Xenharmonic Wiki

Consistency limits of small EDOs: Difference between revisions

← Older editNewer edit →
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
m simplified wiki markup
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,024 edits
No edit summary
Line 1: Line 1:
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 every edo up to 99. "Consistent" gives the consistency level, and "Distinct" the distinct consistency level.
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 every EDO up to 99. "Consistent" gives the consistency level, and "Distinct" the distinct consistency level. "Consistency distance" gives the consistency distance of given odd limit.


{| class="wikitable"
{| class="wikitable sortable mw-collapsible" style="text-align:right"
|-
! rowspan="2" | EDO
! colspan="2" | Consistency level
! colspan="11" | Consistency distance
|-
|-
! EDO
! Consistent
! Consistent
! Distinct
! Distinct
!  3-limit
!  5-limit
!  7-limit
!  9-limit
! 11-limit
! 13-limit
! 15-limit
! 17-limit
! 19-limit
! 21-limit
! 23-limit
|-
|-
| 1
| 1 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 1
|-
|-
| 2
| 2 || 3 || 1 || 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 1
|-
|-
| 3
| 3 || 5 || 3 || 2 || 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 3
|-
|-
| 4
| 4 || 7 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 1
|-
|-
| 5
| 5 || 9 || 3 || 6 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 3
|-
|-
| 6
| 6 || 7 || 3 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 3
|-
|-
| 7
| 7 || 5 || 3 || 5 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 3
|-
|-
| 8
| 8 || 5 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 3
|-
|-
| 9
| 9 || 7 || 5 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 5
|-
|-
| 10
| 10 || 7 || 3 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 3
|-
|-
| 11
| 11 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 12
| 12 || 9 || 5 || 25 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 5
|-
|-
| 13
| 13 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 14
| 14 || 3 || 3 || 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 15
| 15 || 7 || 5 || 2 || 2 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 5
|-
|-
| 16
| 16 || 7 || 5 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 5
|-
|-
| 17
| 17 || 3 || 3 || 8 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 18
| 18 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 5
|-
|-
| 19
| 19 || 9 || 5 || 4 || 4 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 5
|-
|-
| 20
| 20 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 21
| 21 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 22
| 22 || 11 || 7 || 3 || 2 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0
| 11
| 5
|-
|-
| 23
| 23 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 24
| 24 || 5 || 5 || 12 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 25
| 25 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 26
| 26 || 13 || 7 || 2 || 1 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0
| 13
| 5
|-
|-
| 27
| 27 || 9 || 7 || 2 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 7
|-
|-
| 28
| 28 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 29
| 29 || 15 || 5 || 13 || 1 || 1 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0
| 15
| 5
|-
|-
| 30
| 30 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 31
| 31 || 11 || 7 || 3 || 3 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0
| 11
| 7
|-
|-
| 32
| 32 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 33
| 33 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 34
| 34 || 5 || 5 || 4 || 4 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 35
| 35 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 36
| 36 || 7 || 7 || 8 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 37
| 37 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 38
| 38 || 5 || 5 || 2 || 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 39
| 39 || 5 || 5 || 2 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 40
| 40 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 41
| 41 || 15 || 9 || 30 || 2 || 2 || 2 || 1 || 1 || 1 || 0 || 0 || 0 || 0
| 15
| 9
|-
|-
| 42
| 42 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 43
| 43 || 7 || 7 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 44
| 44 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 45
| 45 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 46
| 46 || 13 || 9 || 5 || 2 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0
| 13
| 9
|-
|-
| 47
| 47 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 48
| 48 || 5 || 5 || 6 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 49
| 49 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 50
| 50 || 9 || 7 || 2 || 2 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 7
|-
|-
| 51
| 51 || 3 || 3 || 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 52
| 52 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 53
| 53 || 9 || 9 || 165 || 8 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 9
|-
|-
| 54
| 54 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 55
| 55 || 5 || 5 || 2 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 56
| 56 || 7 || 7 || 2 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 57
| 57 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 58
| 58 || 17 || 11 || 6 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 0 || 0 || 0
| 17
| 11
|-
|-
| 59
| 59 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 60
| 60 || 9 || 9 || 5 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 9
|-
|-
| 61
| 61 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 62
| 62 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 63
| 63 || 7 || 7 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 64
| 64 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 65
| 65 || 5 || 5 || 22 || 5 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 66
| 66 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 67
| 67 || 3 || 3 || 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 68
| 68 || 9 || 9 || 2 || 2 || 2 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 9
|-
|-
| 69
| 69 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 70
| 70 || 9 || 9 || 9 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 9
|-
|-
| 71
| 71 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 72
| 72 || 17 || 11 || 4 || 2 || 2 || 2 || 2 || 1 || 1 || 1 || 0 || 0 || 0
| 17
| 11
|-
|-
| 73
| 73 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 74
| 74 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 75
| 75 || 5 || 5 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 76
| 76 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 77
| 77 || 9 || 9 || 11 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 9
|-
|-
| 78
| 78 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 79
| 79 || 5 || 5 || 2 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 80
| 80 || 19 || 11 || 2 || 2 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 0 || 0
| 19
| 11
|-
|-
| 81
| 81 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 82
| 82 || 9 || 9 || 15 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 9
|-
|-
| 83
| 83 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 84
| 84 || 9 || 9 || 3 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 9
|-
|-
| 85
| 85 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 86
| 86 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 87
| 87 || 15 || 13 || 4 || 4 || 1 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0
| 15
| 13
|-
|-
| 88
| 88 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 89
| 89 || 11 || 11 || 8 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0
| 11
| 11
|-
|-
| 90
| 90 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 91
| 91 || 9 || 9 || 2 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 9
|-
|-
| 92
| 92 || 5 || 5 || 2 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 93
| 93 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 94
| 94 || 23 || 13 || 36 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| 23
| 13
|-
|-
| 95
| 95 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7
| 7
|-
|-
| 96
| 96 || 5 || 5 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 97
| 97 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5
| 5
|-
|-
| 98
| 98 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 3
| 3
|-
|-
| 99
| 99 || 9 || 9 || 5 || 3 || 3 || 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9
| 9
|}
|}


Retrieved from "https://en.xen.wiki/w/Consistency_limits_of_small_EDOs"