Minimal consistent EDOs: Difference between revisions

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An [[EDO|edo]] N is [[consistent]] with respect to a set of rational numbers s if the [[Patent_val|patent val]] mapping of every element of s is the nearest N-edo approximation. It is ''uniquely consistent'' if every element of s is mapped to a unique value. If the set s is the q [[Odd_limit|odd limit]], we say N is q-limit consistent and q-limit uniquely consistent, respectively. Below is a table of the least consistent, and least uniquely consistent, edo for every odd number up to 135.
An [[EDO]] N is [[consistent]] with respect to a set of rational numbers s if the [[Patent_val|patent val]] mapping of every element of s is the nearest N-edo approximation. It is ''uniquely consistent'' if every element of s is mapped to a unique value. If the set s is the q [[Odd_limit|odd limit]], we say N is q-limit consistent and q-limit uniquely consistent, respectively. Below is a table of the least consistent, and least uniquely consistent, edo for every odd number up to 135.


{| class="wikitable"
{| class="wikitable"
|-
|-
! | Odd limit
! Odd limit
! | Smallest consistent
! Smallest consistent
! | Smallest uniquely consistent
! Smallest uniquely consistent
|-
|-
| | 1
| 1
| | 1
| 1
| | 1
| 1
|-
|-
| | 3
| 3
| | 1
| 1
| | 3
| 3
|-
|-
| | 5
| 5
| | 3
| 3
| | 9
| 9
|-
|-
| | 7
| 7
| | 4
| 4
| | 27
| 27
|-
|-
| | 9
| 9
| | 5
| 5
| | 41
| 41
|-
|-
| | 11
| 11
| | 22
| 22
| | 58
| 58
|-
|-
| | 13
| 13
| | 26
| 26
| | 87
| 87
|-
|-
| | 15
| 15
| | 29
| 29
| | 111
| 111
|-
|-
| | 17
| 17
| | 58
| 58
| | 149
| 149
|-
|-
| | 19
| 19
| | 80
| 80
| | 217
| 217
|-
|-
| | 21
| 21
| | 94
| 94
| | 282
| 282
|-
|-
| | 23
| 23
| | 94
| 94
| | 282
| 282
|-
|-
| | 25
| 25
| | 282
| 282
| | 388
| 388
|-
|-
| | 27
| 27
| | 282
| 282
| | 388
| 388
|-
|-
| | 29
| 29
| | 282
| 282
| | 1323
| 1323
|-
|-
| | 31
| 31
| | 311
| 311
| | 1600
| 1600
|-
|-
| | 33
| 33
| | 311
| 311
| | 1600
| 1600
|-
|-
| | 35
| 35
| | 311
| 311
| | 1600
| 1600
|-
|-
| | 37
| 37
| | 311
| 311
| | 1600
| 1600
|-
|-
| | 39
| 39
| | 311
| 311
| | 2554
| 2554
|-
|-
| | 41
| 41
| | 311
| 311
| | 2554
| 2554
|-
|-
| | 43
| 43
| | 17461
| 17461
| | 17461
| 17461
|-
|-
| | 45
| 45
| | 17461
| 17461
| | 17461
| 17461
|-
|-
| | 47
| 47
| | 20567
| 20567
| | 20567
| 20567
|-
|-
| | 49
| 49
| | 20567
| 20567
| | 20567
| 20567
|-
|-
| | 51
| 51
| | 20567
| 20567
| | 20567
| 20567
|-
|-
| | 53
| 53
| | 20567
| 20567
| | 20567
| 20567
|-
|-
| | 55
| 55
| | 20567
| 20567
| | 20567
| 20567
|-
|-
| | 57
| 57
| | 20567
| 20567
| | 20567
| 20567
|-
|-
| | 59
| 59
| | 253389
| 253389
| | 253389
| 253389
|-
|-
| | 61
| 61
| | 625534
| 625534
| | 625534
| 625534
|-
|-
| | 63
| 63
| | 625534
| 625534
| | 625534
| 625534
|-
|-
| | 65
| 65
| | 625534
| 625534
| | 625534
| 625534
|-
|-
| | 67
| 67
| | 625534
| 625534
| | 625534
| 625534
|-
|-
| | 69
| 69
| | 759630
| 759630
| | 759630
| 759630
|-
|-
| | 71
| 71
| | 759630
| 759630
| | 759630
| 759630
|-
|-
| | 73
| 73
| | 759630
| 759630
| | 759630
| 759630
|-
|-
| | 75
| 75
| | 2157429
| 2157429
| | 2157429
| 2157429
|-
|-
| | 77
| 77
| | 2157429
| 2157429
| | 2157429
| 2157429
|-
|-
| | 79
| 79
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 81
| 81
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 83
| 83
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 85
| 85
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 87
| 87
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 89
| 89
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 91
| 91
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 93
| 93
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 95
| 95
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 97
| 97
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 99
| 99
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 101
| 101
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 103
| 103
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 105
| 105
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 107
| 107
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 109
| 109
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 111
| 111
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 113
| 113
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 115
| 115
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 117
| 117
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 119
| 119
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 121
| 121
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 123
| 123
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 125
| 125
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 127
| 127
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 129
| 129
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 131
| 131
| | 2901533
| 2901533
| | 2901533
| 2901533
|-
|-
| | 133
| 133
| | 70910024
| 70910024
| | 70910024
| 70910024
|-
|-
| | 135
| 135
| | 70910024
| 70910024
| | 70910024
| 70910024
|-
|-
| | 137
| 137
| | 5407372813
| 5407372813
| | 5407372813
| 5407372813
|-
|-
| | 139
| 139
| | 5407372813
| 5407372813
| | 5407372813
| 5407372813
|-
|-
| | 141
| 141
| | 5407372813
| 5407372813
| | 5407372813
| 5407372813
|-
|-
| | 143
| 143
| | 5407372813
| 5407372813
| | 5407372813
| 5407372813
|-
|-
| | 145
| 145
| | 5407372813
| 5407372813
| | 5407372813
| 5407372813
|-
|-
| | 147
| 147
| | 5407372813
| 5407372813
| | 5407372813
| 5407372813
|-
|-
| | 149
| 149
| | 5407372813
| 5407372813
| | 5407372813
| 5407372813
|-
|-
| | 151
| 151
| | 5407372813
| 5407372813
| | 5407372813
| 5407372813
|-
|-
| | 153
| 153
| | 5407372813
| 5407372813
| | 5407372813
| 5407372813
|-
|-
| | 155
| 155
| | 5407372813
| 5407372813
| | 5407372813
| 5407372813
|}
|}


