Every Possible Scale in 12-Tone Equal Temperament

a reference by Aaron Freed

Contents

1. Brief introduction
   1. Key to scale identifiers
   2. Key to interval listings
   3. Key to interval inversions
2. The scales
   1. 1 note (octave)
   2. 2 notes (11 scales)
   3. 3 notes (55 scales)
   4. Tetratonic (4 notes, 165 scales)
   5. Pentatonic (5 notes, 330 scales)
   6. Hexatonic (6 notes, 462 scales)
   7. Heptatonic (7 notes, 462 scales)
   8. Octatonic (8 notes, 330 scales)
   9. Enneatonic (9 notes, 165 scales)
   10. Decatonic (10 notes, 55 scales)
   11. Hendecatonic (11 notes, 11 scales)
   12. Dodecatonic (12 notes, chromatic scale)

Brief notes

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

Key to scale identifiers

I list three forms of identification for each scale:

1. CID (“Carnatic ID”) uses a numbering system inspired by South India’s Carnatic Numbered Melakarta, where the more front-loaded a scale’s intervals are, the earlier it appears in the numbering.
2. 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.
3. 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

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

• 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 (11 scales)

2-note scales (11)

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3 notes (55 scales)

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Tetratonic (4 notes, 165 scales)

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Pentatonic (5 notes, 330 scales)