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Line 1: Line 1:
== Fifthspan examples ==
{| class="wikitable center-all"
{| class="wikitable center-all"
!
!
Line 12: Line 14:
! 23
! 23
|-
|-
! EDO <br> ↓
! rowspan="2" | EDO <br> ↓
! Mapping <br> ↓
! rowspan="2" | Unique fifth <br> (mapping) <br> ↓
| (1/1)
! colspan="9" | ↓ ([[octave-reduced]] corresponding primary intervals) ↓
| (3/2)
|-
| (5/4)
! (1/1)
| (7/4)
! (3/2)
| (11/8)
! (5/4)
| (13/8)
! (7/4)
| (17/16)
! (11/8)
| (19/16)
! (13/8)
| (23/16)
! (17/16)
! (19/16)
! (23/16)
|-
! [[1edo]]
| N/A
|-
! [[2edo]]
| 1\2
| 0
| +1
| +1
| 0
| +1
| +1
| 0
| 0
| +1
|-
! [[3edo]]
| 2\3
| 0
| +1
| -1
| +1
| -1
| +1
| 0
| -1
| +1
|-
! [[4edo]]
| N/A (2x 2edo)
|-
|-
! [[5edo]]
! [[5edo]]
Line 34: Line 68:
| 0
| 0
| +2
| +2
| +1
|-
! [[6edo]]
| N/A (2x 3edo)
|-
|-
! [[7edo]]
! [[7edo]]
Line 43: Line 81:
| -1
| -1
| +3
| +3
| +2
| -3
| +1
|-
|-
! [[8edo]]
! [[8edo]]
Line 54: Line 95:
| -3
| -3
| +2
| +2
| +4
|-
|-
! [[9edo]]
! [[9edo]]
Line 65: Line 107:
| +2
| +2
| +4
| +4
| +1
|-
! [[10edo]]
| N/A (2x 5edo)
|-
|-
! [[11edo]]
! [[11edo]]
Line 76: Line 122:
| +2
| +2
| -5
| -5
| +1
|-
|-
! [[12edo]]
! [[12edo]]
Line 87: Line 134:
| -5
| -5
| -3
| -3
| +6
|-
|-
! [[13edo]]
! [[13edo]]
Line 92: Line 140:
| 0
| 0
| +1
| +1
| +7
| -6
| -2
| -2
| +4
| +4
Line 99: Line 147:
| +2
| +2
| -4
| -4
|-
! [[14edo]]
| N/A (2x 7edo)
|-
! [[15edo]]
| N/A (3x 5edo)
|-
|-
! [[16edo]]
! [[16edo]]
Line 110: Line 164:
| -7
| -7
| +4
| +4
| +8
|-
|-
! [[17edo]]
! [[17edo]]
Line 131: Line 186:
| +4
| +4
| -7
| -7
| -8
| +2
| +9
|-
|-
! [[19edo]]
! [[19edo]]
Line 140: Line 198:
| +6
| +6
| -4
| -4
| -5
| -3
| -6
| -6
|-
! [[20edo]]
| N/A (4x 5edo)
|-
! [[21edo]]
| N/A (3x 7edo)
|-
|-
! [[22edo]]
! [[22edo]]
Line 149: Line 215:
| -2
| -2
| -6
| -6
| -4
| -9
| -10
| -10
| -3
| +6
|-
|-
! [[23edo]]
! [[23edo]]
Line 163: Line 231:
| +4
| +4
| +8
| +8
|-
! [[24edo]]
| N/A (2x 12edo)
|-
! [[25edo]]
| N/A (5x 5edo)
|-
|-
! [[26edo]]
! [[26edo]]
Line 186: Line 260:
| -10
| -10
| -8
| -8
| +11
|-
! [[28edo]]
| N/A (4x 7edo)
|-
|-
! [[29edo]]
! [[29edo]]
Line 195: Line 273:
| +11
| +11
| +8
| +8
| +7
| -3
| +6
|-
! [[30edo]]
| N/A (6x 5edo)
|-
! [[31edo]]
| 18\31
| 0
| +1
| +4
| +10
| -13
| +15
| -5
| -3
| -6
|-
! [[32edo]]
| 19\32
| 0
| +1
| +14
| -2
| -11
| -14
| -15
| -8
| +11
|-
! [[33edo]]
