Ambiguity: results

I'll post some lists of ambiguous equal temperaments here. Note that
I'm not proposing "ambiguous" as a technical term with a precise
meaning, which would miss the point. However, for my website, there
has to be an algorithm for determining ambiguity so that warts can be
added or alternatives presented. I've tried to cover all reasonable
choices of mappings so it must at least be close to what a subjective
sense of ambiguity would be. I'm only considering less than 100
divisions of the octave here, and only consecutive prime limits. The
same rules can be applied in other cases, and are on the website.

For the take home results, here are some EDOs that have ambiguous
mappings from low limits:

5-limit
1 2 6 11 13 14 17 20 21 32 33 40 51 52 54 59 64 66 67 76 85 86 88 98
7-limit
1 2 3 7 8 11 13 14 17 20 23 24 25 28 30 32 33 34 36 38 39 47 48 51 52
54 55 59 61 64 65 66 67 69 71 76 79 83 85 86 88 92 96 97 98

Here are some EDOs that are unambiguous in higher limits (ambiguous
divisions are in the majority here):

11-limit
2 6 9 12 15 16 22 26 29 31 37 40 41 43 46 49 50 53 56 57 58 62 63 72
73 74 77 78 80 81 87 89 94 95
13-limit
9 15 16 26 29 31 37 41 43 46 50 53 56 58 62 63 72 74 77 78 80 84 87 94 95
17-limit
10 16 26 31 43 46 50 56 58 62 72 78 80 87 94
19-limit
26 31 43 50 56 57 62 72 77 80 87 94
23-limit
31 50 62 77 80 87 94
29-limit
62 77 80 87
31-limit
62 87

Note that although unambiguous ETs become rare, they also tend to be good.

The first rule for determining ambiguity is that the patent val
(nearest-primes mapping) disagrees with the TE optimal mapping (the
one that gives the lowest TE optimal error). Here are lower limit
cases where they disagree:

5-limit
1 11 14 17 20 32 33 51 52 54 64 66 67 85 86 98
7-limit
1 3 7 8 11 13 14 17 20 23 28 30 33 34 38 39 54 61 64 66 67 69 71 79 85
86 92 96 98
11-limit
1 3 4 7 8 11 13 14 17 18 20 23 25 27 28 30 33 34 38 39 42 45 47 52 54
60 61 64 65 66 67 69 70 71 75 76 79 82 85 86 90 91 92 96 97 98 99

and higher limit cases where they agree:

13-limit
7 9 10 15 16 19 21 22 24 26 29 31 37 40 41 43 44 46 50 52 53 56 57 58
61 62 63 67 68 70 72 74 77 78 79 80 84 87 89 93 94 95 97
17-limit
7 9 10 16 21 22 24 26 31 37 40 41 43 44 46 50 53 56 57 58 62 63 67 68
70 72 74 77 78 79 80 84 87 89 93 94
19-limit
7 9 16 21 24 26 31 36 37 40 41 43 44 46 48 50 53 56 57 62 63 67 68 72
74 77 80 84 87 89 93 94

The next rule is that the TOP mapping (the one with the lowest
TOP(-max) error) should agree with the TE optimal mapping. Most of
the time they're in agreement. Here are the cases where they disagree
(there are none <100 in the 5-limit):

7-limit
7 13 20 24 32 48 52 65 79
11-limit
5 7 11 13 20 24 32 48 52 59 61 79 83 91 93 97
13-limit
2 3 13 20 22 23 24 36 47 48 51 52 55 61 67 83 89 93
17-limit
2 3 11 13 19 20 22 23 24 25 33 35 36 37 45 47 48 51 52 55 61 63 66 67
69 83 89 93
19-limit
2 3 8 11 13 19 20 23 24 25 32 33 37 40 45 47 48 51 52 55 61 63 65 66
67 69 70 79 89 93

The final rule is that there should be no other mapping with the same
octave division that has a TE optimal error within 20% of that of the
TE optimal mapping. Here are ambiguous EDOs by this rule:

5-limit
2 6 11 13 17 21 32 40 51 52 59 64 66 76 85 86 88
7-limit
1 2 7 11 13 20 24 25 32 34 36 38 47 48 51 52 55 59 61 64 65 66 67 69
76 79 83 88 92 96 97
11-limit
4 5 7 10 11 13 19 20 21 24 25 28 30 32 33 35 36 42 44 47 48 51 52 54
55 59 61 64 65 66 67 68 70 75 76 79 82 83 84 85 88 91 92 93 96 97 98

and unambiguous EDOs in higher limits:

13-limit
8 9 12 15 16 17 18 26 27 29 31 34 37 38 41 43 45 46 49 50 53 56 58 60
62 63 66 69 72 74 77 78 80 84 87 88 91 92 94 95 96
17-limit
8 10 16 17 18 26 27 29 31 34 38 43 46 49 50 56 58 60 62 65 72 78 80 87 94 95 96
19-limit
18 24 26 27 29 31 34 38 39 43 49 50 56 57 62 64 72 77 80 87 94 95 99

Note that this rule almost makes the one about TOP and TE agreement
redundant, in that the TOP mapping is very likely to have a TE optimal
error within 20% of that of the TE optimal mapping. I forgot to
record the disagreements, but I think there was only one 2-digit EDO
in the 19-limit (maybe 94). Increasing the margin to 25% gave only
one 3-digit disagreement in the 19-limit. Of course, there are a lot
of EDOs that are deemed ambiguous by this rule that weren't otherwise.

Graham

Great work Graham, thanks for doing this. The three rules are
reasonable and breaking out results by rule was informative too.

