., Ionian and Mixolydian both have I, ii, IV, and vi chords). For each mode rooted on C, we get the following chords:

Chords for C harmonic minor’s parallel modes
ModeMode1234567
1C Aeolian 7 C harmonic minor C minD dim E augF min G maj A majB dim
2C Locrian 6 C Maqam Tarznauyn C dim D augE min F maj G majA dimB min
3C Ionian 5 C Ionian augmented C augD min E maj F majG dimA minB dim
4C Dorian 4 C Lydian diminished C min D maj E majF dimG minA dim B aug
5C Phrygian 3 C Phrygian dominant C maj D majE dimF minG dim A augB min
6C Lydian 2 C Aeolian harmonic C majD dimE minF dim G augA min B maj
7C Mixolydian 1 C Super-Locrian 7 C dimD minE dim F augG min A maj B𝄫 maj

Meanwhile, the chords for C harmonic minor’s relative modes are as follows:

Chords for C harmonic minor’s relative modes
ModeMode1234567
1C Aeolian 7 C harmonic minor C minD dim E augF min G maj A majB dim
2D Locrian 6 D Maqam Tarznauyn D dim E augF min G maj A majB dimC min
3E Ionian 5 E Ionian augmented E augF min G maj A majB dimC minD dim
4F Dorian 4 F Lydian diminished F min G maj A majB dimC minD dim E aug
5G Phrygian 3 G Phrygian dominant G maj A majB dimC minD dim E augF min
6A Lydian 2 A Aeolian harmonic A majB dimC minD dim E augF min G maj
7B Mixolydian 1 B Super-Locrian 7 B dimC minD dim E augF min G maj A maj

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Melodic minor

Melodic minor is perhaps better related to the modes starting with Ionian. I haven’t drawn borders this time, because… well, it’s easier to show the table, then explain.

Melodic minor vs. modes from Ionian (rooted on C, linear order)
Mode 1 234567Intervals
Ionian CDEFGAB 11½111½
Ionian 3 Dorian 7Melodic minor CDEFGAB 1½1111½
Dorian CDEFGAB 1½111½1
Dorian 2 Phrygian 6Jazz minor inverse CDEFGAB ½1111½1
Phrygian CDEFGAB ½111½11
Phrygian 1 Lydian 5Lydian augmented CDEFGAB 1111½1½
Lydian CDEFGAB 111½11½
Lydian 7 Mixolydian 4Lydian dominant CDEFGAB 111½1½1
Mixolydian CDEFGAB 11½11½1
Mixolydian 6 Aeolian 3Aeolian dominant CDEFGAB 11½1½11
Aeolian CDEFGAB 1½11½11
Aeolian 5 Locrian 2Half-diminished CDEFGAB 1½1½111
Locrian CDEFGAB ½11½111
Locrian 4 Ionian 1Super-Locrian CDEFGAB ½1½1111

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The dual mnemonics for each mode of melodic minor in this table effectively show how we can derive melodic minor and each of its modes in two different ways from two different modes of Ionian:

  1. Melodic minor is Ionian with a flat third or Dorian with a sharp seventh.
  2. Jazz minor inverse is Dorian with a flat second or Phrygian with a sharp sixth.
  3. Lydian augmented is Phrygian with a flat first* or Lydian with a sharp fifth.
  4. Lydian dominant is Lydian with a flat seventh or Mixolydian with a sharp fourth.
  5. Aeolian dominant is Mixolydian with a flat sixth or Aeolian with a sharp third.
  6. Half-diminished is Aeolian with a flat fifth or Locrian with a sharp second.
  7. Super-Locrian is Locrian with a flat fourth or Ionian with a sharp first*.

Asterisks are necessary for the first scale degree when transposing every scale degree to C. When improvising on an existing scale, the same principles apply as with harmonic minor’s Mixolydian 1 – the mode’s root will already be transposed within the scale you’re playing, so you just have to bear that in mind when thinking of what notes to play above it. This may be clearer in the following table, which shows how the above modes relate to C melodic minor and C Ionian:

Melodic minor vs. modes from Ionian (rooted on scale, linear order)
Mode 1 234567Intervals
Ionian CDEFGAB 11½111½
Ionian 3Melodic minor CDEFGAB 1½1111½
Dorian DEFGABC 1½111½1
Dorian 2Jazz minor inverse DEFGABC ½1111½1
Phrygian EFGABCD ½111½11
Phrygian 1Lydian augmented EFGABCD 1111½1½
Lydian FGABCDE 111½11½
Lydian 7Lydian dominant FGABCDE 111½1½1
Mixolydian GABCDEF 11½11½1
Mixolydian 6Aeolian dominant GABCDEF 11½1½11
Aeolian ABCDEFG 1½11½11
Aeolian 5Half-diminished ABCDEFG 1½1½111
Locrian BCDEFGA ½11½111
Locrian 4Super-Locrian BCDEFGA ½1½1111

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In short, E Phrygian 1’s root note is already flat in its parent scale – you don’t have to flat it again!

“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 Mixolydian 1 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.

Melodic minor vs. modes from Ionian (rooted on C, in “circle of fifths” order)
Mode 1 234567Intervals
Lydian CDEFGAB 111½11½
Lydian 7Lydian dominant CDEFGAB 111½1½1
Ionian CDEFGAB 11½111½
Ionian 3Melodic minor CDEFGAB 1½1111½
Mixolydian CDEFGAB 11½11½1
Mixolydian 6Aeolian dominant CDEFGAB 11½1½11
Dorian CDEFGAB 1½111½1
Dorian 2Jazz minor inverse CDEFGAB ½1111½1
Aeolian CDEFGAB 1½11½11
Aeolian 5Half-diminished CDEFGAB 1½1½111
Phrygian CDEFGAB ½111½11
Phrygian 1Lydian augmented CDEFGAB 1111½1½
Locrian CDEFGAB ½11½111
Locrian 4Super-Locrian CDEFGAB ½1½1111

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Phrygian mode corresponds to Ionian’s third degree; that’s the note 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.

Melodic minor vs. modes from Ionian (rooted on scale, in “circle of fifths” order)
Mode 1 234567Intervals
Lydian FGABCDE 111½11½
Lydian 7Lydian dominant FGABCDE 111½1½1
Ionian CDEFGAB 11½111½
Ionian 3Melodic minor CDEFGAB 1½1111½
Mixolydian GABCDEF 11½11½1
Mixolydian 6Aeolian dominant GABCDEF 11½1½11
Dorian DEFGABC 1½111½1
Dorian 2Jazz minor inverse DEFGABC ½1111½1
Aeolian ABCDEFG 1½11½11
Aeolian 5Half-diminished ABCDEFG 1½1½111
Phrygian EFGABCD ½111½11
Phrygian 1Lydian augmented EFGABCD 1111½1½
Locrian BCDEFGA ½11½111
Locrian 4Super-Locrian BCDEFGA ½1½1111

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Of course, as the red names in the “rooted on C” table suggest, we can also get melodic minor by raising a different set of notes on a different series of modes. I’ve changed the first set of scale names accordingly, and since this interpretation of melodic minor raises pitches from its parent modes instead of lowering them, I’ve printed it first in this table. Note also Ionian’s different root key here (B major instead of C major).

Melodic minor vs. modes from Dorian (rooted on scale, linear order)
Mode 1 234567Intervals
Dorian 7Melodic minor CDEFGAB 1½1111½
Dorian CDEFGAB 1½111½1
Phrygian 6Jazz minor inverse DEFGABC ½1111½1
Phrygian DEFGABC ½111½11
Lydian 5Lydian augmented EFGABCD 1111½1½
Lydian EFGABCD 111½11½
Mixolydian 4Lydian dominant FGABCDE 111½1½1
Mixolydian FGABCDE 11½11½1
Aeolian 3Aeolian dominant GABCDEF 11½1½11
Aeolian GABCDEF 1½11½11
Locrian 2Half-diminished ABCDEFG 1½1½111
Locrian ABCDEFG ½11½111
Ionian 1Super-Locrian BCDEFGA ½1½1111
Ionian BCDEFGA 11½111½

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Here’s a comparison of these transformations in “circle of fifths” order, rooted to C:

Melodic minor vs. modes from Dorian (rooted on C, in “circle of fifths” order)
Mode 1 234567Intervals
Lydian 5Lydian augmented CDEFGAB 1111½1½
Lydian CDEFGAB 111½11½
Ionian 1Super-Locrian CDEFGAB ½1½1111
Ionian CDEFGAB 11½111½
Mixolydian 4Lydian dominant CDEFGAB 111½1½1
Mixolydian CDEFGAB 11½11½1
Dorian 7Melodic minor CDEFGAB 1½1111½
Dorian CDEFGAB 1½111½1
Aeolian 3Aeolian dominant CDEFGAB 11½1½11
Aeolian CDEFGAB 1½11½11
Phrygian 6Jazz minor inverse CDEFGAB ½1111½1
Phrygian CDEFGAB ½111½11
Locrian 2Half-diminished CDEFGAB 1½1½111
Locrian CDEFGAB ½11½111

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And in “circle of fifths” order rooted on their parent scales:

Melodic minor vs. modes from Dorian (rooted on scale, in “circle of fifths” order)
Mode 1 234567Intervals
Lydian 5Lydian augmented EFGABCD 1111½1½
Lydian EFGABCD 111½11½
Ionian 1Super-Locrian BCDEFGA ½1½1111
Ionian BCDEFGA 11½111½
Mixolydian 4Lydian dominant FGABCDE 111½1½1
Mixolydian FGABCDE 11½11½1
Dorian 7Melodic minor CDEFGAB 1½1111½
Dorian CDEFGAB 1½111½1
Aeolian 3Aeolian dominant GABCDEF 11½1½11
Aeolian GABCDEF 1½11½11
Phrygian 6Jazz minor inverse DEFGABC ½1111½1
Phrygian DEFGABC ½111½11
Locrian 2Half-diminished ABCDEFG 1½1½111
Locrian ABCDEFG ½11½111

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Analysis of chord tonality by scale position. (For reasons that will become clearer in the next section, I’m using Super-Locrian as the top scale of melodic minor’s “circle of fifths” order.)

Chord tonalities by scale position & mode (melodic minor, linear order)
Mode1234567
1Ionian 3 Dorian 7 Melodic minor minmin AUG Maj Majdimdim
2Dorian 2 Phrygian 6 Jazz minor inverse min AUG Maj Majdimdimmin
3Phrygian 1 Lydian 5 Lydian augmented AUG Maj Majdimdimminmin
4Lydian 7 Mixolydian 4 Lydian dominant Maj Majdimdimminmin AUG
5Mixolydian 6 Aeolian 3 Aeolian dominant Majdimdimmindim AUG Maj
6Aeolian 5 Locrian 2 Half-diminished dimdimminmin AUG Maj Maj
7Locrian 4 Ionian 1 Super-Locrian dimminmin AUG Maj Majmin
Chord tonalities by scale position & mode (melodic minor, “circle of fifths” order)
Mode1234567
7Locrian 4 Ionian 1 Super-Locrian dimminmin AUG Maj Majmin
4Lydian 7 Mixolydian 4 Lydian dominant Maj Majdimdimminmin AUG
1Ionian 3 Dorian 7 Melodic minor minmin AUG Maj Majdimdim
5Mixolydian 6 Aeolian 3 Aeolian dominant Majdimdimminmin AUG Maj
2Dorian 2 Phrygian 6 Jazz minor inverse min AUG Maj Majdimdimmin
6Aeolian 5 Locrian 2 Half-diminished dimdimminmin AUG Maj Maj
3Phrygian 1 Lydian 5 Lydian augmented AUG Maj Majdimdimminmin

Oddly, melodic minor’s sucessive modes in circle of fifths order have no chords in common.

Chords for C melodic minor’s parallel modes
Mode1234567
1C melodic minor C minD min E aug F maj G majA dimB dim
2C jazz minor inverse C min D aug E maj F majG dimA dimB min
3C Lydian augmented C aug D maj E majF dimG dimA minB min
4C Lydian dominant C maj D majE dimF dimG minA min B aug
5C Aeolian dominant C majD dimE dimF minG min A aug B maj
6C half-diminished C dimD dimE minF min G aug A maj B maj
7C Super-Locrian C dimD minE min F aug G maj A majB min
Chords for C melodic minor’s relative modes
Mode1234567
1C melodic minor C minD min E aug F maj G majA dimB dim
2D jazz minor inverse D min E aug F maj G majA dimB dimC min
3E Lydian augmented E aug F maj G majA dimB dimC minD min
4F Lydian dominant F maj G majA dimB dimC minD min E aug
5G Aeolian dominant G majA dimB dimC minD min E aug F maj
6A Half-diminished A dimB dimC minD min E aug F maj G maj
7B Super-Locrian B dimC minD min E aug F maj G majA min

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The Ionian scale’s stability

Let’s use a slightly more flexible root to compare melodic minor’s modes to Ionian’s in both directions – Ionian’s modes, melodic minor’s modes, and Dorian’s modes. Note especially how much stabler the Ionian scale’s root is.

Melodic minor vs. Ionian & Dorian (rooted on C±½, “circle of fifths” order)
Mode 1 234567Intervals
CDEFGAB ½111½11
Phrygian 1 Lydian 5Lydian augmented CDEFGAB 1111½1½
Lydian CDEFGAB 111½11½
C Locrian CDEFGAB ½11½111
Locrian 4 Ionian 1 CDEFGAB ½1½1111
C Ionian CDEFGAB 11½111½
Lydian CDEFGAB 111½11½
Lydian 7 Mixolydian 4Lydian dominant CDEFGAB 111½1½1
Mixolydian CDEFGAB 11½11½1
Ionian CDEFGAB 11½111½
Ionian 3 Dorian 7Melodic minor CDEFGAB 1½1111½
Dorian CDEFGAB 1½111½1
Mixolydian CDEFGAB 11½11½1
Mixolydian 6 Aeolian 3Aeolian dominant CDEFGAB 11½1½11
Aeolian CDEFGAB 1½11½11
Dorian CDEFGAB 1½111½1
Dorian 2 Phrygian 6Jazz minor inverse CDEFGAB ½1111½1
Phrygian CDEFGAB ½111½11
Aeolian CDEFGAB 1½11½11
Aeolian 5 Locrian 2Half-diminished CDEFGAB 1½1½111
Locrian CDEFGAB ½11½111
Phrygian CDEFGAB ½111½11
Phrygian 1 Lydian 5 CDEFGAB 1111½1½
C Lydian CDEFGAB 111½11½
Locrian CDEFGAB ½11½111
Locrian 4 Ionian 1Super-Locrian CDEFGAB ½1½1111
CDEFGAB 11½111½

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None of these representations are perfect, but together, they may help illuminate how these scales’ modes are related. As you can see, it’s quite messy – we have to move our root up or down a half-step at points to preserve relationships to the Ionian scale and the circle of fifths. Whether we read melodic minor as Ionian 3 or as Dorian 7, we must move its root three times in a row to keep the circle of fifths progression stable:

  1. Half-diminished to Lydian augmented: lower it a half-step
  2. Lydian augmented to Super-Locrian: raise it a half-step
  3. Super-Locrian to Lydian dominant: lower it a half-step

If the above table continued, its next three modes would be B Lydian, B Lydian dominant, and B Mixolydian – and a case could be made for rewriting its last six rows as B Phrygian, B Lydian augmented, B Lydian, B Locrian, B Super-Locrian, and B Ionian.

One further set of comparisons involves melodic minor and Mixolydian. Oddly, this lines up better in several respects: in particular, it lines up each scale’s symmetrical modes (Aeolian dominant and Dorian) and balances accidentals across the comparison (i.e., C Super-Locrian and C Lydian each have one sharp; C Ionian has no accidentals, while C Lydian dominant has a sharp and a flat).

Melodic minor vs. modes from Mixolydian (rooted on C±½, linear order)
Mode 1 234567Intervals
Ionian CDEFGAB 11½111½
Lydian 7 Mixolydian 4Lydian dominant CDEFGAB 111½1½1
Dorian CDEFGAB 1½111½1
Mixolydian 6 Aeolian 3Aeolian dominant CDEFGAB 11½1½11
Phrygian CDEFGAB ½111½11
Aeolian 5 Locrian 2Half-diminished CDEFGAB 1½1½111
Lydian CDEFGAB 111½11½
Locrian 4 Ionian 1 CDEFGAB ½1½1111
Mixolydian CDEFGAB 11½11½1
Ionian 3 Dorian 7Melodic minor CDEFGAB 1½1111½
Aeolian CDEFGAB 1½11½11
Dorian 2 Phrygian 6Jazz minor inverse CDEFGAB ½1111½1
Locrian CDEFGAB ½11½111
Phrygian 1 Lydian 5 CDEFGAB 1111½1½
Melodic minor vs. modes from Mixolydian (rooted on C±½, “circle of fifths” order)
Mode 1 234567Intervals
Lydian CDEFGAB 111½11½
Locrian 4 Ionian 1 CDEFGAB ½1½1111
Ionian CDEFGAB 11½111½
Lydian 7 Mixolydian 4Lydian dominant CDEFGAB 111½1½1
Mixolydian CDEFGAB 11½11½1
Ionian 3 Dorian 7Melodic minor CDEFGAB 1½1111½
Dorian CDEFGAB 1½111½1
Mixolydian 6 Aeolian 3Aeolian dominant CDEFGAB 11½1½11
Aeolian CDEFGAB 1½11½11
Dorian 2 Phrygian 6Jazz minor inverse CDEFGAB ½1111½1
Phrygian CDEFGAB ½111½11
Aeolian 5 Locrian 2Half-diminished CDEFGAB 1½1½111
Locrian CDEFGAB ½11½111
Phrygian 1 Lydian 5 CDEFGAB 1111½1½

Although this comparison swaps two interval pairs from the Ionian modes we’re comparing them to, it also only swaps two of their notes (which may include the root). There are two ways to read this set of transformations:

  1. Melodic minor swaps Mixolydian’s second interval with its third and its sixth with its seventh.
  2. Melodic minor swaps Mixolydian’s second and third intervals with its sixth and seventh.

The latter is probably the more helpful way to read it. Since both these interval pairs collectively add up to three semitones, only the notes within each interval pair move.

