Consistent tunings

A typical way of evaluating equal temperaments is to measure the smallest
possible tuning deviation of several just intervals of interest. The problem
with this approach is that it does not guarantee that the approximating
intervals are consistent with one another. For example, in 24tet, the
calculation would approximate 5:4 as 8 steps, 7:5 as 12 steps, and 7:4 as 19
steps. But a 7:5:4 chord cannot be tuned with all three intervals as given!

I have looked at all equal tunings up to 100 steps.

For the 5-limit, the following tunings are INCONSISTENT:
1 2 11 13 14 17 20 21 32 33 40 51
52 54 64 66 67 85 86 98

For the 7-limit, the following tunings are INCONSISTENT:
1 2 3 7 8 11 13 14 17 20 21 23
24 25 28 30 32 33 34 38 39 40 44 47
48 51 52 54 55 61 64 65 66 67 69 71
74 75 79 85 86 92 96 97 98 100

For the 9-limit, the following tunings are CONSISTENT:
5 12 19 22 26 27 29 31 41 46 50 53
58 60 68 70 72 77 80 82 84 87 89 91
94 99

For the 11-limit, the following tunings are CONSISTENT:
22 26 29 31 41 46 58 72 80 87 89 94

For the 13-limit, the following tunings are CONSISTENT:
26 29 41 46 58 72 80 87 94

For the 15-limit, the following tunings are CONSISTENT:
29 41 58 72 80 87 94

For the 17-limit, the following tunings are CONSISTENT:
58 72 80 94

For the 19-limit, the following tunings are CONSISTENT:
80 94

For the 21-limit, the following tuning is CONSISTENT:
94

For the 23-limit, the following tuning is CONSISTENT:
94

NO tuning up to 100tet can consistently represent the 25-limit.

So let's remove the inconsistent tunings from the table of "best" equal
temperaments that I posted in TD 683 (Apr 8 '96) :

pitch resolution -> .15% .25% .35% .45% .55% .65% .75% .85% .95% 1.05% 1.15%
harmonic limit
3 41 29 12 12 12 12 12 12 12 7 5
5 53 34 34 19 19 12 12 12 12 12 12
7 72 31 31 31 31 22 22 15 12 12 10
9 53 41 41 31 31 19 19 12 12 12 12
11 72 41 31 31 31 22 22 22 22 22
13 87 41
15 87 41
17 72 72
19
21
23 94


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