-Carl

Graham wrote:

>I'll post some lists of ambiguous equal temperaments here. Note that
>I'm not proposing "ambiguous" as a technical term with a precise
>meaning, which would miss the point. However, for my website, there
>has to be an algorithm for determining ambiguity so that warts can be
>added or alternatives presented. I've tried to cover all reasonable
>choices of mappings so it must at least be close to what a subjective
>sense of ambiguity would be. I'm only considering less than 100
>divisions of the octave here, and only consecutive prime limits. The
>same rules can be applied in other cases, and are on the website.
>
>For the take home results, here are some EDOs that have ambiguous
>mappings from low limits:
>
>5-limit
>1 2 6 11 13 14 17 20 21 32 33 40 51 52 54 59 64 66 67 76 85 86 88 98
>7-limit
>1 2 3 7 8 11 13 14 17 20 23 24 25 28 30 32 33 34 36 38 39 47 48 51 52
>54 55 59 61 64 65 66 67 69 71 76 79 83 85 86 88 92 96 97 98
>
>Here are some EDOs that are unambiguous in higher limits (ambiguous
>divisions are in the majority here):
>
>11-limit
>2 6 9 12 15 16 22 26 29 31 37 40 41 43 46 49 50 53 56 57 58 62 63 72
>73 74 77 78 80 81 87 89 94 95
>13-limit
>9 15 16 26 29 31 37 41 43 46 50 53 56 58 62 63 72 74 77 78 80 84 87 94 95
>17-limit
>10 16 26 31 43 46 50 56 58 62 72 78 80 87 94
>19-limit
>26 31 43 50 56 57 62 72 77 80 87 94
>23-limit
>31 50 62 77 80 87 94
>29-limit
>62 77 80 87
>31-limit
>62 87
>
>Note that although unambiguous ETs become rare, they also tend to be good.
>
>The first rule for determining ambiguity is that the patent val
>(nearest-primes mapping) disagrees with the TE optimal mapping (the
>one that gives the lowest TE optimal error). Here are lower limit
>cases where they disagree:
>
>5-limit
>1 11 14 17 20 32 33 51 52 54 64 66 67 85 86 98
>7-limit
>1 3 7 8 11 13 14 17 20 23 28 30 33 34 38 39 54 61 64 66 67 69 71 79 85
>86 92 96 98
>11-limit
>1 3 4 7 8 11 13 14 17 18 20 23 25 27 28 30 33 34 38 39 42 45 47 52 54
>60 61 64 65 66 67 69 70 71 75 76 79 82 85 86 90 91 92 96 97 98 99
>
>and higher limit cases where they agree:
>
>13-limit
>7 9 10 15 16 19 21 22 24 26 29 31 37 40 41 43 44 46 50 52 53 56 57 58
>61 62 63 67 68 70 72 74 77 78 79 80 84 87 89 93 94 95 97
>17-limit
>7 9 10 16 21 22 24 26 31 37 40 41 43 44 46 50 53 56 57 58 62 63 67 68
>70 72 74 77 78 79 80 84 87 89 93 94
>19-limit
>7 9 16 21 24 26 31 36 37 40 41 43 44 46 48 50 53 56 57 62 63 67 68 72
>74 77 80 84 87 89 93 94
>
>The next rule is that the TOP mapping (the one with the lowest
>TOP(-max) error) should agree with the TE optimal mapping. Most of
>the time they're in agreement. Here are the cases where they disagree
>(there are none <100 in the 5-limit):
>
>7-limit
>7 13 20 24 32 48 52 65 79
>11-limit
>5 7 11 13 20 24 32 48 52 59 61 79 83 91 93 97
>13-limit
>2 3 13 20 22 23 24 36 47 48 51 52 55 61 67 83 89 93
>17-limit
>2 3 11 13 19 20 22 23 24 25 33 35 36 37 45 47 48 51 52 55 61 63 66 67
>69 83 89 93
>19-limit
>2 3 8 11 13 19 20 23 24 25 32 33 37 40 45 47 48 51 52 55 61 63 65 66
>67 69 70 79 89 93
>
>The final rule is that there should be no other mapping with the same
>octave division that has a TE optimal error within 20% of that of the
>TE optimal mapping. Here are ambiguous EDOs by this rule:
>
>5-limit
>2 6 11 13 17 21 32 40 51 52 59 64 66 76 85 86 88
>7-limit
>1 2 7 11 13 20 24 25 32 34 36 38 47 48 51 52 55 59 61 64 65 66 67 69
>76 79 83 88 92 96 97
>11-limit
>4 5 7 10 11 13 19 20 21 24 25 28 30 32 33 35 36 42 44 47 48 51 52 54
>55 59 61 64 65 66 67 68 70 75 76 79 82 83 84 85 88 91 92 93 96 97 98
>
>and unambiguous EDOs in higher limits:
>
>13-limit
>8 9 12 15 16 17 18 26 27 29 31 34 37 38 41 43 45 46 49 50 53 56 58 60
>62 63 66 69 72 74 77 78 80 84 87 88 91 92 94 95 96
>17-limit
>8 10 16 17 18 26 27 29 31 34 38 43 46 49 50 56 58 60 62 65 72 78 80 87 94 95 96
>19-limit
>18 24 26 27 29 31 34 38 39 43 49 50 56 57 62 64 72 77 80 87 94 95 99
>
>Note that this rule almost makes the one about TOP and TE agreement
>redundant, in that the TOP mapping is very likely to have a TE optimal
>error within 20% of that of the TE optimal mapping. I forgot to
>record the disagreements, but I think there was only one 2-digit EDO
>in the 19-limit (maybe 94). Increasing the margin to 25% gave only
>one 3-digit disagreement in the 19-limit. Of course, there are a lot
>of EDOs that are deemed ambiguous by this rule that weren't otherwise.
>
>
> Graham
>
>