To preseve the pattern, harmonic minor’s modes must shift their roots in similar ways to melodic minor’s, except more unpredictably spaced (which feels inevitable, since its intervals are also less evenly spaced):

  1. Ionian augmented (#2) to Super-Locrian 7 (#3)
  2. Super-Locrian 7 (#3) to Lydian diminished (#4)
  3. Maqam Tarznauyn (#7) to Aeolian harmonic (#1)
Harmonic minor’s modes revisited, or, please throw some snacks down this rabbit hole
Mode 1 234567Intervals
Lydian 2Aeolian harmonic CDEFGAB ½1½11½
Lydian CDEFGAB 111½11½
Ionian 5Ionian augmented CDEFGAB 11½½1½
Ionian CDEFGAB 11½111½
Mixolydian 1Super-Locrian 7 CDEFGAB ½1½11½
Mixolydian CDEFGAB 11½11½1
Dorian 4Lydian diminished CDEFGAB 1½½1½1
Dorian CDEFGAB 1½111½1
Aeolian 7Harmonic minor CDEFGAB 1½11½½
Aeolian CDEFGAB 1½11½11
Phrygian 3Phrygian dominant CDEFGAB ½½1½11
Phrygian CDEFGAB ½111½11
Locrian 6Maqam Tarznauyn CDEFGAB ½11½½1
Locrian CDEFGAB ½11½111

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The Ionian scale’s descent is comparatively stable: each transformation lowers only one note of the scale, and it lowers that note only by a semitone. As far as I can ascertain, it is the only heptatonic scale for which this is true. This occurs in part because it comes as close as any heptatonic scale in twelve-tone equal temperament can come to having its notes evenly spaced, without being precisely even. Two whole tones, a semitone, three whole tones, and a semitone. The fact that these notes, in turn, traverse the circle of fifths from F to B is the other part of the puzzle.

Descending through Ionian’s modes in circle of fifths order lowers one note every transformation by a semitone. Few other scale transformations are so simple. Harmonic and melodic minor’s transformations each lower two notes by a semitone and raise a third note by a semitone:

Harmonic minor & melodic minor’s “circle of fifths” progressions
Mode 1 234567 Shift from Previous Note
C D E F G A B −½ 0 0 0 0 0 0
C D E F G A B 0 0 0 −½ 0 0 0
C D E F G A B 0 0 0 0 0 0 −½
C D E F G A B 0 0 −½ 0 0 0 0
C D E F G A B 0 0 0 0 0 −½ 0
C D E F G A B 0 −½ 0 0 0 0 0
C D E F G A B 0 0 0 0 −½ 0 0
C D E F G A B −½ 0 0 0 0 0 0
C D E F G A B −½ 0 0 0 −½ 0
C D E F G A B 0 −½ 0 −½ 0 0
C D E F G A B 0 0 0 −½ 0 −½
C D E F G A B −½ 0 −½ 0 0 0
C D E F G A B 0 0 0 −½ 0 −½
C D E F G A B 0 −½ 0 0 0 −½
C D E F G A B 0 0 −½ 0 −½ 0
C D E F G A B −½ 0 0 0 −½ 0
C D E F G A B 0 −½ 0 −½ 0 0
C D E F G A B𝄫 0 0 0 −½ 0 −½
C D E𝄫 F G A B𝄫 −½ 0 −½ 0 0 0
Phrygian 1 Lydian 5 C Lydian augmented C D E F G A B −½ −½ 0 0 0 0
Locrian 4 Ionian 1 C D E F G A B 0 0 −½ −½ 0 0
Lydian 7 Mixolydian 4 C Lydian dominant C D E F G A B −½ 0 0 0 0 −½
Ionian 3 Dorian 7 C melodic minor C D E F G A B 0 0 −½ −½ 0 0
Mixolydian 6 Aeolian 3 C Aeolian dominant C D E F G A B 0 0 0 0 −½ −½
Dorian 2 Phrygian 6 C jazz minor inverse C D E F G A B 0 −½ −½ 0 0 0
Aeolian 5 Locrian 2 C half-diminished C D E F G A B 0 0 0 −½ −½ 0
Phrygian 1 Lydian 5 C D E F G A B −½ −½ 0 0 0 0
Locrian 4 Ionian 1 C Super-Locrian C D E F G A B 0 0 −½ −½ 0 0

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Note a few additional patterns here:

(At some point, I plan to make equivalents of §3’s charts for at least 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.)

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Mathematical proof of even spacing

Ionian has the most even interval distribution any seven-note scale can have in 12-TET, and I’ll prove it.

  1. In n-tone equal temperament, for an s-note scale, divide n by s to determine the mean interval size, expressed in units of 1/n octave (i.e., the smallest possible interval; in 12-TET, a semitone).

    12/7 is 1.714285714….
  2. If there is a remainder, discard it to get the scale’s lower interval size, and add one to the result to get its larger interval size. If there is no remainder, skip the rest of this proof; you’re done (but, as Oliver Messiaen would’ve already told you, you won’t have any other modes).

    That’s 1. So all intervals in the scale should be at least a semitone (and, in fact, must be).
  3. Take the modulus of the temperament size and the scale size to determine how many intervals will get an extra 1/n octave added to them.

    12 modulo 7 is 5. Thus, we have five leftover half-steps to add to five of the intervals.
  4. Subtract the number of leftovers from the number of notes in the scale to determine how many intervals don’t get an extra 1/n octave.

    7 - 5 is 2. Thus, five intervals have added semitones, two don’t. Five whole steps, two half steps.
  5. To determine the most even interval distribution, first divide the number of occurrences of the more frequent interval by the number of occurrences of the less frequent one.

    • If the divisor is 0, you need to work on your reading comprehension: step 2 already told you to stop.
    • If the less frequent interval occurs multiple times and the quotient is an integer, the scale will repeat the same pattern multiple times per octave and is therefore a mode of limited transposition.
    • If the less frequent interval occurs nearly as often as the more frequent one, you may need to repeat some of these steps, replacing n with the count of the more frequent interval and s with the count of the less frequent one.
    5 / 2 = 2.5. There should be a median of 2.5 occurrences of the more frequent interval in a row.
  6. The interval distribution should take roughly this form: small group of more frequent interval, less frequent interval, large group of more frequent interval, less frequent interval. If there are more than two groups of the less frequent interval, determining the optimal distribution may require repeating some of these steps, replacing with n the more frequent interval’s count and s with the less frequent one’s count.

    Since we can’t have a whole step exactly 2.5 times in a row, we’ll have to have two in one group and three in another. That gives us two whole tones, a semitone, three whole tones, and a semitone. That’s Ionian. I literally just described the Ionian scale. Median number of whole tones in a row: 2.5. Therefore, its semitones are as evenly spread out as they possibly can be between its whole tones.

Surely that also applies to its complement, right? Let’s look at the pentatonic scale.

  1. In n-tone equal temperament, for an s-note scale, divide n by s to determine the mean interval size, expressed in units of 1/n octave (i.e., the smallest possible interval; in 12-TET, a semitone).

    12/5 is 2.4.
  2. If there is a remainder, discard it to get the scale’s lower interval size, and add one to the result to get its larger interval size. If there is no remainder, skip the rest of this proof; you’re done (but, as Oliver Messiaen would’ve already told you, you won’t have any other modes).

    That’s 2. So all intervals in the scale should be at least a whole tone.
  3. Take the modulus of the temperament size and the scale size to determine how many intervals will get an extra 1/n octave added to them.

    12 modulo 5 is 2. So we have two leftover half-steps to add to two of the intervals.
  4. Subtract the number of leftovers from the number of notes in the scale to determine how many intervals don’t get an extra 1/n octave.

    5 - 2 is 3. Thus, two intervals have extra semitones, three won’t. Three whole steps, two minor thirds.
  5. To determine the most even interval distribution, first divide the number of occurrences of the more frequent interval by the number of occurrences of the less frequent one.

    • If the divisor is 0, you need to work on your reading comprehension: step 2 already told you to stop.
    • If the less frequent interval occurs multiple times and the quotient is an integer, the scale will repeat the same pattern multiple times per octave and is therefore a mode of limited transposition.
    • If the less frequent interval occurs nearly as often as the more frequent one, you may need to repeat some of these steps, replacing n with the count of the more frequent interval and s with the count of the less frequent one.
    3 / 2 = 1.5. There should be a median of 1.5 occurrences of the more frequent interval in a row.
  6. The interval distribution should take roughly this form: small group of more frequent interval, less frequent interval, large group of more frequent interval, less frequent interval. If there are more than two groups of the less frequent interval, determining the optimal distribution may require repeating some of these steps, replacing with n the more frequent interval’s count and s with the less frequent one’s count.

    Since we can’t have exactly 1.5 whole steps in a row, we’ll have to have two in one group and one by itself. That gives us two whole tones, a minor third, a whole tone, and a minor third. Which, again, is the pentatonic scale.

Of course, we already knew this. If the Ionian scale has the most even interval distribution a heptatonic scale can have in 12-TET, its complement must also have the most even interval distribution a pentatonic scale can have in 12-TET, by definition. Nonetheless, it’s nice to prove it mathematically.

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Other single-note transformations of Ionian

So far, we’ve almost exclusively explored single-note transformations of Ionian and its modes:

Do other single-note transformations exist? Yes, but fewer than you might expect:

Thus, eight single-note transformations create heptatonic scales that aren’t other modes of Ionian. (I’ve printed the five we haven’t yet explored in bold, blue text.)

Threshold of Transformation
Mode 1 234567 Intervals
Ionian 1CDEFGAB½1½1111
CDEFGAB½½111½
CDEFGAB½½111½
Ionian 3CDEFGAB1½1111½
CDEFGAB11½½1½
Ionian 5CDEFGAB11½½1½
CDEFGAB11½1½½
CDEFGAB11½1½½

Expanding those gives us:

4. Expand, expand, expand. Clear forest, make land, fresh blood on hands
Mode 1 234567 Intervals
Ionian 2 CDEFGAB ½½111½
Dorian 1 CDEFGAB ½111½½
Phrygian 7 CDEFGAB𝄫 ½111½½
Lydian 6 CDEFGAB 111½½½
Mixolydian 5 CDEFGAB 11½½½1
Aeolian 4 CDEFGAB 1½½½11
Locrian 3 CDE𝄫FGAB ½½½111
Ionian 2 CDEFGAB ½½111½
Dorian 1 CDEFGAB ½½111½
Phrygian 7 CDEFGAB ½111½½
Lydian 6 CDEFGAB 111½½½
Mixolydian 5 CDEFGAB 11½½½1
Aeolian 4 CDEFGAB 1½½½11
Locrian 3 CDEFGAB ½½½111
Ionian 5 CDEFGAB 11½½1½
Dorian 4 CDEFGAB 1½½1½1
Phrygian 3 CDE𝄫FGAB ½½1½11
Lydian 2 CDEFGAB ½1½11½
Mixolydian 1 CDEFGAB 1½11½½
Aeolian 7 CDEFGAB𝄫 1½11½½
Locrian 6 CDEFGA𝄫B ½11½½1
Ionian 6 CDEFGAB 11½1½½
Dorian 5 CDEFGAB 1½1½½1
Phrygian 4 CDEFGAB ½1½½11
Lydian 3 CDEFGAB 1½½11½
Mixolydian 2 CDEFGAB ½½11½1
Aeolian 1 CDEFGAB ½½11½1
Locrian 7 CDEFGAB ½11½1½
Ionian 6 CDEFGAB 11½1½½
Dorian 5 CDEFGAB 1½1½½1
Phrygian 4CDEFGAB ½1½½11
Lydian 3 CDEFGAB 1½½11½
Mixolydian 2 CDEFGAB ½½11½1
Aeolian 1CDEFGAB ½11½1½
Locrian 7CDEFGAB𝄫 ½11½1½

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Or, in circle of fifths order:

5. Why just shells? Why limit yourself? She sells seashells; sell oil as well
Mode 1 234567 Intervals
Lydian 6 CDEFGAB 111½½½
Ionian 2 CDEFGAB ½½111½
Mixolydian 5 CDEFGAB 11½½½1
Dorian 1 CDEFGAB ½111½½
Aeolian 4 CDEFGAB 1½½½11
Phrygian 7 CDEFGAB𝄫 ½111½½
Locrian 3 CDE𝄫FGAB ½½½111
Lydian 6 CDEFGAB 111½½½
Ionian 2 CDEFGAB ½½111½
Mixolydian 5 CDEFGAB 11½½½1
Dorian 1 CDEFGAB ½½111½
Aeolian 4 CDEFGAB 1½½½11
Phrygian 7 CDEFGAB ½111½½
Locrian 3 CDEFGAB ½½½111
Lydian 2 CDEFGAB ½1½11½
Ionian 5 CDEFGAB 11½½1½
Mixolydian 1 CDEFGAB 1½11½½
Dorian 4 CDEFGAB 1½½1½1
Aeolian 7 CDEFGAB𝄫 1½11½½
Phrygian 3 CDE𝄫FGAB ½½1½11
Locrian 6 CDEFGA𝄫B ½11½½1
Lydian 3 CDEFGAB 1½½11½
Ionian 6 CDEFGAB 11½1½½
Mixolydian 2 CDEFGAB ½½11½1
Dorian 5 CDEFGAB 1½1½½1
Aeolian 1 CDEFGAB ½½11½1
Phrygian 4 CDEFGAB ½1½½11
Locrian 7 CDEFGAB ½11½1½
Lydian 3 CDEFGAB 1½½11½
Ionian 6 CDEFGAB 11½1½½
Mixolydian 2 CDEFGAB ½½11½1
Dorian 5 CDEFGAB 1½1½½1
Aeolian 1 CDEFGAB ½11½1½
Phrygian 4 CDEFGAB ½1½½11
Locrian 7 CDEFGAB𝄫 ½11½1½

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Note that Dorian 1 is the mathematical inverse of Dorian 1, as is Dorian 5 of Dorian 4. Dorian 5 is likewise the mathematical inverse of Dorian 4, harmonic minor’s fourth mode. To clarify:

6. Guns, sell stocks, sell diamonds, sell rocks, sell water to a fish, sell the time to a clock
Mode 1 234567 Intervals
Lydian 6 CDEFGAB 111½½½
Locrian 3 CDEFGAB ½½½111
Ionian 2 CDEFGAB ½½111½
Phrygian 7 CDEFGAB ½111½½
Mixolydian 5 CDEFGAB 11½½½1
Aeolian 4 CDEFGAB 1½½½11
Dorian 1 CDEFGAB ½111½½
Dorian 1 CDEFGAB ½½111½
Aeolian 4 CDEFGAB 1½½½11
Mixolydian 5 CDEFGAB 11½½½1
Phrygian 7 CDEFGAB𝄫 ½111½½
Ionian 2 CDEFGAB ½½111½
Locrian 3 CDE𝄫FGAB ½½½111
Lydian 6 CDEFGAB 111½½½
Lydian 2 CDEFGAB ½1½11½
Locrian 7 CDEFGAB ½11½1½
Ionian 5 CDEFGAB 11½½1½
Phrygian 4 CDEFGAB ½1½½11
Mixolydian 1 CDEFGAB 1½11½½
Aeolian 1 CDEFGAB ½½11½1
Dorian 4 CDEFGAB 1½½1½1
Dorian 5 CDEFGAB 1½1½½1
Aeolian 7 CDEFGAB𝄫 1½11½½
Mixolydian 2 CDEFGAB ½½11½1
Phrygian 3 CDE𝄫FGAB ½½1½11
Ionian 6 CDEFGAB 11½1½½
Locrian 6 CDEFGA𝄫B ½11½½1
Lydian 3 CDEFGAB 1½½11½
Lydian 3 CDEFGAB 1½½11½
Locrian 6 CDEFGAB ½11½½1
Ionian 6 CDEFGAB 11½1½½
Phrygian 3 CDEFGAB ½½1½11
Mixolydian 2 CDEFGAB ½½11½1
Aeolian 7 CDEFGAB 1½11½½
Dorian 5 CDEFGAB 1½1½½1
Dorian 4 CDEFGAB 1½½1½1
Aeolian 1 CDEFGAB ½11½1½
Mixolydian 1 CDEFGAB ½1½11½
Phrygian 4 CDEFGAB ½1½½11
Ionian 5 CDEFGAB 11½½1½
Locrian 7 CDEFGAB𝄫 ½11½1½
Lydian 2 CDEFGAB ½1½11½

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Scale transformations and symmetry

The above data aren’t very easy to parse. Clearly, some of these transformations produce more symmetrical and (for lack of a better term) stabler scales than others. The question is, why? I’m still piecing together the answer, but a piece of the puzzle has to do with note distributions.

The Ionian scale is internally symmetrical: Dorian mode has the same interval order forwards and backwards. Ionian is tied among heptatonic scales in 12-TET for the smallest number of semitone intervals (only two), and it has fewer tritones than any other heptatonic scale in 12-TET. Also – and this is probably the most important fact here – Ionian has the most uniform note distribution among heptatonic scales in 12-TET, but – and this part is equally important – it isn’t completely uniform. As we’ll see later when we discuss modes of limited transposition, completely uniform scales don’t even have modes.

Transforming a single note can thus completely destroy the scale symmetry. As it happens, of all the single-note transformations that produce symmetrical scales, both sets produce versions of the melodic minor scale, which we’ve already examined. Not coincidentally, this also is the closest heptatonic scale within 12-TET to Ionian’s stability respective to the circle of fifths: while one must transform each note three times for its equivalent of Ionian’s descent through its modes, at least it’s the same note three times in a row.

Another important note is that each transformation of Ionian that raises a note has an equal and opposite transformation that lowers a note and produces the first transformation’s reflection. For twelve of the fourteen transformations that produce modes of melodic minor, that reflection is another mode of melodic minor; for the remaining two, that reflection is itself, but is applied to a different mode of Ionian:

While writing Ionian’s modes this way is pretentious, we can say the same of its own internal transformations:

All single-note transformations of the Ionian scale that don’t create hexatonic scales or modes of melodic minor or Ionian result in scales with enantiomorphs (Attic Greek: ἐναντίος, enantíos, opposite, + μορφή, morphḗ, form), which all appear in different sets of single-note transformations of the Ionian scale. Only scales that cannot be transformed into their inversions by rotation have enantiomorphs. Thus:

In the following table, I’ve taken the liberty of rotating Lydian 5 to the end of the first set of scale comparisons, and Locrian 4 to the start of the second. I had several reasons for this:

  1. This places the symmetrical mode, Aeolian 3 / Mixolydian 6, in the center of the comparisons.
  2. The table’s other comparisons are between two discrete sets of scale transformations, but here, we compare a set of scale transformations to itself. This places our comparison in sync with itself.
  3. These are the table’s only comparisons of single-note transformations that can be derived from two discrete parent modes. Oddly, shifting the scales like this actually approximates our usual circle of fifths order:

    • Both sets now open with Ionian 1 / Locrian 4, between which is Lydian.
    • Both sets now close with Lydian 5 / Phrygian 1, between which is Locrian.
  4. Remember in the pentatonic scale analysis how I said symmetrical modes should be circle-order comparisons’ central rows? Now it is.
  5. This results in a few additional quirks:

    1. The 7×7 interval inset has 180° rotational symmetry.
    2. Both halves also have identical interval distributions, with a pattern that spans across them.

