User:BudjarnLambeth/The intervals I see as important

The following are all the 16-integer-limit consonant intervals available in an octave, simplified to their simplest form (by taking them to a higher octave to simplify the fraction), then sorted with most consonant (ie mathematically simplest) first.

(^in a higher octave)

HYPERCONSONANCES

2/1 (octave)
3/1 (perf 5th^)

CONSONANCES

5/1 (maj 3rd^)
4/3 (perf 4th)
5/3 (maj 6th)
7/1 (submin 7th^)
9/1 (large maj 2nd^)
7/3 (submin 3rd^)

AMBISONANCES

6/5 (min 3rd)

11/1 (undec 4th^)

7/5 (small tritone)
9/5 (min 7th)

11/3 (neu 7th^) 13/1 (neu 6th^)

8/7 (supmaj 2nd)

13/3 (large tridec neu 2nd^)

DISSONANCES 15/1 (maj 7th^)

9/7 (supmaj 3rd)

11/5 (large undec neu 2nd) 10/7 (large tritone) 10/9 (small maj 2nd) 11/7 (undec submin 6th) 13/5 (tridec maj 3rd^) 12/7 (supmaj 6th) 11/9 (undec neu 3rd) 13/7 (tridec maj 7th) 13/9 (tridec tritone) 15/7 (large minor 2nd^) 14/9 (sept submin 6th) 12/11 (small undec neu 2nd) 13/11 (tridec min 3rd) 14/11 (undec maj 3rd) 16/9 (Pythag min 7th) 15/11 (pentdec 4th) 16/11 (undec sub5th) 14/13 (small tridec neu 2nd) 15/13 (tridec semi4th) 16/13 (tridec neu 3rd) 16/15 (small neu 2nd)

Any general-purpose tuning system should approximate all hyperconsonances within 10 cents or less, all consonances within 20 cents, and all ambisonances within 30 cents. It should also try to have as few notes as possible that do not approximate any of those three categories of interval, because every note that isn't approximating one of them is a wolf interval that adds a 'bitter' taste to the tuning, and a tuning cannot survive very much bitterness.