Every Possible Mode in 12-Tone Equal Temperament

a reference by Aaron Freed

Contents

1. Brief introduction
   1. Key to scale identifiers
      1. Letter ordering
      2. IDs of common scales
   2. Key to interval listings & inversions
      1. Interval key
   3. Key to interval inversions
      1. Interval inversion key
2. The scales
   1. 1 note (octave)
   2. 2 notes (6 scales, 11 modes)
   3. 3 notes (19 scales, 55 modes)
   4. Tetratonic (4 notes, 43 scales, 165 modes)
   5. Pentatonic (5 notes, 66 scales, 330 modes)
   6. Hexatonic (6 notes, 80 scales, 462 modes)
   7. Heptatonic (7 notes, 66 scales, 462 modes)
   8. Octatonic (8 notes, 43 scales, 330 modes)
   9. Enneatonic (9 notes, 19 scales, 165 modes)
   10. Decatonic (10 notes, 6 scales, 55 modes)
   11. Hendecatonic (11 notes, 1 scale, 11 modes)
   12. Dodecatonic (12 notes, chromatic scale)

Brief notes

An appendix to my book on musical modes and scales: a complete list of all 2,048 modes of all 351 scales in twelve-tone equal temperament, the primary tuning system used in Western music. The links to each mode go to Ian Ring’s page, an invaluable resource for scale analysis.

I define modes in more detail in my primer on music theory and in my book on modes, but the short definition of modes is that they rotate a scale’s intervals either forward or backward. Thus, the Ionian (major) scale is WWHWWWH; Dorian (WHWWWHW) rotates its intervals one degree left, while Locrian (HWWHWWW) rotates them one degree right. These are all the same scale; they’re just different modes.

Usually, a scale has as many modes as notes, but scales with repeating interval patterns (termed “modes of limited transposition”) have fewer modes. A nice bit of trivia for you Rush fans out there: If none of 12-TET’s 351 scales were modes of limited transposition, it would contain 2,112 modes.

Please contact me if you notice any errors or omissions.

Key to scale identifiers

When reading scale codes, bear in mind that the following letters correspond to the following numbers:

(Lowercase sigma is ς at the end of a word and σ in all other cases.)

I list four forms of identification for each scale:

1. The numbering of MID (“Mode ID”) is inspired by South India’s Carnatic Numbered Melakarta, in a musical equivalent of alphabetical order: the lower a scale’s notes, the lower its number (and, as with alphabetization, it goes from left to right). The mode ID consists of a letter from A to L corresponding to the number of notes in the scale, followed by three digits corresponding to how front-loaded the notes are.

2. SID (“Scale ID”) groups scales by the same principle, using a Latin letter to indicate the number of notes, a two-digit scale number, a full stop, and a Greek letter signifying the scale mode. The numbering depends on the prime mode – the most front-loaded with notes. For instance, Ionian’s prime mode is Locrian (which lowers Ionian’s second, third, fifth, sixth, and seventh notes), and the pentatonic scale’s prime mode is major pentatonic (since its other modes all raise at least one note by a semitone). The scales are in turn numbered by which prime modes are the most front-loaded. This has some interesting consequences:

   • The first scale is always a truncation of the chromatic scale.
   • The last scale always has the most even interval distribution a scale of that size can have in twelve-tone equal temperament and, if possible, is a mode of limited transposition. (Five-note, seven-note, and eleven-note modes of limited transposition are impossible in twelve-tone equal temperament).

   For example, the scale ID G42.η means:

   1. G: Since G is the English alphabet’s seventh letter, this is a seven-note scale.
   2. 42: This is the forty-second seven-note scale.
   3. η: Since eta is the Greek alphabet’s seventh letter, this is the scale’s seventh mode, counting the prime mode as its first. Thus, this is six scale rotations left (or one scale rotation right) of the prime mode.

   G42.η is Rāga Gangatarangini, 33 in the carnatic numbered mēḷakartā, which equates to harmonic major ♯2 (i.e., raise the second note by a semitone. Harmonic major, in turn, is Ionian ♭6).

   Because any equal temperament’s chromatic scale is always a mode of limited transposition in that temperament, no scales in twelve-tone equal temperament have more than eleven modes.

3. RID (“Ring ID”) is derived from a binary representation of which tones are present in a scale. It is so named because it is the numbering system Ring uses on his site. Interestingly, the bits are numbered right to left, which means it follows the Ancient Greeks’ example of representing scales in descending order.
4. IID (“Interval ID”) represents the intervals in the scale (in semitones). It is a hexadecimal representation, so A = 10, B = 11, and C = 12.

IDs of a few common scales:

Key to interval listings & inversions

Intervals on the right are measured in whole steps. Interval key (where WT = whole tones, HST = hexadecimal semitones, DST = decimal semitones):

Key to interval inversions

Each interval of n semitones can be considered to invert the interval of 12 - n semitones. That is, moving up by a perfect fourth and moving down by a perfect fifth land on the same note, just at a different octave. The table above lists interval inversions on the same row.

I consider the augmented fourth and diminished fifth to be inversions of each other because they are semantically different in music theory contexts: in an augmented fourth above the first scale degree, the scale’s fourth degree is raised by a semitone; in a diminished fifth, its fifth degree is lowered by a semitone. This can depend on several factors. For instance, in the C major scale:

• Traveling from F up to B is considered an augmented fourth, since F is only note where traveling three notes up the major scale does not result in a perfect fourth.
• Traveling from F down to B is considered a diminished fifth, since F is the only note where traveling four notes down the major scale does not result in a perfect fifth.

The scales

Octave (1-note scale)

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2 notes (6 scales, 11 modes)

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3 notes (19 scales, 55 modes)

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Tetratonic (4 notes, 43 scales, 165 modes)

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Pentatonic (5 notes, 66 scales, 330 modes)