Other one-note transformations don’t produce symmetrical scales; therefore, they have reflections.

I specifically used Dorian mode for these examples because it’s symmetrical in the base scale, but we can still make similar comparisons for the other six modes, since they each have reflections within the Ionian scale:

Thus, the reflection of a transformation of a non-palindromic mode applies to the parent mode’s reflection:

We can observe all this in the table below.

7. Press on the gas, take your foot off the brakes; then run to be the president of the United States
Mode 1 234567 Intervals
Ionian 1 Locrian 4 CDEFGAB ½1½1111
Mixolydian 4 Lydian 7 CDEFGAB 111½1½1
Dorian 7 Ionian 3 CDEFGAB 1½1111½
Aeolian 3 Mixolydian 6 CDEFGAB 11½1½11
Phrygian 6 Dorian 2 CDEFGAB ½1111½1
Locrian 2 Aeolian 5 CDEFGAB 1½1½111
Lydian 5 Phrygian 1 CDEFGAB 1111½1½
Locrian 4 Ionian 1 CDEFGAB ½1½1111
Lydian 7 Mixolydian 4 CDEFGAB 111½1½1
Ionian 3 Dorian 7 CDEFGAB 1½1111½
Mixolydian 6 Aeolian 3 CDEFGAB 11½1½11
Dorian 2 Phrygian 6 CDEFGAB ½1111½1
Aeolian 5 Locrian 2 CDEFGAB 1½1½111
Phrygian 1 Lydian 5 CDEFGAB 1111½1½
Lydian 2 CDEFGAB ½1½11½
Ionian 5 CDEFGAB 11½½1½
Mixolydian 1 CDEFGAB 1½11½½
Dorian 4 CDEFGAB 1½½1½1
Aeolian 7 CDEFGAB𝄫 1½11½½
Phrygian 3 CDE𝄫FGAB ½½1½11
Locrian 6 CDEFGA𝄫B ½11½½1
Lydian 3 CDEFGAB 1½½11½
Ionian 6 CDEFGAB 11½1½½
Mixolydian 2 CDEFGAB ½½11½1
Dorian 5 CDEFGAB 1½1½½1
Aeolian 1 CDEFGAB ½½11½1
Phrygian 4 CDEFGAB ½1½½11
Locrian 7 CDEFGAB ½11½1½
Lydian 3 CDEFGAB 1½½11½
Ionian 6 CDEFGAB 11½1½½
Mixolydian 2 CDEFGAB ½½11½1
Dorian 5 CDEFGAB 1½1½½1
Aeolian 1 CDEFGAB ½11½1½
Phrygian 4 CDEFGAB ½1½½11
Locrian 7 CDEFGAB𝄫 ½11½1½
Lydian 2 CDEFGAB ½1½11½
Ionian 5 CDEFGAB 11½½1½
Mixolydian 1 CDEFGAB ½1½11½
Dorian 4 CDEFGAB 1½½1½1
Aeolian 7 CDEFGAB 1½11½½
Phrygian 3 CDEFGAB ½½1½11
Locrian 6 CDEFGAB ½11½½1
Lydian 6 CDEFGAB 111½½½
Ionian 2 CDEFGAB ½½111½
Mixolydian 5 CDEFGAB 11½½½1
Dorian 1 CDEFGAB ½111½½
Aeolian 4 CDEFGAB 1½½½11
Phrygian 7 CDEFGAB𝄫 ½111½½
Locrian 3 CDE𝄫FGAB ½½½111
Lydian 6 CDEFGAB 111½½½
Ionian 2 CDEFGAB ½½111½
Mixolydian 5 CDEFGAB 11½½½1
Dorian 1 CDEFGAB ½½111½
Aeolian 4 CDEFGAB 1½½½11
Phrygian 7 CDEFGAB ½111½½
Locrian 3 CDEFGAB ½½½111

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One obvious symmetrical scale can’t be produced with a single-note transformation to the Ionian scale (though we can produce it by swapping two intervals; it’s also equivalent to the whole-tone scale with a note added). Its interval distribution is quite far from uniform, and it’s also all but impossible to relate to any sort of circle of fifths order. We’ve already studied it at length, but I haven’t shown it with every mode rooted on C, so here it is.

8. Big smile, mate; big wave, that’s great; now the truth is overrated; tell lies out the gate
Mode 1 234567 Intervals
Locrian major CDEFGAB 11½½111
Super-Locrian 2 CDEFGAB 1½½1111
Leading whole-tone inverse CDE𝄫FGAB ½½11111
Melodic Phrygian CDEFGAB ½11111½
Leading whole-tone CDEFGAB 11111½½
Lydian dominant augmented CDEFGAB 1111½½1
Lydian dominant 6 CDEFGAB 111½½11

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I’m also bizarrely partial to the Major Phyrgian scale, which has its own fearful symmetry, to coin a phrase. It’s more closely related to the Ancient Greek chromatic genus, which I cover below in the section on Ancient Greek harmony, than it is to the Ionian scale. We’ll therefore revisit it later.

9. Polarise the people; controversy is the game; it don’t matter if they hate you if they all say your name
Mode 1 234567 Intervals
Hungarian Romani minor inverse CDEFGAB ½½½½1
Ionian augmented 2 CDEFGAB ½½½1½
Kanakāngi 5 CDE𝄫FGAB𝄫 ½½½1½
Major Phrygian CDEFGAB ½½1½½
Rasikapriyā CDEFGAB ½1½½½
Ultra-Phrygian CDEFGAB𝄫 ½1½½½
Hungarian Romani minor CDEFGAB 1½½½½

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“What immortal hand or eye
Could frame thy fearful symmetry?”

Rotational symmetry: Modes of limited transposition

Definition

The scales we’ve examined in detail thus far have as many modes as they have notes. A seven-note scale multiplied by twelve notes in the chromatic scale gives us eighty-four possible permutations of modes and root notes; pentatonic scales likewise have sixty possible permutations. However, this is not true of every scale (although, as we will eventually see, it is true of every pentatonic and heptatonic scale in 12-TET specifically).

French composer Oliver Messiaen (1908-1992) deemed a scale scales with fewer modes than notes a mode of limited transposition (acronym: MoLT). Such scales can be “simplified” into repetitions smaller than an octave, so they have rotational symmetry: rotating them by their internal repetition produces the same mode. Thus:

  1. A mode of limited transposition can be transformed into itself by a parallel rotation no larger than a tritone and no smaller than a semitone. (This rotation must not change its interval order in any way.)
  2. Modes of limited transposition reuse the same sets of notes across multiple transpositions.
  3. Modes of limited transposition have fewer modes than notes.

As an example, consider the C augmented chord, C-E-G, a stack of four-semitone intervals.

  1. A four-semitone parallel rotation transforms it into itself.
  2. When transposed to E, it becomes E-G-C, the same set of notes it has on C.
  3. Since its intervals are all the same, it has only one mode.
    • Further explanation: Counting E-G-C as the C augmented chord’s second relative mode would be double-counting modes since, by definition, E-G-C must be its first relative mode on E. Likewise, G-C-E must be its first relative mode on G. Therefore, it has only a single mode.

For all three reasons, it is therefore a mode of limited transposition. By contrast, let’s consider Ionian.

  1. Ionian cannot transform into itself by sub-octave rotations.
  2. Ionian transposes into unique pitch sets for each note of the chromatic scale. (I’ve used circle of fifths order so that the note composition only changes one note per line, and I’ve highlighted the root in green.)
    Ionian’s pitch sets across the chromatic scale
    RootCCDEEFFGGABB
    D C D E F G A B
    A C D E F G A B
    E C D E F G A B
    B C D E F G A B
    F C D E F G A B
    C C D E F G A B
    G C D E F G A B
    D C D E F G A B
    A C D E F G A B
    E C D E F G A B
    B C D E F G A B
    F C D E F G A B
  3. Ionian, of course, has seven notes and seven modes.

Messiaen identified seven possible patterns (beyond the chromatic scale in its entirety); eight “truncations” also remove notes in ways that conform to the patterns, and the chromatic scale itself meets Messiaen’s definition of a mode of limited transposition, making for a total of sixteen.

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Mode 1: The whole-tone scale

Since the whole-tone scale repeats a single interval six times (W-W-W-W-W-W), it has only one mode (i.e., itself) that may only be made from two sets of notes:

Whole-tone note sets
# 1 2 3 4 5 6
1 C D E F G A
2 C D F G A B

Multiplying one mode by six repetitions by two note sets gives us a total of twelve transpositions:

Transpositions of the whole-tone scale
TP123456P123456
1 1 C D E F G A 2 C D F G A B
2 1 D E F G A C 2 D F G A B C
3 1 E F G A C D 2 F G A B C D
4 1 F G A C D E 2 G A B C D F
5 1 G A C D E F 2 A B C D F G
6 1 A C D E F G 2 B C D F G A

The whole-tone scale is the first mode of limited transposition, and the only one that has no other modes. (A few truncations of the modes of limited transposition also have no other modes, as we shall see below.)

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Mode 2: The octatonic scale

The octatonic scale (H-W-H-W-H-W-H-W), Messiaen’s second mode of limited transposition, repeats a two-note pattern every three half-steps. Since its pattern has only two notes, it has only two modes:

The second mode’s modes
# 1 2 3 4 5 6 7 8
1 ½ 1 ½ 1 ½ 1 ½ 1
2 1 ½ 1 ½ 1 ½ 1 ½

And since its pattern spans three half-steps, there are only three possible note sets:

The second mode’s notes
# 1 2 3 4 5 6 7 8
1 C C D E F G A A
2 C D E F G G A B
3 D D F F G A B C

Does it make sense why we have to stop counting here? C octatonic’s third mode would start on D, but it would contain exactly the same notes as D octatonic’s first mode, in exactly the same order! We can’t count them both, so the octatonic scale has six total permutations of modes and note sets.

But if we have to stop counting modes at the end of each cluster, how do we calculate the number of discrete transpositions of the scale and its modes? As far as I can work out, the calculation is quite simple:

  1. (8 / 4 = 2) Divide the scale’s note count by its repetitions per octave to count its modes.
  2. (2 × 12 = 24) Multiply by the number of transpositions (which is always 12 in 12-TET).

As we see below, the octatonic scale indeed has twenty-four total transpositions:

Transposing the second mode of limited transposition
M T P 1 2 3 4 5 6 7 8 M T P 1 2 3 4 5 6 7 8
1 1 C C D E F G A A 1 1 C D E F G A A C
2 C D E F G G A B 2 D E F G G A B C
3 D D F F G A B C 3 D F F G A B C D
2 1 D E F G A A C C 2 1 E F G A A C C D
2 E F G G A B C D 2 F G G A B C D E
3 F F G A B C D D 3 F G A B C D D F
3 1 F G A A C C D E 3 1 G A A C C D E F
2 G G A B C D E F 2 G A B C D E F G
3 G A B C D D F F 3 A B C D D F F G
4 1 A A C C D E F G 4 1 A C C D E F G A
2 A B C D E F G G 2 B C D E F G G A
3 B C D D F F G A 3 C D D F F G A B

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Mode 3: The triple chromatic scale

The third mode of limited transposition repeats a three-interval pattern across four half-steps; thus, it has three unique modes that may be constructed from four possible sets of notes, for twelve total permutations of notes per four-half-step cluster:

The third mode of limited transposition
M S 1 23456789Intervals
1 1 CDD EFG GAB 1½½ 1½½ 1½½
2 CDE FGG ABC
3 DEF FGA ACC
4 DFF GAA BCD
2 1 DDE FGG ABC ½½1 ½½1 ½½1
2 DEF GGA BCC
3 EFF GAA CCD
4 FFG AAB CDD
3 1 DEF GGA BCD ½1½ ½1½ ½1½
2 EFG GAB CCD
3 FFG AAC CDE
4 FGA ABC DDF

I leave filling in the rest of the table as an exercise for the reader. A quick hint: You should wind up with three sets of twelve scales that each walk up the chromatic scale by half-steps, for a total of thirty-six.

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Mode 4: The double chromatic scale

Modes 4 through 7 all repeat patterns of various lengths twice an octave. Since the fourth mode of limited transposition has eight total notes, it has forty-eight possible transpositions.

The fourth mode of limited transposition
M S 1 2345678Intervals
1 1 CCDF FGGB ½½½½½½
2 CDDF GGAC
3 DDEG GAAC
4 DEFG AABD
5 EFFA ABCD
6 FFGA BCCE
2 1 CCEFFGAB ½½½ ½½½
2 CDFFGGBC
3 DDFGGACC
4 DEGGAACD
5 EFGAABDD
6 FFAABCDE
3 1 CDEFFAAB ½½½ ½½½
2 CEFFGABC
3 DFFGGBCC
4 DFGGACCD
5 EGGAACDD
6 FGAABDDE
4 1 CCDDFGGA ½½½ ½½½
2 CDDEGGAA
3 DDEFGAAB
4 DEFFAABC
5 EFFGABCC
6 FFGGBCCD

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Mode 5: The tritone chromatic scale

The fifth mode has six notes and repeats twice an octave; therefore, it has thirty-six possible transpositions.

The fifth mode of limited transposition
M S 1 23456Intervals
1 1 CCFFGB ½2½ ½2½
2 CDFGGC
3 DDGGAC
4 DEGAAD
5 EFAABD
6 FFABCE
2 1 CEFFAB 2½½2½½
2 CFFGBC
3 DFGGCC
4 DGGACD
5 EGAADD
6 FAABDE
3 1CCDFGG ½½2½½2
2 CDDGGA
3 DDEGAA
4 DEFAAB
5 EFFABC
6 FFGBCC

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Mode 6: The whole-tone chromatic scale

Messiaen’s sixth mode of limited transposition repeats a four-note, six-half-step pattern. Four modes, six note combinations per mode, twenty-four note permutations per cluster, two clusters per octave, forty-eight permutations of roots and modes across the entire chromatic scale.

The sixth mode of limited transposition
M S 1 2345678Intervals
1 1 CDEFFGAB 11½½11½½
2 CDFFGABC
3 DEFGGACC
4 DFGGABCD
5 EFGAACDD
6 FGAABCDE
2 1 CDDEFGAA 1½½11½½1
2 CDEFGAAB
3 DEFFGABC
4 DFFGABCC
5 EFGGACCD
6 FGGABCDD
3 1 CCDEFGGA ½½11½½11
2 CDDFGGAB
3 DDEFGAAC
4 DEFGAABC
5 EFFGABCD
6 FFGABCCD
4 1 CCDFFGAB ½11½½11½
2 CDEFGGAC
3 DDFGGABC
4 DEFGAACD
5 EFGAABCD
6 FFGABCDE

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Mode 7: Duplex genus secundum inverse

This decatonic scale repeats the same pattern twice an octave; thus, it has sixty possible transpositions.

The seventh mode of limited transposition
M S 1 2345678910Intervals
1 1 CCDDF FGGAB ½½½1½ ½½½1½
2 CDDEF GGAAC
3 DDEFG GAABC
4 DEFFG AABCD
5 EFFGA ABCCD
6 FFGGA BCCDE
2 1 CCDEF FGGAB ½½1½½½½1½½
2 CDDFF GGABC
3 DDEFG GAACC
4 DEFGG AABCD
5 EFFGA ABCDD
6 FFGAA BCCDE
3 1 CCDEF FGAAB ½1½½½½1½½½
2 CDEFF GGABC
3 DDFFG GABCC
4 DEFGG AACCD
5 EFGGA ABCDD
6 FFGAA BCDDE
4 1 CDDEF FGAAB 1½½½½1½½½½
2 CDEFF GAABC
3 DEFFG GABCC
4 DFFGG ABCCD
5 EFGGA ACCDD
6 FGGAA BCDDE
5 1 CCDDE FGGAA ½½½½1½½½½1
2 CDDEF GGAAB
3 DDEFF GAABC
4 DEFFG AABCC
5 EFFGG ABCCD
6 FFGGA BCCDD

Truncations & implications

To illustrate MoLT truncation, let’s strip every other note of the first two MoLTs. The results may look familiar:

  1. Whole-tone: An augmented chord.

    Transposing the augmented chord
    P/T 1 2 3 P/T 1 2 3 P/T 1 2 3 P/T 1 2 3
    A/1 C E G A/2 C F A A/3 D F A A/4 D G B
    B/1 E G C B/2 F A C B/3 F A D B/4 G B D
    C/1 G C E C/2 A C F C/3 A D F C/4 B D G
  2. Octatonic: A diminished seventh chord.

    Transposing the diminished seventh chord
    P/T 1 2 3 4 P/T 1 2 3 4 P/T 1 2 3 4 P/T 1 2 3 4
    A/1 C D F A B/1 D F A C C/1 F A C D D/1 A C D F
    A/2 C E G A B/2 E G A C C/2 G A C E D/2 A C E G
    A/3 D F G B B/3 F G B D C/3 G B D F D/3 B D F G

(Note: Each arrangement of the three diminished seventh chords appears on the same line as its rotations; by contrast, the augmented chord’s three-note sets are 3×3 squares. For each chord, A/2 precedes B/1.)

Some additional notes:

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Mode 0: The chromatic scale

The chromatic scale itself merits further discussion. It repeats one interval twelve times (H-H-H-H-H-H-H-H-H-H-H-H). Thus, it has only one mode, which in turn may only be made from one permutation of notes. The chromatic scale can therefore be transposed in twelve different ways across the entire, um, chromatic scale:

Permuatations of the chromatic scale
#123456789101112
1CCDDEFFGGAAB
2CDDEFFGGAABC
3DDEFFGGAABCC
4DEFFGGAABCCD
5EFFGGAABCCDD
6FFGGAABCCDDE
7FGGAABCCDDEF
8GGAABCCDDEFF
9GAABCCDDEFFG
10AABCCDDEFFGG
11ABCCDDEFFGGA
12BCCDDEFFGGAA

By extension, we might consider the chromatic scale the Zeroth Mode of Limited Transposition: all other scales in twelve-tone equal temperament are truncations of it.

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All modes of limited transposition in 12-TET

In 12-TET, sixteen scales meet all necessary criteria for modes of limited transposition, with thirty-eight modes between them (in all, they’re missing sixty-four modes). I listed intervals in semitones and only listed one mode per scale. §10 (Scale counts by size) lists all modes of each.