=OEIS integer sequences links=
== OEIS integer sequences links ==
 
* {{OEIS|A116474|Equal divisions of the octave with progressively increasing consistency levels}}
* {{OEIS|A116474|Equal divisions of the octave with progressively increasing consistency levels}}
* {{OEIS|A116475|Equal divisions of the octave with progressively increasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit}}
* {{OEIS|A116475|Equal divisions of the octave with progressively increasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit}}

Revision as of 20:14, 22 November 2020

An EDO N is consistent with respect to a set of rational numbers s if the patent val mapping of every element of s is the nearest N-edo approximation. It is uniquely consistent if every element of s is mapped to a unique value. If the set s is the q odd limit, we say N is q-limit consistent and q-limit uniquely consistent, respectively. Below is a table of the least consistent, and least uniquely consistent, edo for every odd number up to 135.

Odd limit Smallest consistent Smallest uniquely consistent
1 1 1
3 1 3
5 3 9
7 4 27
9 5 41
11 22 58
13 26 87
15 29 111
17 58 149
19 80 217
21 94 282
23 94 282
25 282 388
27 282 388
29 282 1323
31 311 1600
33 311 1600
35 311 1600
37 311 1600
39 311 2554
41 311 2554
43 17461 17461
45 17461 17461
47 20567 20567
49 20567 20567
51 20567 20567
53 20567 20567
55 20567 20567
57 20567 20567
59 253389 253389
61 625534 625534
63 625534 625534
65 625534 625534
67 625534 625534
69 759630 759630
71 759630 759630
73 759630 759630
75 2157429 2157429
77 2157429 2157429
79 2901533 2901533
81 2901533 2901533
83 2901533 2901533
85 2901533 2901533
87 2901533 2901533
89 2901533 2901533
91 2901533 2901533
93 2901533 2901533
95 2901533 2901533
97 2901533 2901533
99 2901533 2901533
101 2901533 2901533
103 2901533 2901533
105 2901533 2901533
107 2901533 2901533
109 2901533 2901533
111 2901533 2901533
113 2901533 2901533
115 2901533 2901533
117 2901533 2901533
119 2901533 2901533
121 2901533 2901533
123 2901533 2901533
125 2901533 2901533
127 2901533 2901533
129 2901533 2901533
131 2901533 2901533
133 70910024 70910024
135 70910024 70910024
137 5407372813 5407372813
139 5407372813 5407372813
141 5407372813 5407372813
143 5407372813 5407372813
145 5407372813 5407372813
147 5407372813 5407372813
149 5407372813 5407372813
151 5407372813 5407372813
153 5407372813 5407372813
155 5407372813 5407372813

OEIS integer sequences links

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