Minimum-ambiguity DE scales

These distributionally even scales have maximum distinguishability, where "distinguishability" is defined as the minimum separation between different notes or intervals. If "ambiguity" (confusability) is defined as the inverse of distinguishability, then they have minimum ambiguity.

The scale that has minimum ambiguity between notes is always an equal scale.

The DE scale of N notes that has strictly minimum ambiguity between intervals is always a subscale of (2N-P)-EDO, where P is the number of periods per octave. (Octave-repeating scales are assumed here, but in fact the minimum ambiguity property is insensitive to any stretching or compressing, so the same scale patterns apply as equal divisions of any interval whatsoever.)

Scales are also listed that have some totally ambiguous (coinciding) intervals, but have minimum ambiguity over all other intervals.

3 notes

Minimum ambiguity between notes: 3edo

Minimum ambiguity between intervals: 3-note DE scales in 5edo

• 1 2 2
• 1 1 3

One ambiguous interval allowed: 1 1 2 in 4edo

4 notes

Minimum ambiguity between notes: 4edo (always N-edo for N notes; hereafter omitted)

Minimum ambiguity between intervals:

• 2 periods per octave in 6edo
  • 1 2 1 2
• 1 period per octave in 7edo
  • 1 2 2 2 (a neutral third scale)
  • 1 1 1 4

One ambiguous interval allowed:

• 1 period per octave in 6edo
  • 1 1 1 3

5 notes

Minimum ambiguity between intervals: 9edo

• 1 2 2 2 2 (superpelog-like)
• 1 1 3 1 3 (mavila- or pelog-like)
• 1 1 1 1 5

One ambiguous interval allowed: 8edo

• 1 2 1 2 2 (father-like)
• 1 1 1 1 4

Two ambiguous intervals allowed: 7edo

• 1 1 2 1 2 (2s ambiguous with L, 2L+s ambiguous with L+3s)
• 1 1 1 1 3

6 notes

Minimum ambiguity between intervals:

• 3 periods per octave in 9edo
  • 1 2 1 2 1 2
• 2 periods per octave in 10edo
  • 1 2 2 1 2 2 (decimal-like)
  • 1 1 3 1 1 3
• 1 period per octave in 11edo
  • 1 2 2 2 2 2 (machine-like)
  • 1 1 1 1 1 6

7 notes

Minimum ambiguity between intervals: 13edo

• 1 2 2 2 2 2 2 (archeotonic)
• 1 1 3 1 3 1 3
• 1 1 1 4 1 1 4
• 1 1 1 1 1 1 7

One ambiguous interval allowed: 12edo

• 1 2 2 1 2 2 2 (familiar meantone-like diatonic scale)
• 1 1 1 1 1 1 6

Two ambiguous intervals allowed: 11edo)

• 1 2 1 2 1 2 2 (orgone-like)
• 1 1 1 3 1 1 3
• 1 1 1 1 1 1 5

8 notes

Minimum ambiguity between intervals:

• 4 periods per octave in 12edo
  • 1 2 1 2 1 2 1 2 (familiar diminished-like octatonic scale)
• 2 periods per octave in 14edo
  • 1 2 2 2 1 2 2 2 (hedgehog-like)
  • 1 1 1 4 1 1 1 4
• 1 period per octave in 15edo
  • 1 2 2 2 2 2 2 2 (porcupine-like)
  • 1 1 1 1 1 1 1 8

One ambiguous interval allowed:

• 1 period per octave in 14edo
  • 1 1 3 1 1 3 1 3
  • 1 1 1 1 1 1 1 7

Two ambiguous intervals allowed:

• 2 periods per octave in 12edo
  • 1 1 1 3 1 1 1 3 (pajara-like)
• 1 period per octave in 13edo
  • 1 2 1 2 2 1 2 2 (oneirotonic)
  • 1 1 1 1 1 1 1 6

9 notes

Minimum ambiguity between intervals:

• 3 periods per octave in 15edo
  • 1 2 2 1 2 2 1 2 2 (triforce-like)
  • 1 1 3 1 1 3 1 1 3
• 1 period per octave in 17edo
  • 1 2 2 2 2 2 2 2 2 (bleu-like)
  • 1 1 3 1 3 1 3 1 3
  • 1 1 1 1 6 1 1 1 6
  • 1 1 1 1 1 1 1 1 9

One ambiguous interval allowed:

• 1 period per octave in 16edo
  • 1 2 2 2 2 1 2 2 2 (mavila-like)
  • 1 1 1 1 1 1 1 1 8

Two ambiguous intervals allowed:

• 1 period per octave in 15edo
  • 1 1 1 1 4 1 1 1 4
  • 1 1 1 1 1 1 1 1 7