MoLTs at a glance
Notes     Intervals     Modes
12 111111111111 1
10 1111211112 5
9 112112112 3
8 12121212 2
11131113 4
11221122 4
6 222222 1
131313 2
114114 3
123123 3
132132 3
4 3333 1
1515 2
2424 2
3 444 1
2 66 1
MoLTs at a glance
Notes Intervals
(semitones)
Modes NotesIntervals
(semitones)
Modes NotesIntervals
(semitones)
Modes NotesIntervals
(semitones)
Modes
12 111111111111 1 10 1111211112 5 9 112112112 3 8 12121212 2
8 11131113 4 8 11221122 4 6 222222 1 6 131313 2
6 114114 3 6 123123 3 6 132132 3 4 3333 1
4 1515 2 4 2424 2 3 444 1 2 66 1

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These scales have the following note distributions across their parallel modes:

Note distributions of parallel modes of limited transposition
ScaleCCDDEFFGGAAB
Mode 0 (111111111111)111111111111
Mode 7 (1112111121)544444544444
Mode 3 (211211211)322232223222
Mode 4 (11311131)432223432223
Mode 6 (22112211)423232423232
Mode 2 (12121212)211211211211
Mode 5 (141141)321012321012
Mode 2 Truncation 1 (321321)311211311211
Mode 3 Truncation (131313)210121012101
Mode 2 Truncation 2 (231231)311211311211
Mode 1 (222222)101010101010
Mode 5 Truncation 1 (1515)210001210001
Mode 6 Truncation (2424)201010201010
Diminished Seventh (3333)100100100100
Augmented Chord (444)100010001000
Tritone (66)100000100000

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Adjusting these numbers to account for the missing modes gives us the following distributions:

Note distributions of parallel modes of limited transposition (adjusted for inflation)
ScaleCCDDEFFGGAAB
Mode 0 (111111111111)121212121212121212121212
Mode 7 (1112111121)10888881088888
Mode 3 (211211211)966696669666
Mode 4 (11311131)864446864446
Mode 6 (22112211)846464846464
Mode 2 (12121212)844844844844
Mode 5 (141141)642024642024
Mode 2 Truncation 1 (321321)622422622422
Mode 3 Truncation (131313)630363036303
Mode 2 Truncation 2 (231231)622422622422
Mode 1 (222222)606060606060
Mode 5 Truncation 1 (1515)420002420002
Mode 6 Truncation (2424)402040204020
Diminished Seventh (3333)400400400400
Augmented Chord (444)300030003000
Tritone (66)200000200000

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Single-note transformations of modes of limited transposition

I mentioned that I’d return to melodic Phrygian, alternating heptamode, alternating heptamode inverse, apathetic minor, and Pacific after we’d covered modes of limited transposition. 12-TET cannot contain heptatonic modes of limited transposition, but melodic Phrygian, alternating heptamode, and alternating heptamode inverse are single-note transformations of modes of limited transposition. For instance:

The scale transformations we apply to each of these scales to produce their modes consist solely of moving the added or deleted note by the interval at which their parent modes of limited transposition repeat, which is why they don’t take us through the entire chromatic scale. Here’s a note distribution comparison, adjusted for inflation limited transposition (i.e., the whole-tone scale is short 5 modes and the octatonic scale is short 3, so to facilitate 1:1 comparisons, I’ve multiplied the whole-tone scale’s values by 6 and the octatonic scale’s values by 4).

Note distributions across parallel modes (…again‽ But that trick never works!)
ScaleCCDDEFFGGAAB
Ionian 7 2 5 4 3 6 2 6 3 4 5 2
Chromatic heptatonic 7 6 5 4 3 2 2 2 3 4 5 6
Melodic Phrygian 7 2 6 2 6 2 6 2 6 2 6 2
Whole-tone 6 0 6 0 6 0 6 0 6 0 6 0
(Difference) 1 2 0 2 0 2 0 2 0 2 0 2
Whole-tone chromatic 8 4 6 4 6 4 8 4 6 4 6 4
Melodic Phrygian 7 2 6 2 6 2 6 2 6 2 6 2
(Difference) 1 2 0 2 0 2 2 2 0 2 0 2
Octatonic 8 4 4 8 4 4 8 4 4 8 4 4
Alternating heptamode 7 3 3 6 3 3 6 3 3 6 3 3
(Difference) 1 1 1 2 1 1 2 1 1 2 1 1
Double chromatic 8 6 4 4 4 6 8 6 4 4 4 6
Apathetic minor 7 5 3 2 3 5 6 5 3 2 3 5
(Difference) 1 1 1 2 1 1 2 1 1 2 1 1

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Note also that the whole-tone scale is rotationally symmetrical by two semitones, the octatonic scale is rotationally symmetrical by three semitones, and the double chromatic and whole-tone chromatic scales are rotationally symmetrical by six semitones. That is, the whole-tone scale’s distribution repeats every two semitones, the octatonic scale’s every three semitones, and the double chromatic and whole-tone chromatic scales’ every six semitones. This is axiomatically equivalent to their being modes of limited transposition.

Now, note how melodic Phrygian, alternating heptamode, and apathetic minor come to duplicating those distributions: in fact, disregarding the root note, alternating heptamode has exactly (octatonic’s adjusted distributions × ¾), and melodic Phrygian has exactly ((whole-tone’s adjusted distributions × ⅔) + 2). The root note is the outlier here because, by definition, it must always occur in every mode of any scale.

It should be fairly obvious why melodic Phrygian and its modes are 12-TET’s only heptatonic single-note transformations of the whole-tone scale: we may only add notes into six gaps, which are all whole steps; thus, all such transformations must change one whole step into two half-steps.

12-TET contains three different octatonic modes of limited transposition, namely Messiaen’s modes 2, 4, and 6. Can we derive other similar heptatonic scales in similar ways? That is, can deleting a single note of the octatonic, double chromatic, or whole-tone chromatic scales create other scales whose modes can in turn all be derived through single-note transformations to themselves? I believe I’ve exhausted all the possibilities, but I haven’t mathematically proved that yet.

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Microtonal corollaries

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Achiral scales: Reflective & translational symmetry

Reflective symmetry

Rotational symmetry is not the only kind of symmetry a scale may possess. As it turns out, counting the octave and the chromatic scale, sixty-four modes of fifty-one scales in twelve-tone equal temperament have interval distributions with reflective symmetry. They are listed below.

(The scale names are based on interval ordering: the opening letter refers to the number of notes within the scale. The three-digit number orders every mode of every scale with that number of notes by “alphabetizing” the intervals. The two-digit number orders the scales based on their lowest-numbered modes. The Greek letters order the modes, where the most left-aligned is α; subsequent Greek letters each rotate the scale one interval to the left. “SC” means “Scale Complement”. I haven’t yet listed it for every scale, but I’ll get to it soon™.

I may replace this numbering with the now-ubiquitous pitch class sets theorist Allen Forte (1926-2014) gave them in his book The Structure of Atonal Music (1973), since it’s especially convenient for analysis that non-hexatonic scales have the same Forte numbers as their complements, but to be honest, I don’t fully understand Forte’s numbering system yet.)

One-note reflective symmetry: the octave
Mode 1 Intervals SC
A01.αA001C6K01

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Two-note reflective symmetry: the tritone
Mode 12 Intervals SC
B06.αB006CF33J06

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Three-note reflective symmetry (five scales, five modes)
Mode 123 Intervals SC
C01.βC010CCB½ 5½I01
C10.βC018CDA1 41I05
C16.βC025CDA3I13
C19.αC031CEG2 22I19
C13.βC036CFG1 I16

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Tetratonic reflective symmetry (three scales, five modes)
Mode 1234 Intervals SC
D05.βD035CCFB½½H21
D36.βD070CDFA1221H36
D43.αD097CDFAH43
D36.δD117CEFG2112H36
D05.δD131CFFG½½H21

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Pentatonic reflective symmetry (ten scales, ten modes)
Mode 12345 Intervals SC
E01.γE036CCDAB½½4½½G01
E13.βE063CCDAB½131½G09
E34.γE146CDDAA1½3½1G11
E17.βE082CCEGB½2½G44
E55.γE220CDEGA½2½G55
E20.βE094CCFGB½212½G45
E31.εE266CEFGG2½1½2G37
E64.γE165CDEGA11211G49
E66.βE177CDFGA111G66
E65.εE232CDFGA111G64

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Hexatonic reflective symmetry (six scales, ten modes)
Mode 123456 Intervals SC
F04.γF074CCDFAB½½22½½F22
F47.βF179CCFFGB½2½½2½F47
F04.ζF407CEFFGG2½½½½2F22
F65.γF254CDDFAA1½½1F60
F42.εF304CDFFGA1½½1F67
F33.εF373CDFFGA1½½1F58
F33.βF129CCDFAB½11½F58
F42.βF162CCEFGB½11½F67
F65.ζF356CDEFGA½11½F60
F80.αF287CDEFGA111111F80

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Heptatonic reflective symmetry (ten scales, ten modes)
Mode 1234567 Intervals SC
G01.δG056CDE𝄫F𝄫G𝄪AB½½½3½½½E01
G09.γG090CDE𝄫FGAB½½121½½E13
G44.βG159CDEFGAB½1½2½1½E17
G45.δG284CDEFGAB1½½2½½1E20
G11.γG107CDE𝄫FGAB½½1½½E34
G55.βG209CDEFGAB½½1½½E55
G37.ηG389CDEFGAB𝄫½½1½½E31
G64.ζG334CDEFGAB11½1½11E65
G66.γG301CDEFGAB1½111½1E66
G49.βG176CDEFGAB½11111½E64

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Octatonic reflective symmetry (five scales, ten modes)
Mode 12345678 Intervals SC
H03.δH065CCDDFAAB½ ½ ½ ½ ½ ½ D09
H17.γH111CCDFFGAB½ ½ ½ ½ ½ ½ D22
H17.ηH185CCEFFGGB½ ½ ½ ½ ½ ½ D22
H03.θH295CDEFFGGA ½ ½ ½ ½ ½ ½ D09
H41.δH237CDDEFGAA1 ½ ½ 1 1 ½ ½ 1 D41
H40.ζH249CDDFFGAA1 ½ 1 ½ ½ 1 ½ 1 D40
H15.ηH268CDEFFGGA1 1 ½ ½ ½ ½ 1 1 D34
H41.βH166CCDFFGAB½ 1 1 ½ ½ 1 1 ½ D41
H40.βH154CCDEFGAB½ 1 ½ 1 1 ½ 1 ½ D40
H15.γH099CCDEFGAB½ ½ 1 1 1 1 ½ ½ D34

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Enneatonic reflective symmetry (five scales, five modes)
Mode 123456789 Intervals SC
I01.εI035CCDDEGAAB½½½½2½½½½C01
I05.δI049CCDDFGAAB½½½111½½½C10
I13.γI069CCDEFGGAB½½1½1½1½½C16
I19.βI096CCDEFGGAB½1½½1½½1½C19
I16.ιI131CDDEFGGAA1½½½1½½½1C13

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Decatonic reflective symmetry (three scales, five modes)
Mode 123456789A Intervals
J02.εJ020CCDDEFGAAB½½½½11½½½½
J04.δJ025CCDDFFGAAB½½½1½½1½½½
J06.γJ031CCDEFFGGAB½½1½½½½1½½
J04.ιJ038CCDEFFGGAB½1½½½½½½1½
J02.κJ046CDDEFFGGAA1½½½½½½½½1

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Hendecatonic reflective symmetry (one scale, one mode)
Mode 123456789AB Intervals
K01.ζK006CCDDEFGGAAB½½½½½1½½½½½

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Dodecatonic reflective symmetry (the chromatic scale)
Mode 123456789ABC Intervals
L01.αL001CCDDEFFGGAAB½½½½½½½½½½½½

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There are 351 discrete scales in twelve-tone equal temperament. Of these, 256 are chiral scales that cannot be transformed into their inverses by rotation. As a result, another scale serves as their enantiomorph (Greek: ἐναντίος, enantíos, opposite, and μορφή, morphḗ, form). The most familiar achiral scale is surely harmonic minor. Fittingly, its enantiomorph, Mixolydian 2, is harmonic major’s fifth mode; in turn, of course, harmonic major’s enantiomorph, Phrygian dominant, is harmonic minor’s fifth mode.

By contrast, achiral scales can be transformed into their inverses by rotation and thus have no enantiomorphs: every reflection of any mode of an achiral scale is either itself (if the mode itself is symmetrical) or another of its modes. For instance, Ionian’s reflection is Phrygian; Dorian’s reflection is itself. The classic metaphor is that achiral scales are socks that fit either foot, while chiral scales are mittens that only fit one hand.

Every symmetrical heptatonic scale in twelve-tone equal temperament is a complement of a symmetrical pentatonic scale. The following table shows the interval distributions of symmetrical pentatonic scales and their complements side-by-side, first using my order for the heptatonics, then my order for the pentatonics. You may have noticed that I ordered the scales above so that their interval distributions would form aesthetic patterns. An interesting consequence of this is that their complements’ symmetrical modes form their own aesthetic patterns, which are quite dissimilar from the originals, but quite similar to each other.

(I should clarify that, while these tables list these scales’ symmetrical modes side-by-side, scale complements apply on a scale-wide basis, not a modal basis, simply because the process of forming a scale complement requires rotating the scale, and not necessarily by a constant amount. Moreover, in 12-TET, only hexatonic scales have complements of the same size; thus, 1:1 modal relationships between complements cannot always exist.)

Symmetrical scale complements (5 & 7)
heptatonic order
PentatonicIntervalsHeptatonicIntervals
E01.γ E036 ½ ½ 4 ½ ½ G01.δ G056 ½ ½ ½ 3 ½ ½ ½
E13.β E063 ½ 1 3 1 ½ G09.γ G090 ½ ½ 1 2 1 ½ ½
E17.β E082 ½ 2 ½ G44.β G159 ½ 1 ½ 2 ½ 1 ½
E20.β E094 ½ 2 1 2 ½ G45.δ G284 1 ½ ½ 2 ½ ½ 1
E34.γ E146 1 ½ 3 ½ 1 G11.γ G107 ½ ½ 1 ½ ½
E55.γ E220 ½ 2 ½ G55.β G209 ½ ½ 1 ½ ½
E31.ε E266 2 ½ 1 ½ 2 G37.η G389 ½ ½ 1 ½ ½
E65.ε E232 1 1 1 G64.ζ G334 1 1 ½ 1 ½ 1 1
E66.β E177 1 1 1 G66.γ G301 1 ½ 1 1 1 ½ 1
E64.γ E165 1 1 2 1 1 G49.β G176 ½ 1 1 1 1 1 ½
pentatonic order
PentatonicIntervalsHeptatonicIntervals
E01.γ E036 ½ ½ 4 ½ ½ G01.δ G056 ½ ½ ½ 3 ½ ½ ½
E13.β E063 ½ 1 3 1 ½ G09.γ G090 ½ ½ 1 2 1 ½ ½
E34.γ E146 1 ½ 3 ½ 1 G11.γ G107 ½ ½ 1 ½ ½
E17.β E082 ½ 2 ½ G44.β G159 ½ 1 ½ 2 ½ 1 ½
E55.γ E220 ½ 2 ½ G55.β G209 ½ ½ 1 ½ ½
E20.β E094 ½ 2 1 2 ½ G45.δ G284 1 ½ ½ 2 ½ ½ 1
E31.ε E266 2 ½ 1 ½ 2 G37.η G389 ½ ½ 1 ½ ½
E64.γ E165 1 1 2 1 1 G49.β G176 ½ 1 1 1 1 1 ½
E66.β E177 1 1 1 G66.γ G301 1 ½ 1 1 1 ½ 1
E65.ε E232 1 1 1 G64.ζ G334 1 1 ½ 1 ½ 1 1

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A similar pattern occurs with three-note scales and the enneatonics:

Symmetrical scale complements (3 & 9)
Three-noteIntervalsEnneatonicIntervals
C01.β C010 ½5½ I01.ε I035 ½½½½2½½½½
C10.β C018 141 I05.δ I049 ½½½111½½½
C16.β C025 3 I13.γ I069 ½½1½1½1½½
C19.α C031 222 I19.β I096 ½1½½1½½1½
C13.β C036 1 I16.ι I131 1½½½1½½½1

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Translational symmetry

I’ve mentioned that 12-TET contains 351 scales, that 256 scales are chiral, and that 51 scales have reflectively symmetrical modes. If you’ve done the math, you’ve already figured out that I haven’t accounted for 44 scales.

The scale complement pattern I noted above is only so straightforward with odd-numbered scales – even-numbered scales are more complicated. The reason is that reflective symmetry is a subset of a broader type of symmetry called translational symmetry. The 44 remaining scales are translationally symmetrical (and thus achiral), but do not have any reflectively symmetrical modes.

A scale possesses translational symmetry if it can be transformed into its inverse by rotation. Reflective symmetry occurs on notes (like Dorian’s symmetry); however, another form of translational symmetry occurs between notes. This can only exist if a scale’s note count is a multiple of 2. I’ll henceforth call such scales even scales (antonym: odd scales) for brevity’s sake.

In 12-TET, all achiral scales have achiral complements (caveat: the chromatic scale’s complement is… um… er… what‽), but not all reflectively symmetrical scales have reflectively symmetrical complements. However, for the same reason that translational symmetry between notes can only occur in even scales, all reflectively symmetrical odd scales have reflectively symmetrical complements.

Among scales with translational but not reflective symmetry, five have two notes, twelve are tetratonic, fourteen hexatonic, ten octatonic, and three decatonic. Since literally all two-note and decatonic scales are translationally symmetrical, I see little point in printing them here, but the tetratonics, hexatonics, and octatonics are:

Tetratonic translational symmetry (twelve additional scales)
Mode 1234 IntervalsSC
D001D01.αCCDD½½½H01
D009D01.βCCDB½½½
D045D01.γCCAB½½½
D165D01.δCAAB½½½
D010D09.αCCDE½1½4H03
D053D09.βCDDB1½4½
D043D09.γCCAA½4½1
D163D09.δCGAB4½1½
D047D15.δCDDF1½1H07
D016D15.αCCDA½11
D080D15.βCDAB11½
D159D15.γCGAA1½1
D018D16.αCCEF½½H08
D088D16.βCDEB½½
D040D16.γCCGA½½
D158D16.δCGGB½½
D025D22.αCCFF½2½3H17
D115D22.βCEFB2½3½
D036D22.γCCGG½3½2
D149D22.δCFGB3½2½
D031D27.αCCFG½½H31
D135D27.βCFFB½½
D055D34.αCDEF1113H15
D059D34.βCDEA1131
D077D34.γCDGA1311
D151D34.δCFGA3111
D062D38.αCDFG11H28
D093D38.βCDFA11
D073D38.γCDGA11
D138D38.δCFGA11
D068D41.αCDFG1212H41
D119D41.βCEFA2121
D084D20.δCDEG½H26
D022D20.αCCEA½
D106D20.βCDGB½
D140D20.γCFGA½
D091D40.δCDFG12H40
D064D40.αCDFA12
D102D40.βCDGA21
D122D40.γCEGA21
D113D24.δCEFA2½2H38
D027D24.αCCFG½22
D124D24.βCEGB22½
D100D24.γCDGG2½2

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Hexatonic translational symmetry (fourteen additional scales)
Mode 123456 Intervals SC
F001F01.αCCDDEF½½½½½F01
F007F01.βCCDDEB½½½½½
F028F01.γCCDDAB½½½½½
F084F01.δCCDAAB½½½½½
F210F01.εCCGAAB½½½½½
F462F01.ζCGGAAB½½½½½
F029F22.αCCDEFF½½1½½3F04
F090F22.βCCDEFB½1½½3½
F231F22.γCDDEAB1½½3½½
F081F22.δCCDGAA½½3½½1
F207F22.εCCGGAB½3½½1½
F459F22.ζCFGGAB3½½1½½
F212F11.ζCDDEFG1½½½1F11
F012F11.αCCDDFA½½½11
F048F11.βCCDEAB½½11½
F139F11.γCCDGAB½11½½
F335F11.δCDGAAB11½½½
F452F11.εCFGGAA1½½½1
F091F55.αCCDEFG½1½1½F14
F236F55.βCDDFFB1½1½½
F103F55.γCCDEAA½1½½1
F264F55.δCDDGAB1½½1½
F199F55.εCCFGAA½½1½1
F449F55.ζCFFGAB½1½1½
F050F37.αCCDFFG½½½½F16
F145F37.βCCEFFB½½½½
F351F37.γCDEFAB½½½½
F075F37.δCCDGGA½½½½
F198F37.εCCFGGB½½½½
F447F37.ζCFFGAB½½½½
F111F67.αCCDFGG½111½2F42
F285F67.βCDEFGB111½2½
F279F67.γCDEFAA11½2½1
F258F67.δCDDGGA1½2½11
F181F67.εCCFFGA½2½111
F421F67.ζCEFGAB2½111½
F233F58.ζCDDFFG1½1½12F33
F094F58.αCCDEFA½1½121
F245F58.βCDDFAB1½121½
F134F58.γCCDGAA½121½1
F329F58.δCDFGAB121½1½
F430F58.εCEFGAA21½1½1
F410F16.ζCEFFGB2½½½2½F37
F019F16.αCCDDGG½½½2½2
F068F16.βCCDFGB½½2½2½
F185F16.γCCFFAB½2½2½½
F426F16.δCEFAAB2½2½½½
F176F16.εCCFFGG½2½½½2
F339F14.ζCDEFFA½½½F55
F016F14.αCCDDFA½½½
F061F14.βCCDFGB½½½
F171F14.γCCEGAB½½½
F401F14.δCDFAAB½½½
F392F14.εCDFGGA½½½
F273F71.εCDEFGA11½11F71
F239F71.ζCDDFGA1½111
F116F71.αCCDFGA½1111
F296F71.βCDEGAB1111½
F315F71.γCDFGAA111½1
F383F71.δCDFGGA11½11
F354F60.ζCDEFGA½1½1F65
F097F60.αCCDEGA½1½1
F253F60.βCDDFGB1½1½
F163F60.γCCEFAA½1½1
F389F60.δCDFGAB1½1½
F306F60.εCDFFGA1½1½
F310F73.βCDFFAB1½1½F74
F365F73.γCDEGAA½1½1
F157F73.δCCEFGA½1½1
F378F73.εCDFFGB1½1½
F241F73.ζCDDFGA1½1½
F123F73.αCCDFGA½1½1
F125F74.αCCDFGA½11½F73
F314F74.βCDFGGB11½½
F379F74.γCDFFAA1½½1
F248F74.δCDDFGA1½½1
F148F74.εCCEFGA½½11
F360F74.ζCDEFAB½11½
F150F76.αCCEFGA½½½F76
F364F76.βCDEGGB½½½

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Octatonic translational symmetry (ten additional scales)
Mode 12345678 Intervals SC
H001H01.αCCDDEFFG½½½½½½½D01
H005H01.βCCDDEFFB½½½½½½½
H015H01.γCCDDEFAB½½½½½½½
H035H01.δCCDDEAAB½½½½½½½
H070H01.εCCDDGAAB½½½½½½½
H126H01.ζCCDGGAAB½½½½½½½
H210H01.ηCCFGGAAB½½½½½½½
H330H01.θCFFGGAAB½½½½½½½
H036H21.αCCDDFFGG½½½1½½½2D05
H074H21.βCCDEFFGB½½1½½½2½
H136H21.γCCDEFFAB½1½½½2½½
H230H21.δCDDEFAAB1½½½2½½½
H066H21.εCCDDGGAA½½½2½½½1
H122H21.ζCCDFGGAB½½2½½½1½
H206H21.ηCCFFGGAB½2½½½1½½
H326H21.θCEFFGAAB2½½½1½½½
H056H31.αCCDDFGGA½½½½½½D27
H108H31.βCCDFFGGB½½½½½½
H188H31.γCCEFFGAB½½½½½½
H304H31.δCDEFFAAB½½½½½½
H212H07.θCDDEFFGA1½½½½½1D15
H008H07.αCCDDEFGA½½½½½11
H024H07.βCCDDEFAB½½½½11½
H054H07.γCCDDFGAB½½½11½½
H104H07.δCCDEGAAB½½11½½½
H181H07.εCCDFGAAB½11½½½½
H293H07.ζCDFGGAAB11½½½½½
H316H07.ηCDFFGGAA1½½½½½1
H131H26.ηCCDEFFGA½1½½½1½D20
H223H26.θCDDEFGGB1½½½1½½
H043H26.αCCDDFFAA½½½1½½1
H088H26.βCCDEFGAB½½1½½1½
H159H26.γCCDEGGAB½1½½1½½
H264H26.δCDDFGAAB1½½1½½½
H193H26.εCCEFGGAA½½1½½½1
H311H26.ζCDEFGGAB½1½½½1½
H081H38.αCCDEFGGA½½1½1½½D24
H149H38.βCCDEFGGB½1½1½½½
H251H38.γCDDFFGAB1½1½½½½
H143H38.δCCDEFGAA½1½½½½1
H242H38.εCDDEGGAB1½½½½1½
H109H38.ζCCDFFGAA½½½½1½1
H190H38.ηCCEFFGAB½½½1½1½
H307H38.θCDEFGGAB½½1½1½½
H297H08.θCDEFFGGB½½½½½½D16
H010H08.αCCDDEFGA½½½½½½
H028H08.βCCDDEGGB½½½½½½
H061H08.γCCDDFGAB½½½½½½
H115H08.δCCDFFAAB½½½½½½
H197H08.εCCEFGAAB½½½½½½
H315H08.ζCDEGGAAB½½½½½½
H183H08.ηCCEFFGGA½½½½½½
H170H28.γCCDFFGAB½11½11½½D38
H280H28.δCDEFGAAB11½11½½½
H256H28.εCDDFGGAA1½11½½½1
H163H28.ζCCDFFGGA½11½½½11
H271H28.ηCDEFFGAB11½½½11½
H224H28.θCDDEFGAA1½½½11½1
H047H28.αCCDDFGGA½½½11½11
H095H28.βCCDEFGAB½½11½11½
H150H43.αCCDEFGAA½1½1½1½1D43
H253H43.βCDDFFGAB1½1½1½1½
H232H36.θCDDEFGGA1½½1½½11D36
H076H36.αCCDEFFGA½½1½½111
H141H36.βCCDEFGAB½1½½111½
H239H36.γCDDEFGAB1½½111½½
H097H36.δCCDEFGAA½½111½½1
H173H36.εCCDFGGAB½111½½1½
H284H36.ζCDEFGGAB111½½1½½
H273H36.ηCDEFFGAA11½½1½½1

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Counting achiral scales

As I mentioned, all achiral scales have achiral complements. I also mentioned a caveat: the chromatic scale’s complement, the null set, shares some characteristics of a divide by zero error. In the context of scale complements, however, I’m willing to count it as one. Perhaps more importantly, Forte counts it as a pitch class. Bearing that in mind, the achiral scales break down as follows:

Achiral scales by size
NotesScales
0111121
2106
395
4815
5710
620
Total96

Forte grouped pitch classes into 224 discrete sets. 128 of these are the 256 chiral scales; the other 96 are the achiral scales, for a total 352 scales (or 351, if we exclude the null set).

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Self-complementary scales, symmetry, and scale generators

Only six scales in 12-TET are their own complements. In the table below, “RS” means “rotation size”, namely how many semitones we must rotate the scale after flipping its bits to wind up with its original note composition. Importantly, in each case, rotating the scale up and rotating it down will produce the same mode.

Self-complementing scales
ID Forte Notes (prime form) Intervals Binary form RS
F01 6-1 CC DD EF ½ ½ ½ ½ ½ 111111000000 6
F47 6-7 CCDFGG ½½2½½2 111000111000 3
F76 6-20 CCEFGA½½½ 110011001100 2
F80 6-35 CDEFGA111111 101010101010 1
F71 6-32 CDEFGA 11½11 101011010100 6
F11 6-8 CDDEFG 1½½½1 101111010000 6

For presumably obvious reasons, such scales must be hexatonic, and for less obvious reasons, they must also be achiral. With only twenty achiral hexatonic scales, it’s little surprise that so few self-complementing scales exist. We’ll explore why self-complements must be achiral once we’ve examined some other interesting commonalities between the self-complements:

As long as we’re returning to the subject of generators, let’s also observe that F71 truncates Ionian by a single note. We’ve already analyzed how the pentatonic scale both truncates and complements Ionian, so this should surprise no one. In fact, I hypothesize that, in any t-tone equal temperament, for a given note count of c:

Since 12-TET contains a very obvious dearth of test cases, I’ll need to write better tools for analysis in other equal temperaments to ascertain whether I’m correct.

Now, why do self-complementing scales need to be achiral? Let’s take Rāga Syamalam as an example. (I’ll produce more detailed representations of this data soon™.)

If we flip its bits, we get 010011000111. Arbitrarily, I’m going to rotate that five semitones to the left, producing Rāga Gangatarangini.

Clearly, not the same scale. This is an inevitable result of XORing an asymmetrical sequence of ones and zeroes.

Perhaps confusingly, we haven’t wound up with Rāga Syamalam’s enantiomorph, although it is quite similar:

Effectively, the 0011 and the 01 swap positions between the two scales. For what it’s worth, Forte considers these scales’ set classes to be “zygotic”, or twinned: their interval distribution is so similar that they share a large number of characteristics. Forte assigned Rāga Gangatarangini the number 6-Z17 and Rāga Syamalam (and its enantiomorph) the number 6-Z43, where Z stands for “zygotic”.

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A crash course in Ancient Greek harmony

Etymology

Ionian and its modes are named for places in or near ancient Greece and/or ancient Greek tribes:

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 discrete regions ⟨en.wikipedia.org/wiki/Locris⟩ of ancient Greece

However, they have little 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.

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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 what we would consider descending order. But then, the ancient Greek metaphor for time literally inverted the modern one: they saw the past as receding away in front of us, continually getting ever more distant, and the future as creeping up from behind us. I actually find their metaphor far more apt than ours: who can actually see the future? And our memories of the past get more distant every day.

I don’t know if this metaphor affected how they described changes over time. I may be overthinking this, but if they thought of the past as in front of them, they may not have perceived this as a descent. I don’t have enough information to know if concrete proof exists one way or the other. Certainly, where the ancient Greeks refer to time, translators must be aware of their metaphor, and anyone who reads translated Greek writing that concerns time should take the differences into account (and even ask if the translator knew of them).

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 alteration⁽⁸⁾).

The same ancient Greek tonos could actually use any of three genera (singular: genus). Each tonos contained two tetrachords, four-note sets spanning a 4:3 ratio, or perfect fourth. The note spacing varied between genera, but all three 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

The tetrachords and synaphai’s positions, meanwhile, varied between tonoi, with notable consequences:

Since tetrachords spanned 4:3 ratios, synaphai were mathematically constrained to 9:8, or major seconds:

Note that enharmonic has a different meaning in twelve-tone equal temperament than it has in ancient Greek harmony (or any tuning system that uses microtonality). In 12-TET, enharmonic means two tones have the same pitch. In ancient Greek harmony, it refers to an interval spacing smaller than a semitone and to the tuning system that used it. As stated, I’ll approximate this interval in 24-TET with quarter-tones.

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Pythagoras with the looking glass: Comparing interval ratios

One point in twelve-tone equal temperament’s favor is how closely it approximates every foundational interval of Pythagorean tuning. In the chart headers below, P stands for Pythagorean, M for Modern (i.e., 12-TET), R for Ratio, and Q for Quotient. The final column shows 12-TET’s difference from Pythagorean tuning.

(The name “Pythagorean” is a misnomer – their eponymous tuning system is actually much older. The oldest known description of it appears on a Mesopotamian clay tablet from the 19th century BCE.)

A comparison of Pythagorean tuning & twelve-tone equal temperament
IntervalPRPQMRMQMQPQ
Octave2:122¹⁄₁=2=+0
Perfect fifth3:21.52⁷⁄₁₂≈1.49830707688≈−0.00169292312
Perfect fourth4:31.333333333…2⁵⁄₁₂≈1.33483985417+0.00150652083
Major third81:641.2656252¹⁄₃≈1.25992104989≈−0.00570395011
Diminished fourth8192:6561≈1.248590153942¹⁄₃≈1.259921049890.01133089595
Minor third32:271.185185185…2¹⁄₄≈1.18920711500+0.00402192982
Major second9:81.1252¹⁄₆≈1.12246204831≈−0.00253795169
Minor second256:243≈1.053497942392¹⁄₁₂≈1.05946309436+0.00596515197

So 12-TET is within about 0.0015 of the Pythagorean ratios for perfect fourths and fifths; its major second is off by about 0.0025; its minor third is off by about 0.004; and its major third and minor second are off by about 0.006. Only an incredibly well-trained ear could discern any of these differences. (The difference becomes vastly more noticeable when we stack the interval several times; 12-TET ultimately became standard because it avoids dissonances that are inherent to Pythagorean tuning, notably its so-called wolf intervals).

The only interval where 12-TET is off by more than 0.01 is the diminished fourth – which doesn’t really exist in 12-TET. It’s the result of subtracting two Pythagorean minor thirds from an octave. Since Pythagorean note spacing wasn’t even, an octave minus two minor thirds produced a different interval than a major third. In 12-TET, it’s just a major third. Unlike many diminished intervals, the Pythagorean diminished fourth is quite consonant – in fact it’s actually closer to a 5:4 ratio than the major third is, so this may not be surprising. In any case, both the modern and Pythagorean major third and minor third closely approximate 5:4 (1.25) and 6:5 (1.2) ratios, which may be part of why our ears find them so harmonically pleasing. (In just intonation, a minor third would use a 6:5 ratio and a major third would use a 5:4 ratio; however, Pythagorean tuning does not use any numbers that are not exact powers of three or exact powers of two.)

Incidentally, all Pythagorean interval ratios are based on powers of two and three, and Pythagorean intervals smaller than perfect fifths can be derived through a sequence of ratio division. To wit:

Pythagorean interval division (or is it subtraction?)
DividendDivisorQuotient
Octave2¹:32:1
Perfect fifth3¹:2¹3:2
Octave2¹:32:1Perfect fifth3¹:2¹3:2Perfect fourth2²:3¹4:3
Perfect fifth3¹:2¹3:2Perfect fourth2²:3¹4:3Major second3²:2³9:8
Perfect fourth2²:3¹4:3Major second3²:2³9:8Minor third2⁵:3³32:27
Minor third2⁵:3³32:27Major second3²:2³9:8Minor second2⁸:3256:243
Minor third2⁵:3³32:27Minor second2⁸:3256:243Diminished fourth2¹³:38192:6561
Perfect fourth2²:3¹4:3Minor second2⁸:3256:243Major third3⁴:281:64

Note that dividing pitch ratios effectively equates to subtracting intervals. Since pitch is a binary logarthmic scale, raising a note an octave doubles its pitch. In mathematical terms, log₂ (⁄ᵦ) equals the number of octaves separating the notes with pitches α and β, which will be fractional if they’re not separated by an exact multiple of an octave, zero if α = β, and negative if β > α. As a result, dividing 2:1 by 3:2 gives us a 4:3 ratio, but it subtracts a perfect fifth from an octave, giving us a perfect fourth. Try not to think about it too hard and you might not get a headache.

One interesting footnote to the above table: Remember the Fibonacci sequence? Start with 0 and 1, and repeatedly add the previous two numbers together. Thus, the sequence’s next numbers are:

  1. 0 + 1 = 1
  2. 1 + 1 = 2
  3. 1 + 2 = 3
  4. 2 + 3 = 5
  5. 5 + 3 = 8
  6. 8 + 5 = 13
  7. 13 + 8 = 21
  8. 21 + 13 = 34
  9. 34 + 21 = 55

Now, look again at the ratios above. 0, 1, 2, 3, 5, 8, and 13 – the sequence’s first eight numbers – appear both as exponents and, in 2 and 3’s cases, as bases. The major third is, in fact, the only interval in the above table that doesn’t use Fibonacci numbers as exponents. The Fibonacci spiral truly is everywhere – although the Indian mathematician Acharya Pingala (Sanskrit: आचार्य पिङ्गल) is the first writer known to have explicitly described the sequence (ca. 200 BCE), there it is in Pythagorean tuning. (In fact, it’s even older than that: the system dates back to the Mesopotamians.)

It may seem rather haphazard as to whether the ratio starts with a power of 3 or a power of 2; it may also seem rather haphazard as to whether we’re dividing the most recent interval by the second-most recent interval or vice versa. In both these cases, the answer simply depends on which is larger: the larger number appears first in the ratio, and the larger ratio serves as the dividend. Either way, the power of 2 is always one Fibonacci number ahead of the power of 3. Another way to think of the sequence is as follows:

The Fibonacci sequence in Pythagorean tuning
RatioInterval jumpExample
2¹:3an octave higherC3toC4
2¹:3¹a perfect fifth lowerC4toF3
2²:3¹a perfect fourth higherF3toB3
2³:3²a major second lowerB3toA3
2⁵:3³a minor third higherA3toB3
2⁸:3a minor second higherB3toC4
2¹³:3a diminshed fourth higherC4toE4

If the sequence continued, the next interval would be 2²¹:3¹³, or 2,097,152:1,594,323 (≈1.31538715806). As far as I know, this was not a Pythagorean interval, which probably won’t surprise anyone. Nor is it likely to surprise anyone that Fibonacci had his own tuning system, which I’ll undoubtedly write about soon™.

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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
Lydian Enharmonic ¼ 2 ¼ ¼ 2 1¼
Lydian Chromatic ½ ½  ½ 1½
Lydian Diatonic 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 Enharmonic ¼ ¼ 2 1¼ ¼ 2
Dorian Chromatic ½  ½ 1½ ½
Dorian Diatonic 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
Hypophrygian Enharmonic 2 1¼ ¼ 2 ¼ ¼
HypophrygianChromatic 1½ ½ ½ ½
HypophrygianDiatonic 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

The above table is quite abstract, so to follow it up, here are the actual scales. There is no particularly well-established standard for 24-TET notation. I’ve chosen to use ʌ to mean “raise this note by a quarter tone” and v to mean “lower this note by a quarter tone.” 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 synaphe (a bit more so, even, because it will become a bit less legible shortly).

Greek enharmonic tonoi (C roots, linear order)
Enharmonic Tonos 1 2 3 4567Intervals
Mixolydian CCFFB ¼ ¼ 2 ¼ ¼ 2 1
Lydian CFv FCv ¼ 2 ¼ ¼ 2 1¼
Phrygian CEFv FAB Cv 2 ¼ ¼ 2 1¼ ¼
Dorian CCFG G¼ ¼ 2 1¼ ¼ 2
Hypolydian CFv Gv G Cv ¼ 2 1¼ ¼ 2 ¼
Hypophrygian CEF Gv G B Cv 2 1¼ ¼ 2 ¼ ¼
Hypodorian CD DG G1¼ ¼ 2 ¼ ¼ 2

Here are the chromatic tonoi rooted in C and, for the sake of representing what medieval Europeans might have thought they were, their inversions.

Greek chromatic tonoi & their inversions (C roots, linear order)
Chromatic Tonos 1 234567Intervals
Mixolydian C D E𝄫 F G A𝄫 B ½ ½ ½ ½ 1
Lydian C D E F G A B ½ ½ ½ 1½
Phrygian C D E F G A B ½ ½ 1½ ½
Dorian C D E𝄫 F G A B𝄫 ½ ½ 1½ ½
Hypolydian C D E F G A B ½ 1½ ½ ½
Hypophrygian C D E F G A B 1½ ½ ½ ½
Hypodorian C D E F G A B𝄫 1½ ½ ½ ½
Mixolydian inverse C D E F G A B 1 ½ ½ ½ ½
Lydian inverse C D E F G A B ½ 1 ½ ½ ½
Phrygian inverse C D E𝄫 F G A B𝄫 ½ ½ 1 ½ ½
Dorian inverse C D E F G A B ½ ½ 1 ½ ½
Hypolydian inverse C D E F G A B ½ ½ ½ 1 ½
Hypophrygian inverse C D E𝄫 F G A𝄫 B𝄫 ½ ½ ½ ½ 1
Hypodorian inverse C D E F G A B ½ ½ 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 234567Intervals
Mixolydian BCDEFGA ½½½½1
Lydian CDEFGAB ½½½1½
Phrygian DEFGABC ½½1½½
Dorian EFGABCD ½½1½½
Hypolydian FGABCDE ½1½½½
Hypophrygian GABCDEF 1½½½½
Hypodorian ABCDEFG 1½½½½
Mixolydian inverse FGABCDE 1½½½½
Lydian inverse EFGABCD ½1½½½
Phrygian inverse DEFGABC ½½1½½
Dorian inverse CDEFGAB ½½1½½
Hypolydian inverse BCDEFGA ½½½1½
Hypophrygian inverse ABCDEFG ½½½½1
Hypodorian inverse GABCDEF ½½½½1

The ancient Greek tonoi’s “circle of fifths” order 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.

I’m reversing the inverted scales’ order in the next table, since as its predecessor clearly demonstrates, they’re actually moving in the opposite direction from their namesakes. Also, I’m reintroducing Major Phrygian and its modes here, since they’re the midway point between the chromatic scales and their inversions.

Greek chromatic tonoi & their variants (mode-based roots, “circle of fifths” order)
Chromatic Tonos 1 234567Intervals
Hypolydian FGABCDE ½1½½½
Lydian CDEFGAB ½½½1½
Hypophrygian GABCDEF 1½½½½
Phrygian DEFGABC ½½1½½
Hypodorian ABCDEFG 1½½½½
Dorian EFGABCD ½½1½½
Mixolydian BCDEFGA ½½½½1
Ultra-Phrygian EFGABCD ½1½½½
Kanakāngi 5 BCDEFGA ½½½1½
Hungarian Romani minor FGABCDE 1½½½½
Major Phrygian CDEFGAB ½½1½½
Hungarian Romani minor inverse GABCDEF ½½½½1
Rasikapriyā DEFGABC ½1½½½
Ionian augmented 2 ABCDEFG ½½½1½
Mixolydian inverse FGABCDE 1½½½½
Dorian inverse CDEFGAB ½½1½½
Hypodorian inverse GABCDEF ½½½½1
Phrygian inverse DEFGABC ½½1½½
Hypophrygian inverse ABCDEFG ½½½½1
Lydian inverse EFGABCD ½1½½½
Hypolydian inverse BCDEFGA ½½½1½

So effectively, Lydian and Hypolydian 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 Ionian scale in this way shows us the same thing occurring with it:

Greek diatonic tonoi (C roots, circle of fifths order)
ModernAncient1 234567Intervals
Lydian Hypolydian CDEF GAB1 1 1½ 1 1 ½
Ionian Lydian CDEFGAB1 1 ½ 1 1 1½
MixolydianHypophrygianCDEFGAB 1 1½ 1 1 ½ 1
Dorian Phrygian CDE FGAB 1 ½ 1 1 1½ 1
Aeolian Hypodorian CDE FGA B 1½ 1 1 ½ 1 1
Phrygian Dorian CD E FGA B ½ 1 1 1½ 1 1
Locrian Mixolydian CD E FG 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 Ionian 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.

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Chromatic tonos analysis: The circle of fifths

There’s no obvious equivalent of the circle of fifths progression for the chromatic genus, though; for reasons explained above, that’s a special property of the diatonic genus’ mathematical regularity. Rotating most scales requires making more changes to their intervals. Let’s see the scales on C again, this time with the tetrachord placement standardized around Major Phrygian’s layout (since it centers the synaphe within the middle row).

I’ve numbered the scales so I can more clearly explain patterns. The Greek letters refer to scales: α denotes chromatic tonoi, β denotes a mode of Major Phrygian, and γ denotes inverse chromatic tonoi. The number refers to Greek mode order (Mixolydian first, Hypodorian last). I used the same synaphe positions for all three, so the inverse chromatic scales reverse the order (Hypodorian first, Mixolydian last). I’ll retain these numbers throughout my analysis of these scales.

Greek chromatic tonoi & their variants (C roots, linear order)
#Chromatic Tonos 1 234567Intervals
α.1Mixolydian CD E𝄫 FG A𝄫 B ½ ½ ½ ½ 1
α.2Lydian CD EFG AB½ ½ ½ 1½
α.3Phrygian CD EFG A B½ ½ 1½ ½
α.4Dorian CD E𝄫 FGA B𝄫 ½ ½ 1½ ½
α.5Hypolydian CD EF GA B½ 1½ ½ ½
α.6Hypophrygian CD E F GA B1½ ½ ½ ½
α.7Hypodorian CDE F GA B𝄫 1½ ½ ½ ½
β.1Hungarian Romani minor inverse CD EFG AB ½ ½ ½ ½ 1
β.2Ionian augmented 2 CD EFG AB ½ ½ ½ 1½
β.3Kanakāngi 5 CD E𝄫 FG A B𝄫 ½ ½ ½ 1½
β.4Major Phrygian CD EFGA B½ ½ 1½ ½
β.5Rasikapriyā CD EF GA B ½ 1½ ½ ½
β.6Ultra-Phrygian CD E F GA B𝄫 ½ 1½ ½ ½
β.7Hungarian Romani minor CDE F GA B1½ ½ ½ ½
γ.1Hypodorian inverse CD EFG AB ½ ½ ½ ½ 1
γ.2Hypophrygian inverse CD E𝄫 FG A𝄫 B𝄫 ½ ½ ½ ½ 1
γ.3Hypolydian inverse CD EFG A B½ ½ ½ 1 ½
γ.4Dorian inverse CD EFGA B ½ ½ 1 ½ ½
γ.5Phrygian inverse CD E𝄫 F GA B𝄫 ½ ½ 1 ½ ½
γ.6Lydian inverse CD E F GA B½ 1 ½ ½ ½
γ.7Mixolydian inverse CDE F GA B1 ½ ½ ½ ½

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Their “circle of fifths” order looks like this:

Greek chromatic tonoi & their variants (C roots, “circle of fifths” order)
#Chromatic Tonos 1 234567Intervals
α.6Hypophrygian CD E F GA B1½ ½ ½ ½
α.3Phrygian CD EFG A B½ ½ 1½ ½
α.7Hypodorian CDE F GA B𝄫 1½ ½ ½ ½
α.4Dorian CD E𝄫 FGA B𝄫 ½ ½ 1½ ½
α.1Mixolydian CD E𝄫 FG A𝄫 B ½ ½ ½ ½ 1
α.5Hypolydian CD EF GA B½ 1½ ½ ½
α.2Lydian CD EFG AB½ ½ ½ 1½
β.6Ultra-Phrygian CD E F GA B𝄫 ½ 1½ ½ ½
β.3Kanakāngi 5 CD E𝄫 FG A B𝄫 ½ ½ ½ 1½
β.7Hungarian Romani minor CDE F GA B1½ ½ ½ ½
β.4Major Phrygian CD EFGA B½ ½ 1½ ½
β.1Hungarian Romani minor inverse CD EFG AB ½ ½ ½ ½ 1
β.5Rasikapriyā CD EF GA B ½ 1½ ½ ½
β.2Ionian augmented 2CD EFG AB ½ ½ ½ 1½
γ.6Lydian inverse CD E F GA B½ 1 ½ ½ ½
γ.3Hypolydian inverse CD EFG A B½ ½ ½ 1 ½
γ.7Mixolydian inverse CDE F GA B1 ½ ½ ½ ½
γ.4Dorian inverse CD EFGA B ½ ½ 1 ½ ½
γ.1Hypodorian inverse CD EFG AB ½ ½ ½ ½ 1
γ.5Phrygian inverse CD E𝄫 F GA B𝄫 ½ ½ 1 ½ ½
γ.2Hypophrygian inverse CD E𝄫 FG A𝄫 B𝄫 ½ ½ ½ ½ 1

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It fascinates me how these three sets of modes complement each other. Note the latter chart’s accidental distributions: the inversions have sharps bunched in the middle, the chromatic scales have flats bunched in the middle, and Major Phrygian has flats bunched above it and sharps below it. Of course, the interval distribution explains why that might have happened:

In short, the accidental distribution results directly from the interval distribution. Comparing the same position across sets (i.e., α.1, β.1, γ.1, then α.2, β.2, γ.2, then α.3, β.3, γ.3, etc.) may clarify this:

Greek chromatic tonoi & their variants (C roots, “aligned synaphai” order)
#Chromatic Tonos 1 234567Intervals
α.1Mixolydian CDE𝄫FGA𝄫B ½ ½ ½ ½ 1
β.1Hungarian Romani minor inverse CDE FGA B ½ ½ ½ ½ 1
γ.1Hypodorian inverse CDE FGA B ½ ½ ½ ½ 1
α.2Lydian CDE FGA B ½ ½ ½ 1½
β.2Ionian augmented 2CDE FGA B ½ ½ ½ 1½
γ.2Hypophrygian inverse CDE𝄫FGA𝄫B𝄫½ ½ ½ ½ 1
α.3Phrygian CDE FGA B ½ ½ 1½ ½
β.3Kanakāngi 5 CDE𝄫FGA B𝄫½ ½ ½ 1½
γ.3Hypolydian inverse CDE FGA B ½ ½ ½ 1 ½
α.4Dorian CDE𝄫FGA B𝄫½ ½ 1½ ½
β.4Major Phrygian CDE FGA B ½ ½ 1½ ½
γ.4Dorian inverse CDE FGA B ½ ½ 1 ½ ½
α.5Hypolydian CDE FGA B ½ 1½ ½ ½
β.5Rasikapriyā CDE FGA B ½ 1½ ½ ½
γ.5Phrygian inverse CDE𝄫FGA B𝄫½ ½ 1 ½ ½
α.6Hypophrygian CDE FGA B 1½ ½ ½ ½
β.6Ultra-Phrygian CDE FGA B𝄫½ 1½ ½ ½
γ.6Lydian inverse CDE FGA B ½ 1 ½ ½ ½
α.7Hypodorian CDE FGA B𝄫1½ ½ ½ ½
β.7Hungarian Romani minor CDE FGA B 1½ ½ ½ ½
γ.7Mixolydian inverse CDE FGA B 1 ½ ½ ½ ½

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Chromatic tonos analysis: Tetrachord swap

So let’s run an experiment. Let’s allow ourselves to swap scales with the same Greek letters, while keeping the Arabic numerals the same. That is to say, without moving any tetrachords, let’s swap scales between sets. The first set of scales will front-load the largest intervals, and the third set will back-load the largest intervals; balanced scales will go into the second set. Here’s the result:

Greek chromatic tonoi & their variants (C roots, “cyclical tetrachord swap” order)
#Chromatic Tonos 1 234567Intervals
α.6Hypophrygian CDE FGA B 1½ ½ ½ ½
α.3Phrygian CDE FGA B ½ ½ 1½ ½
γ.7Mixolydian inverse CDE FGA B 1 ½ ½ ½ ½
γ.4Dorian inverse CDE FGA B ½ ½ 1 ½ ½
γ.1Hypodorian inverse CDE FGA B ½ ½ ½ ½ 1
β.5Rasikapriyā CDE FGA B ½ 1½ ½ ½
β.2Ionian augmented 2CDE FGA B ½ ½ ½ 1½
γ.6Lydian inverse CDE FGA B ½ 1 ½ ½ ½
γ.3Hypolydian inverse CDE FGA B ½ ½ ½ 1 ½
β.7Hungarian Romani minor CDE FGA B 1½ ½ ½ ½
β.4Major Phrygian CDE FGA B ½ ½ 1½ ½
β.1Hungarian Romani minor inverse CDE FGA B ½ ½ ½ ½ 1
α.5Hypolydian CDE FGA B ½ 1½ ½ ½
α.2Lydian CDE FGA B ½ ½ ½ 1½
β.6Ultra-Phrygian CDE FGA B𝄫½ 1½ ½ ½
β.3Kanakāngi 5 CDE𝄫FGA B𝄫½ ½ ½ 1½
α.7Hypodorian CDE FGA B𝄫1½ ½ ½ ½
α.4Dorian CDE𝄫FGA B𝄫½ ½ 1½ ½
α.1Mixolydian CDE𝄫FGA𝄫B ½ ½ ½ ½ 1
γ.5Phrygian inverse CDE𝄫FGA B𝄫½ ½ 1 ½ ½
γ.2Hypophrygian inverse CDE𝄫FGA𝄫B𝄫½ ½ ½ ½ 1

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Neat. Now what if we put the scales back into something resembling linear order?

Greek chromatic tonoi & their variants (C roots, “linear tetrachord swap” order)
Chromatic Tonos 1 234567Intervals
γ.1Hypodorian inverse CDE FGA B ½ ½ ½ ½ 1
β.2Ionian augmented 2CDE FGA B ½ ½ ½ 1½
α.3Phrygian CDE FGA B ½ ½ 1½ ½
γ.4Dorian inverse CDE FGA B ½ ½ 1 ½ ½
β.5Rasikapriyā CDE FGA B ½ 1½ ½ ½
α.6Hypophrygian CDE FGA B 1½ ½ ½ ½
γ.7Mixolydian inverse CDE FGA B 1 ½ ½ ½ ½
β.1Hungarian Romani minor inverse CDE FGA B ½ ½ ½ ½ 1
α.2Lydian CDE FGA B ½ ½ ½ 1½
γ.3Hypolydian inverse CDE FGA B ½ ½ ½ 1 ½
β.4Major Phrygian CDE FGA B ½ ½ 1½ ½
α.5Hypolydian CDE FGA B ½ 1½ ½ ½
γ.6Lydian inverse CDE FGA B ½ 1 ½ ½ ½
β.7Hungarian Romani minor CDE FGA B 1½ ½ ½ ½
α.1Mixolydian CDE𝄫FGA𝄫B ½ ½ ½ ½ 1
γ.2Hypophrygian inverse CDE𝄫FGA𝄫B𝄫½ ½ ½ ½ 1
β.3Kanakāngi 5 CDE𝄫FGA B𝄫½ ½ ½ 1½
α.4Dorian CDE𝄫FGA B𝄫½ ½ 1½ ½
γ.5Phrygian inverse CDE𝄫FGA B𝄫½ ½ 1 ½ ½
β.6Ultra-Phrygian CDE FGA B𝄫½ 1½ ½ ½
α.7Hypodorian CDE FGA B𝄫1½ ½ ½ ½

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The above table clarifies why the group distribution breaks down as it does: we’ve swapped scales in such a way that across all three scale sets, we cycle between γ, β, and α, in order, seven times in a row. This sorts scales 6 and 3 to one group; scales 7, 4, and 1 to another; and scales 5 and 2 to a third.

Grouping like numbers in the above table together produces similar results to our first “aligned synaphai” table, but as one might expect, we’ve consistently front-loaded large intervals in the first scale of each trio and back-loaded them in the third. The minor thirds also move more consistently, always moving two positions earlier when they cross synaphai and one when they don’t.

(Aside: English needs an equivalent of quadrant, quintant, sextant, septant, octant, etc. for the number three.)

Greek chromatic tonoi & their variants (C roots, “aligned tetrachord swap” order)
#Chromatic Tonos 1 234567Intervals
α.1Mixolydian CDE𝄫FGA𝄫B ½ ½ ½ ½ 1
β.1Hungarian Romani minor inverse CDE FGA B ½ ½ ½ ½ 1
γ.1Hypodorian inverse CDE FGA B ½ ½ ½ ½ 1
γ.2Hypophrygian inverse CDE𝄫FGA𝄫B𝄫½ ½ ½ ½ 1
α.2Lydian CDE FGA B ½ ½ ½ 1½
β.2Ionian augmented 2CDE FGA B ½ ½ ½ 1½
β.3Kanakāngi 5 CDE𝄫FGA B𝄫½ ½ ½ 1½
γ.3Hypolydian inverse CDE FGA B ½ ½ ½ 1 ½
α.3Phrygian CDE FGA B ½ ½ 1½ ½
α.4Dorian CDE𝄫FGA B𝄫½ ½ 1½ ½
β.4Major Phrygian CDE FGA B ½ ½ 1½ ½
γ.4Dorian inverse CDE FGA B ½ ½ 1 ½ ½
γ.5Phrygian inverse CDE𝄫FGA B𝄫½ ½ 1 ½ ½
α.5Hypolydian CDE FGA B ½ 1½ ½ ½
β.5Rasikapriyā CDE FGA B ½ 1½ ½ ½
β.6Ultra-Phrygian CDE FGA B𝄫½ 1½ ½ ½
γ.6Lydian inverse CDE FGA B ½ 1 ½ ½ ½
α.6Hypophrygian CDE FGA B 1½ ½ ½ ½
α.7Hypodorian CDE FGA B𝄫1½ ½ ½ ½
β.7Hungarian Romani minor CDE FGA B 1½ ½ ½ ½
γ.7Mixolydian inverse CDE FGA B 1 ½ ½ ½ ½

So, as we can see, each interval’s relative position has a massive impact on a scale’s accidental distribution. This makes intuitive sense, but it still might be hard to understand how much it underpins a scale’s entire composition without seeing it laid out like this.

I only noticed after numbering the scales that each trio starts on the same letter that closed out the previous trio, then cycles through the others in ascending order (resetting to α after γ). That actually explains a lot.

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Chromatic tonos analysis: Minor third position

Next set of comparisons: by shifting Hypophrygian and Phrygian to the end of the chromatic tonoi, and Phrygian inverse and Hypophrygian inverse to the start of the inverse chromatic tonoi, we align the minor thirds instead of the synaphai. This resembles the first set of scales while increasing the similarity of the scale ordering between sets. I also swapped the inverse chromatic and chromatic scales’ positions from the “circle of fifths” order comparisons; I’ll explain why below.

Greek chromatic tonoi & their variants (C roots, “cyclical aligned minor thirds” order)
#Chromatic Tonos 1 234567Intervals
γ.5Phrygian inverse CDE𝄫FGA B𝄫½ ½ 1 ½ ½
γ.2Hypophrygian inverse CDE𝄫FGA𝄫B𝄫½ ½ ½ ½ 1
γ.6Lydian inverse CDE FGA B ½ 1 ½ ½ ½
γ.3Hypolydian inverse CDE FGA B ½ ½ ½ 1 ½
γ.7Mixolydian inverse CDE FGA B 1 ½ ½ ½ ½
γ.4Dorian inverse CDE FGA B ½ ½ 1 ½ ½
γ.1Hypodorian inverse CDE FGA B ½ ½ ½ ½ 1
β.6Ultra-Phrygian CDE FGA B𝄫½ 1½ ½ ½
β.3Kanakāngi 5 CDE𝄫FGA B𝄫½ ½ ½ 1½
β.7Hungarian Romani minor CDE FGA B 1½ ½ ½ ½
β.4Major Phrygian CDE FGA B ½ ½ 1½ ½
β.1Hungarian Romani minor inverse CDE FGA B ½ ½ ½ ½ 1
β.5Rasikapriyā CDE FGA B ½ 1½ ½ ½
β.2Ionian augmented 2CDE FGA B ½ ½ ½ 1½
α.7Hypodorian CDE FGA B𝄫1½ ½ ½ ½
α.4Dorian CDE𝄫FGA B𝄫½ ½ 1½ ½
α.1Mixolydian CDE𝄫FGA𝄫B ½ ½ ½ ½ 1
α.5Hypolydian CDE FGA B ½ 1½ ½ ½
α.2Lydian CDE FGA B ½ ½ ½ 1½
α.6Hypophrygian CDE FGA B 1½ ½ ½ ½
α.3Phrygian CDE FGA B ½ ½ 1½ ½

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Between the three sets of scales, this set of comparisons simply moves each synaphe one position to the left (e.g., Phrygian inverse to Ultra-Phrygian, Ultra-Phrygian to Hypodorian). This becomes especially clear when we rearrange them into linear order:

Greek chromatic tonoi & their variants (C roots, “linear aligned minor thirds” order)
#Chromatic Tonos 1 234567Intervals
γ.7Mixolydian inverse CDE FGA B 1 ½ ½ ½ ½
γ.1Hypodorian inverse CDE FGA B ½ ½ ½ ½ 1
γ.2Hypophrygian inverse CDE𝄫FGA𝄫B𝄫½ ½ ½ ½ 1
γ.3Hypolydian inverse CDE FGA B ½ ½ ½ 1 ½
γ.4Dorian inverse CDE FGA B ½ ½ 1 ½ ½
γ.5Phrygian inverse CDE𝄫FGA B𝄫½ ½ 1 ½ ½
γ.6Lydian inverse CDE FGA B ½ 1 ½ ½ ½
β.1Hungarian Romani minor inverse CDE FGA B ½ ½ ½ ½ 1
β.2Ionian augmented 2CDE FGA B ½ ½ ½ 1½
β.3Kanakāngi 5 CDE𝄫FGA B𝄫½ ½ ½ 1½
β.4Major Phrygian CDE FGA B ½ ½ 1½ ½
β.5Rasikapriyā CDE FGA B ½ 1½ ½ ½
β.6Ultra-Phrygian CDE FGA B𝄫½ 1½ ½ ½
β.7Hungarian Romani minor CDE FGA B 1½ ½ ½ ½
α.2Lydian CDE FGA B ½ ½ ½ 1½
α.3Phrygian CDE FGA B ½ ½ 1½ ½
α.4Dorian CDE𝄫FGA B𝄫½ ½ 1½ ½
α.5Hypolydian CDE FGA B ½ 1½ ½ ½
α.6Hypophrygian CDE FGA B 1½ ½ ½ ½
α.7Hypodorian CDE FGA B𝄫1½ ½ ½ ½
α.1Mixolydian CDE𝄫FGA𝄫B ½ ½ ½ ½ 1

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Now let’s compare across sets with the minor thirds aligned. This is why I swapped the chromatic scales and their inversions: if I hadn’t, we’d be moving the synaphai right rather than left.

Greek chromatic tonoi & their variants (C roots, “doubly aligned minor thirds” order)
Chromatic Tonos 1 234567Intervals
γ.7Mixolydian inverse CDE FGA B 1 ½ ½ ½ ½
β.1Hungarian Romani minor inverse CDE FGA B ½ ½ ½ ½ 1
α.2Lydian CDE FGA B ½ ½ ½ 1½
γ.1Hypodorian inverse CDE FGA B ½ ½ ½ ½ 1
β.2Ionian augmented 2CDE FGA B ½ ½ ½ 1½
α.3Phrygian CDE FGA B ½ ½ 1½ ½
γ.2Hypophrygian inverse CDE𝄫FGA𝄫B𝄫½ ½ ½ ½ 1
β.3Kanakāngi 5 CDE𝄫FGA B𝄫½ ½ ½ 1½
α.4Dorian CDE𝄫FGA B𝄫½ ½ 1½ ½
γ.3Hypolydian inverse CDE FGA B ½ ½ ½ 1 ½
β.4Major Phrygian CDE FGA B ½ ½ 1½ ½
α.5Hypolydian CDE FGA B ½ 1½ ½ ½
γ.4Dorian inverse CDE FGA B ½ ½ 1 ½ ½
β.5Rasikapriyā CDE FGA B ½ 1½ ½ ½
α.6Hypophrygian CDE FGA B 1½ ½ ½ ½
γ.5Phrygian inverse CDE𝄫FGA B𝄫½ ½ 1 ½ ½
β.6Ultra-Phrygian CDE FGA B𝄫½ 1½ ½ ½
α.7Hypodorian CDE FGA B𝄫1½ ½ ½ ½
γ.6Lydian inverse CDE FGA B ½ 1 ½ ½ ½
β.7Hungarian Romani minor CDE FGA B 1½ ½ ½ ½
α.1Mixolydian CDE𝄫FGA𝄫B ½ ½ ½ ½ 1

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Now, we’ve effectively just reversed the first “aligned synaphai” table’s pattern. That table moved the minor third one degree left twice in a row, then moved it one degree right and the synaphe one degree left. This one moves the synaphe one degree left twice in a row, then moves it one degree right and the minor third one degree left. Another way to say this is that by rotating Mixolydian to the end of the chromatic tonoi and Mixolydian inverse to the front of the inverse chromatic tonoi, we’ve swapped the synaphai and tetrachords’ movement patterns.

One curiosity here is how much Mixolydian and Mixolydian inverse differ from the two scales immediately below and above them, respectively. Most other scale trios remain fairly consistent in note composition; these two are the exceptions. In fact, Mixolydian really behaves more like the third and sixth trios, and Mixolydian inverse really behaves more like the second and fifth. I understand why, but it’s still slightly surreal to see it laid out like this. I can think of at least three explanations that clarify why this occurs:

  1. Mixolydian inverse is one of this table’s only two scales that start with a consecutive minor third and major second, in either order, and Mixolydian is one of its only two scales that end with those intervals.
  2. We can just refer back to the “linear tetrachord swap” table and see that its first seven scales consist of this table’s second trio, fifth trio, and Mixolydian inverse; its last seven scales consist of Mixolydian followed by this table’s third trio and sixth trio; and its middle seven scales consist of this table’s second scale, third scale, middle three scales, third-from-last scale, and second-from-last scale;.
  3. We can just look at the interval distributions here. The first and last scale trios are the only ones where a major second moves from the front of the scale to the end, or vice versa. Thus, all the other scale trios have much more consistent note distributions.

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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 about some aspects of Greek harmony. There are actually multiple possible sources of this, and I’m not totally sure which one was at fault, but I’ll present a couple of ways a person could wind up with the scale names they got.

One possibility is that they erroneously thought the Greeks described their tetrachords in ascending order. Four of our modern modes also had multiple names, and they borrowed three of these from the Ancient Greek tonoi: their Hypodorian was our Aeolian, their Hypophrygian was our Locrian, and their Hypolydian was our Ionian.

Medieval names for the Greek diatonic tonoi
AncientMedievalModern1 234567Intervals
Mixolydian HypophrygianLocrian CDEFGAB½ 1 1 ½ 1 1 1
Lydian Hypolydian Ionian CDEFGAB1 1 ½ 1 1 1½
Phrygian Dorian Dorian CDEFGAB1 ½ 1 1 1½ 1
Dorian Phrygian Phrygian CDEFGAB½ 1 1 1½ 1 1
Hypolydian Lydian Lydian CDEFGAB1 1 1½ 1 1 ½
HypophrygianMixolydian MixolydianCDEFGAB1 1½ 1 1 ½ 1
Hypodorian Hypodorian Aeolian CDEFGAB1½ 1 1 ½ 1 1

So, what if we were to purchase fast food and disguise it as our own cooking reverse the order of the notes within each tetrachord?

Inverting the Greek diatonic tetrachord
OriginalReversedModern1 234567Intervals
Mixolydian Mixolydian Mixolydian CDEFGAB1 1 ½ 1 1 ½ 1
Lydian Hypodorian Aeolian CDEFGAB1 ½ 1 1 ½ 1 1
Phrygian Hypophrygian Locrian CDEFGAB½ 1 1 ½ 1 1 1
Dorian Hypolydian Ionian CDEFGAB1 1 ½ 1 1 1 ½
Hypolydian Dorian Dorian CDEFGAB1 ½ 1 1 1 ½ 1
Hypophrygian Phrygian Phrygian CDEFGAB½ 1 1 1 ½ 1 1
Hypodorian Lydian Lydian CDEFGAB1 1 1 ½ 1 1 ½

So, effectively, for Mixolydian, they put the tetrachords in the right parts of the scale, but they put the intervals within each tetrachord in the wrong order. Then, they put the remaining modes in the opposite of the Ancient Greeks’ order, likely assuming that they were rotating the scale in the opposite direction.

Remember, the ancient Greeks’ metaphor for time inverted the modern one: they thought of the past as being in front and the future as being behind them. I don’t know if this was the source of medieval Europeans’ confusion, but it wouldn’t entirely shock me if it were.

Now, remember Chekhov’s table near the start of this document? This is another potential source of their confusion. I’ll reprint a variant, this time depicting both the blueshifted (ascending) and redshifted (descending) scale transformations. In this case, we care about the redshift, since Greek harmony went in descending order.

Mode transformations re-re-revisited
# ModeMode 12 3 4 5 6 7
1Ionian Ionian
2Dorian Dorian 3 7
3Phrygian Phrygian 23 67
4Lydian Lydian 4
5Mixolydian Mixolydian 7
6Aeolian Aeolian 3 67
7Locrian Locrian 23 567

The renamed modes aside, each modern name is the Greek name for the mode of the note we redshift in circle of fifths order to get the first mode. That is:

Modes and the notes they redshift
# Modern Medieval Greek Redshift
1 Ionian Hypolydian Lydian 4Lydian
2 Dorian Dorian Phrygian 3Phrygian
3 Phrygian Phrygian Dorian 2Dorian
4 Lydian Lydian Hypolydian 1Ionian
5 Mixolydian Mixolydian Hypophrygian 7Locrian
6 Aeolian Hypodorian Hypodorian 6Aeolian
7 Mixolydian Locrian Hypophrygian 5Mixolydian

Perhaps medieval scholars were confused on this point; perhaps it’s just coincidence.

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Applied Greek harmony: Tetrachords in modern scales

We’ve already seen how the Ionian 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 Ionian scale’s familiarity. Symmetrical scales built on two of the same tetrachord (e.g., double harmonic minor 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 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. (Melodic minor does possess a symmetrical mode, Aeolian dominant [W-W-H-W-H-W-W]; harmonic minor does not.)

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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, via Cris Forester’s book on the subject ⟨chrysalis-foundation.org/musical-mathematics-pages/philolaus-and-euclid⟩, here’s 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 Ionian 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 a direct ancestor of our Ionian 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.

I also wish to acknowledge a few resources for the Greek language itself. Wiktionary is low key one of the best online resources for learning languages; it contains a wealth of information on Greek declensions, conjugations, and vocabulary. Λογεῖον and the Liddell, Scott, Jones wiki capably filled gaps in Wiktionary’s coverage. I’m by no means fluent in Attic Greek, but I’ve managed to write lyrics in it that don’t completely embarrass me. (Here’s the song itself if you want to listen to it.) I’d never have managed that without such comprehensive lexicons.

(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.)

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Appendix 1: Greek musical terminology

It might seem like overkill to include a table this repetitive, but Google Translate is not great at parsing Ancient Greek. My hope is that this will help.

This table focuses exclusively on musical meanings of terms; many have other meanings as well. For instance, the lyre’s three strings are named after the three Muses.

Λεξικογραφία Ἑλληνῐκῶν μουσῐκῶν ῐ̓́δῐογλῶσσῐῶν
Lexikographía Hellēnĭkôn mousĭkôn ĭ́dĭóglôssĭôn
A lexicography of Ancient Greek musical idioglossia
Ἀττικός Ἑλληνική
Attĭkós Hellēnĭkḗ
Attic Greek
Ῥωμᾰῐ̈σμένη
Rhṓmăĭ̈sméni
Romanized
Μετάφρασις
Metáphrasis
Translation
ἐναρμόνιος μιξολῡ́δῐος τόνος
enarmónios mixolū́dĭos tónos
enharmonic Mixolydian tonos
ἐναρμόνιος Λῡ́δῐος τόνος enarmónios Lū́dĭos tónos enharmonic Lydian tonos
ἐναρμόνιος Φρῠ́γῐος τόνος enarmónios Phrŭ́gios tónos enharmonic Phrygian tonos
ἐναρμόνιος Δώριος τόνος enarmónios Dṓrios tónos enharmonic Dorian tonos
ἐναρμόνιος ὑπολύδῐος τόνος enarmónios hŭpolū́dĭos tónos enharmonic Hypolydian tonos
ἐναρμόνιος ὑποφρῠ́γῐος τόνος enarmónios hŭpophrŭ́gios tónos enharmonic Hypophrygian tonos
ἐναρμόνιος ὑποδώριος τόνος enarmónios hŭpodṓrios tónos enharmonic Hypodorian tonos
χρωμᾰτῐκός μιξολῡ́δῐος τόνος
khrōmătĭkós mixolū́dĭos tónos
chromatic Mixolydian tonos
χρωμᾰτῐκός Λῡ́δῐος τόνος khrōmătĭkós Lū́dĭos tónos chromatic Lydian tonos
χρωμᾰτῐκός Φρῠ́γῐος τόνος khrōmătĭkós Phrŭ́gios tónos chromatic Phrygian tonos
χρωμᾰτῐκός Δώριος τόνος khrōmătĭkós Dṓrios tónos chromatic Dorian tonos
χρωμᾰτῐκός ὑπολύδῐος τόνος khrōmătĭkós hŭpolū́dĭos tónos chromatic Hypolydian tonos
χρωμᾰτῐκός ὑποφρῠ́γῐος τόνος khrōmătĭkós hŭpophrŭ́gios tónos chromatic Hypophrygian tonos
χρωμᾰτῐκός ὑποδώριος τόνος khrōmătĭkós hŭpodṓrios tónos chromatic Hypodorian tonos
διατονικός μιξολῡ́δῐος τόνος
diatonikós mixolū́dĭos tónos
diatonic Mixolydian tonos
διατονικός Λῡ́δῐος τόνος diatonikós Lū́dĭos tónos diatonic Lydian tonos
διατονικός Φρῠ́γῐος τόνος diatonikós Phrŭ́gios tónos diatonic Phrygian tonos
διατονικός Δώριος τόνος diatonikós Dṓrios tónos diatonic Dorian tonos
διατονικός ὑπολύδῐος τόνος diatonikós hŭpolū́dĭos tónos diatonic Hypolydian tonos
διατονικός ὑποφρῠ́γῐος τόνος diatonikós hŭpophrŭ́gios tónos diatonic Hypophrygian tonos
διατονικός ὑποδώριος τόνος diatonikós hŭpodṓrios tónos diatonic Hypodorian tonos
μουσικά mousiká music
μουσικός mousikós musically skilled, musical
ἁρμονίᾱ harmoníā harmony
διαπασῶν diapasôn octave (lit. “through all”)
διπλόος diplóos double, 2:1 ratio
δῐοξειῶν dĭoxeiôn perfect fifth
ἡμιόλιος hēmiólios 1½, 3:2 ratio
σῠλλᾰβή sŭllăbḗ perfect fourth
ἐπίτριτος epítritos 1⅓, 4:3 ratio
τρῐ́τος, τρῐ́τη trĭ́tos, trĭ́tē third
ἐπόγδοος epógdoos 1⅛, 9:8 ratio
δίεσις díesis a scale’s smallest interval
ὑπάτη hupátē lyre’s lowest-pitched string
παραμέση paramésē second-lowest-pitched string
μέση mésē lyre’s middle string
νήτη, νεάτη nḗtē, neátē lyre’s highest-pitched string

…OK, fine, I completely made up the declension of «ῐ̓́δῐογλῶσσῐῶν», but to be fair, it wouldn’t have sufficed at all to have used a modern declension when all the surrounding language is Attic.

(For the time being, a complete explanation of declensions is beyond my scope, but I may eventually find myself unable to resist writing one.)

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Appendix 2: Interval ratios of 12- and 24-tone equal temperament

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

24-tone equal temperament’s interval ratios
#IntervalExactApproximateLPT
1 Quarter tone, infra second 2¹⁄₂₄ =²⁴2 1.0293022366424
2 Minor second 2²⁄₂₄ = ¹²2 1.0594630943612
3 Neutral second 2³⁄₂₄ =2 1.09050773267 8
4 Major second 2⁴⁄₂₄ =2 1.12246204831 6
5 Ultra second, infra third 2⁵⁄₂₄ =²⁴32 1.1553526968724
6 Minor third 2⁶⁄₂₄ =2 1.18920711500 4
7 Neutral third 2⁷⁄₂₄ =²⁴128 1.2240535433024
8 Major third 2⁸⁄₂₄ = ³2 1.25992104989 3
9 Ultra third, narrow fourth 2⁹⁄₂₄ =8 1.29683955465 8
10Perfect fourth 2¹⁰⁄₂₄= ¹²32 1.3348398541712
11Wide fourth 2¹¹⁄₂₄ =²⁴2,048 1.3739536474624
12Tritone 2¹²⁄₂₄ =2 1.41421356237 2
13Narrow fifth 2¹³⁄₂₄ =²⁴8,192 1.4556531828424
14Perfect fifth 2¹⁴⁄₂₄= ¹²128 1.4983070768812
15Wide fifth, infra sixth 2¹⁵⁄₂₄=32 1.54221082541 8
16Minor sixth 2¹⁶⁄₂₄= ³4 1.58740105197 3
17Neutral sixth 2¹⁷⁄₂₄=²⁴131,072 1.6339154532424
18Major sixth 2¹⁸⁄₂₄=8 1.68179283051 4
19Ultra sixth, infra seventh 2¹⁹⁄₂₄=²⁴524,288 1.7310731220124
20Minor seventh 2²⁰⁄₂₄=32 1.78179743628 6
21Neutral seventh 2²¹⁄₂₄ =128 1.83400808641 8
22Major seventh 2²²⁄₂₄ = ¹²2,048 1.8877486253612
23Ultra seventh, narrow octave2²³⁄₂₄ =²⁴8,388,6081.9430638823124
24Octave 2²⁴⁄₂₄= 2 2 1

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Embryonic surveys of other musical traditions

Back to Babylon, or, the Mesopotamian major scale

The Greeks didn’t even invent the Ionian scale: the Hurrian songs from ca. 1400 BCE Mesopotamia (which survive in fragments) conform to its interval spacing, and surviving Mesopotamian literature on tuning (from 1800 BCE and earlier) is so voluminous that scholars even believe they know the Mesopotamian modes’ correspondence to ours. Musicologist and Assyriologist Anne Draffkorn Kilmer (1931-2023), who deciphered the oldest and most complete surviving Hurrian song, identified the modes in 2014 as:

Mesopotamian modes
AkkadianTranslationGreekModern
IšartuNormalLydianIonian
EmbūbuReed pipePhrygianDorian
Nīd qabliFall of the middleDorianPhrygian
QablītuMiddleHypolydianLydian
KitmuClosedHypophrygianMixolydian
PītuOpenHypodorianAeolian
Nīš tuhriRise of the Achilles tendonMixolydianLocrian

Ironically, until roughly 1990, scholars had apparently repeated medieval Europeans’ Greek scale order mistake: as Kilmer explained, Mesopotamians, like the Greeks, defined scales in descending order, and scholars had once again assumed they were in ascending order. (I understand a fair amount of Greek, but not a single word of Akkadian, so I can do little but summarize others’ scholarship in this subsection.) Kilmer writes that scholars had previously read a fragmented treatise on tuning to mean that strings should be loosened, then tightened, but it was later reread to mean they should be tightened, then loosened, which clarified that it was describing notes in descending order. Thus:

The resulting changes in translation […] turned Gurney’s (1968) “If the harp is (tuned as) X, and the interval Y is not clear; you alter the (string) N, and then Y will be…” into his new (Gurney 1994) rendering (lines 1–12), “If the instrument is (tuned as) X, and the (interval) Y is not clear, you tighten the (string) N, and then Y will be clear.” [Editor’s note: The Mesopotamians called the tritone impure or unclear (Akkadian: la zakû).] The preceding procedures were summed up as “tightening.” The second tuning section of the same text is now translated as follows: (lines 13–20) “If the instrument is (tuned as) X, and you have played an (unclear) interval Y, you loosen the string N and the instrument will be (in the tuning) Z.” The second section was presumably and logically summed up as: “[loosening]”. This newer interpretation is generally accepted today.

Ironically, this means that what caused me to start writing this entire book was independently rediscovering what had been the cornerstone of Mesopotamian tuning practice four millennia earlier. Kilmer continues:

That the seven Mesopotamian musical scales (at least as early as ca. 1800 bce) were heptatonic-diatonic scales has been proven to the satisfaction of cuneiformists and musicologists alike. It should be noted here that, thanks to the observations of Wulstan (1968) and Kümmel (1970), it was recognized that the ancient Mesopotamian musicians/“musicologists” knew what we call today the Pythagorean series of fifths, and that the series could be accomplished within a single octave by means of “inversion.” Kümmel taught us that the names given to the seven tunings/scales were derived from the specific intervals on which the tuning procedure started. For example, if the tuning procedure started with the interval išartu “normal”, the resulting scale was called išartu.

However, the change in our recognizing of the scales as moving downward rather than upward means that išartu 2–6 was no longer string 2 (RE) up to string 6 (LA), but rather 2–6 as 2 (TI) down to 6 (MI). As Crocker (1997: 195) emphasized, “The principal difference brought about by Krispijn’s restoration (of nu-su-h[u-um]) is that the seven octave segments (or intervals) receive different names” (than those they bore earlier). Thus, nīd qabli “fall from the middle” was, before Krispijn (1990), the scale from C-C (ascending). After 1990, it is E-E (descending).

Note that, as of this writing (2025-10-09), Wikipedia’s article on Mesopotamian music theory still relies on sources that predate this re-reading, so it uses the incorrect ascending modes (and nīš tuhri was still read as nīš gabarî, “rise of the duplicate”). I plan to fix it once I’ve had the chance to read Kilmer’s sources.

The Chinese independently invented a twelve-note chromatic scale called Shi’er lü (十二律) somewhere between 600 BCE and 250 BCE; it uses the same ratios as the Mesopotamian and Pythagorean scales (3:2, 4:3, 9:8, 32:27, 81:64, etc.) Shi’er lü was not used as a scale in its own right, but it was used as a basis on which to construct other scales, more comparable to the way in which Indian classical music constructs rāgas. On which note…

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The carnatic numbered mēḷakartā

A numbered set of fundamental rāgasa (musical scales) that originated in carnatic music (South Indian classical music). They must obey a few rules:

Many of these correspond exactly to frequently used Western scales (e.g., #8 is Phrygian, #20 is Aeolian, #21 is harmonic minor, #22 is Dorian, #23 is melodic minor, #28 is Mixolydian, #29 is Ionian, #65 is Lydian); however, many others are virtually unique to Indian music, and some fundamental modes of Western music (e.g., Locrian mode) are absent here, as they break some of the fundamental rules of mēḷakartā.

Indian music uses multiple scales, and although there are commonly held to be 22 shruti per octave, this remains a matter of some debate, since in practice, pitch tends to vary somewhat. To avoid confusion, I’m therefore printing the swaras’ Western names.

The carnatic numbered mēḷakartā
#Mode 1 234567Intervals
1Kanakangi CDE𝄫FGAB𝄫½ ½ 1 ½ ½
2Ratnāngi CDE𝄫FGAB ½ ½ 1 ½ 1 1
3Gānamūrti CDE𝄫FGAB ½ ½ 1 ½ ½
4Vanaspati CDE𝄫FGAB ½ ½ 1 1 ½ 1
5Mānavati CDE𝄫FGAB ½ ½ 1 1 1 ½
6Tānarūpi CDE𝄫FGAB ½ ½ 1 ½ ½
7Senāvati CDE FGAB𝄫½ 1 1 1 ½ ½
8Hanumatodi CDE FGAB ½ 1 1 1 ½ 1 1
9Dhenukā CDE FGAB ½ 1 1 1 ½ ½
10Nātakapriyā CDE FGAB ½ 1 1 1 1 ½ 1
11Kokilapriya CDE FGAB ½ 1 1 1 1 1 ½
12Rūpavati CDE FGAB ½ 1 1 1 ½ ½
13Gāyakapriyā CDE FGAB𝄫½ ½ 1 ½ ½
14Vakuḷābharaṇam CDE FGAB ½ ½ 1 ½ 1 1
15Māyāmāḻavagowla CDE FGAB ½ ½ 1 ½ ½
16Chakravākam CDE FGAB ½ ½ 1 1 ½ 1
17Sūryakāntam CDE FGAB ½ ½ 1 1 1 ½
18Hātakāmbari CDE FGAB ½ ½ 1 ½ ½
19Jhankāradhvani CDE FGAB𝄫1 ½ 1 1 ½ ½
20Naṭabhairavi CDE FGAB 1 ½ 1 1 ½ 1 1
21Kīravāṇi CDE FGAB 1 ½ 1 1 ½ ½
22Kharaharapriyā CDE FGAB 1 ½ 1 1 1 ½ 1
23Gourimanohari CDE FGAB 1 ½ 1 1 1 1 ½
24Varuṇapriyā CDE FGAB 1 ½ 1 1 ½ ½
25Māraranjani CDE FGAB𝄫1 1 ½ 1 ½ ½
26Chārukesi CDE FGAB 1 1 ½ 1 ½ 1 1
27Sarasāngi CDE FGAB 1 1 ½ 1 ½ ½
28Harikāmbhōji CDE FGAB 1 1 ½ 1 1 ½ 1
29DhīraśankarābharaṇamCDE FGAB 1 1 ½ 1 1 1 ½
30Nāganandini CDE FGAB 1 1 ½ 1 ½ ½
31Yāgapriyā CDE FGAB𝄫½ ½ 1 ½ ½
32Rāgavardhini CDE FGAB ½ ½ 1 ½ 1 1
33Gāngeyabhuśani CDE FGAB ½ ½ 1 ½ ½
34Vāgadhīśvari CDE FGAB ½ ½ 1 1 ½ 1
35Śūlini CDE FGAB ½ ½ 1 1 1 ½
36Chalanāṭa CDE FGAB ½ ½ 1 ½ ½
37Sālagam CDE𝄫FGAB𝄫½ ½ 2 ½½ ½
38Jalārnavam CDE𝄫FGAB ½ ½ 2 ½½ 1 1
39Jhālavarāḷi CDE𝄫FGAB ½ ½ 2 ½½ ½
40Navanītam CDE𝄫FGAB ½ ½ 2 ½1 ½ 1
41Pāvani CDE𝄫FGAB ½ ½ 2 ½1 1 ½
42Raghupriyā CDE𝄫FGAB ½ ½ 2 ½½ ½
43Gavāmbhodi CDE FGAB𝄫½ 1 ½½ ½
44Bhavapriyā CDE FGAB ½ 1 ½½ 1 1
45Śubhapantuvarāḷi CDE FGAB ½ 1 ½½ ½
46Shaḍvidamārgini CDE FGAB ½ 1 ½1 ½ 1
47Suvarnāngi CDE FGAB ½ 1 ½1 1 ½
48Divyamaṇi CDE FGAB ½ 1 ½½ ½
49Dhavaḻāmbari CDE FGAB𝄫½ 1 ½½ ½
50Nāmanārāyaṇi CDE FGAB ½ 1 ½½ 1 1
51Kāmavardhini CDE FGAB ½ 1 ½½ ½
52Rāmapriyā CDE FGAB ½ 1 ½1 ½ 1
53Gamanāśrama CDE FGAB ½ 1 ½1 1 ½
54Viśvambari CDE FGAB ½ 1 ½½ ½
55Śāmaḻāngi CDE FGAB𝄫1 ½ ½½ ½
56Śanmukhapriyā CDE FGAB 1 ½ ½½ 1 1
57Simhendramadhyamam CDE FGAB 1 ½ ½½ ½
58Hemavati CDE FGAB 1 ½ ½1 ½ 1
59Dharmavati CDE FGAB 1 ½ ½1 1 ½
60Nītimati CDE FGAB 1 ½ ½½ ½
61Kāntāmaṇi CDE FGAB𝄫1 1 1 ½½ ½
62Riśabhapriyā CDE FGAB 1 1 1 ½½ 1 1
63Latāngi CDE FGAB 1 1 1 ½½ ½
64Vāchaspati CDE FGAB 1 1 1 ½1 ½ 1
65Mechakalyāni CDE FGAB 1 1 1 ½1 1 ½
66Chitrāmbari CDE FGAB 1 1 1 ½½ ½
67Sucharitrā CDE FGAB𝄫½ 1 ½½ ½
68Jyotisvarupini CDE FGAB ½ 1 ½½ 1 1
69Dhāthuvardhani CDE FGAB ½ 1 ½½ ½
70Nāsikābhūśaṇi CDE FGAB ½ 1 ½1 ½ 1
71Kōsalam CDE FGAB ½ 1 ½1 1 ½
72Rasikapriyā CDE FGAB ½ 1 ½½ ½

The numbering consistently obeys several patterns:

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Scale counts in 12-TET by scale size

How to count scales

In twelve-tone equal temperament, by definition, heptatonic scales contain the root and six of the eleven other pitches. Since (as we proved above) heptatonic and pentatonic scales can’t be modes of limited transposition in 12-TET, we can thus calculate the total number of scales using the formula (11
6
) = 11!
6!5!
= 11·10·9·8·7
5·4·3·2
= 11·7·3·2 = 462. Since 462 / 7, there are 66 discrete scales with seven modes each.

Meanwhile, the number of pentatonic scales in 12-TET is (11
4
) = 11!
4!7!
= 11·10·9·8
4·3·2
= 11·5·3·2 = 330. If we discount modes, 330 / 5 also leaves us with 66 discrete pentatonic scales in 12-TET. This is no coincidence: since 12-TET’s heptatonic and pentatonic scales can’t be modes of limited transposition, every pentatonic scale in 12-TET has five modes, every heptatonic scale in 12-TET has seven modes, and every pentatonic scale in 12-TET is, by definition, the scale complement of a heptatonic scale.

As it turns out, 12-tone equal temperament contains the following mode counts for each scale size:

12-TET mode counts
NotesModes
1121
21111
31055
49165
58330
67462
Total2,048

Because sixteen scales are modes of limited transposition, however, we can’t simply divide each of those scale counts by the number of notes in the scale. The following table’s “−” column denotes the number of missing modes. As mentioned above, counting truncations and the chromatic scale, 12-TET contains 16 discrete modes of limited transposition with 38 modes between them; if they were not modes of limited transposition, they’d have a total of 102 modes, so they’re short by 64. (2,048 + 64 = 2,112 ⟨youtu.be/w5jwxrTqoEA⟩.)

All modes of limited transpositionModes of limited transposition
#Intervals#Intervals
12 111111111111 11 6 222222 5
10 1111211112
1112111121
1121111211
1211112111
2111121111
5 131313
313131
4
114114
141141
411411
3
9 112112112
121121121
211211211
6 123123
231231
312312
3
8 12121212
21212121
6 132132
213213
321321
3
11131113
11311131
13111311
31113111
4
4 3333 3
1515
5151
2
11221122
12211221
22112211
21122112
4 2424
4242
2
3 444 2
2 66 1
12 111111111111 11
10 1111211112
1112111121
1121111211
1211112111
2111121111
5
9 112112112
121121121
211211211
6
8 12121212
21212121
6
11131113
11311131
13111311
31113111
4
11221122
12211221
22112211
21122112
4
6 222222 5
131313
313131
4
114114
141141
411411
3
123123
231231
312312
3
132132
213213
321321
3
4 3333 3
1515
5151
2
2424
4242
2
3 444 2
2 66 1

Thus, with modes excluded, 12-tone equal temperament’s discrete scales break down as follows:

12-TET discrete scales
NotesScales
111121
2106
3919
4843
5766
680
Total351

I’m not yet sure if it’s coincidental that modes of limited transposition are responsible for 64 missing modes and that 64 scale modes possess internal reflective symmetry. I have a very strong hunch that it is not.

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All 2,048 modes of all 351 scales in 12-TET

I’ve listed all 2,048 modes of 12-tone equal temperament’s 351 scales. Due to the sheer quantity of data, I’ve put this on its own page, but I consider it an extension of this book. (As of 2025-10-09, the scale list is ≈2,200 lines, ≈34,000 words, and ≈135,000 characters; this page is ≈4,750 lines, ≈67,000 words, and ≈350,000 characters. This adds up to ≈6,950 lines, ≈101,000 words, and ≈485,000 characters between both.)

  1. 1 scale, 1 mode
  2. 6 scales, 11 modes
  3. 19 scales, 55 modes
  4. 43 scales, 165 modes
  5. 66 scales, 330 modes
  6. 80 scales, 462 modes
  7. 66 scales, 462 modes
  8. 43 scales, 330 modes
  9. 19 scales, 165 modes
  10. 6 scales, 55 modes
  11. 1 scale, 11 modes
  12. 1 scale, 1 mode

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