| 19\33
| 0
| +1
| +11
| -9
| +6
| -4
| -12
| -10
| -13
|-
! [[34edo]]
| N/A (2x 17edo)
|-
! [[35edo]]
| N/A (5x 7edo)
|-
! [[36edo]]
| N/A (3x 12edo)
|-
! [[37edo]]
| 22\37
| 0
| +1
| +14
| -2
| -11
| +18
| -15
| -8
| +16
|-
! [[38edo]]
| N/A (2x 19edo)
|-
! [[39edo]]
| 23\39
| 0
| +1
| -13
| -19
| -6
| -9
| +12
| +14
| -11
|-
! [[40edo]]
| 23\40
| 0
| +1
| +11
| -16
| +6
| -4
| -19
| -10
| -13
|-
! [[41edo]]
| 24\41
| 0
| +1
| -8
| -14
| -18
| +20
| +7
| -3
| +6
|-
! [[42edo]]
| 25\42
| 0
| +1
| +14
| -2
| -11
| -19
| -20
| -8
| +16
|-
! [[43edo]]
| 25\43
| 0
| +1
| +4
| +10
| +18
| -16
| -5
| -3
| -18
|-
! [[44edo]]
| N/A (2x 22edo)
|-
! [[45edo]]
| 26\45
| 0
| +1
| +4
| -9
| +6
| +22
| +14
| +16
| -6
|-
! [[46edo]]
| 27\46
| 0
| +1
| +21
| +15
| +11
| +8
| -22
| -3
| +6
|-
! [[47edo]]
| 27\47
| 0
| +1
| +11
| -16
| +13
| -4
| -19
| -10
| -13
|-
! [[48edo]]
| N/A (4x 12edo)
|-
! [[49edo]]
| 29\49
| 0
| +1
| +9
| -2
| +16
| +13
| -10
| +19
| -16
|-
! [[50edo]]
| 29\50
| 0
| +1
| +4
| +10
| -13
| +15
| -24
| -22
| -6
|-
! [[51edo]]
| N/A (3x 17edo)
|-
! [[52edo]]
| N/A (2x 26edo)
|-
! [[53edo]]
| 31\53
| 0
| +1
| -8
| -14
| +23
| +20
| +7
| -3
| +18
|-
! [[54edo]]
| N/A (2x 27edo)
|-
! [[55edo]]
| 32\55
| 0
| +1
| +4
| +22
| -25
| +27
| -5
| -3
| -18
|-
! [[56edo]]
| 33\56
| 0
| +1
| +26
| -19
| -6
| -9
| -27
| +14
| -11
|-
! [[57edo]]
| N/A (3x 19edo)
|-
! [[58edo]]
| N/A (2x 29edo)
|-
! [[59edo]]
| 35\59
| 0
| +1
| -18
| -2
| +21
| -14
| +17
| -8
| +11
|-
! [[60edo]]
| N/A (5x 12edo)
|-
! [[61edo]]
| 36\61
| 0
| +1
| -13
| +20
| -6
| +30
| +12
| -25
| +28
|-
! [[62edo]]
| N/A (2x 31edo)
|-
! [[63edo]]
| 37\63
| 0
| +1
| -25
| +15
| +11
| +8
| +24
| -20
| +6
|-
! [[64edo]]
| 37\64
| 0
| +1
| -15
| -28
| +25
| -23
| +14
| +16
| -6
|-
! [[65edo]]
| 38\65
| 0
| +1
| -8
| -26
| -30
| +32
| +7
| -3
| +18
|-
! [[66edo]]
| N/A (3x 22edo)
|-
! [[67edo]]
| 39\67
| 0
| +1
| +4
| +22
| +30
| -28
| -5
| -3
| -18
|-
! [[68edo]]
| N/A (4x 17edo)
|-
! [[69edo]]
| 40\69
| 0
| +1
| +4
| +29
| -13
| +15
| -24
| -22
| -6
|-
! [[70edo]]
| 41\70
| 0
| +1
| +33
| +27
| -18
| -21
| -34
| -3
| -23
|-
! [[71edo]]
| 42\71
| 0
| +1
| +9
| -24
| +16
| +35
| -10
| -30
| +33
|-
! [[72edo]]
| N/A (6x 12edo)
|-
! [[73edo]]
| 43\73
| 0
| +1
| -30
| -19
| -6
| -9
| +29
| +14
| -11
|-
! [[74edo]]
| 43\74
| 0
| +1
| +4
| +10
| +18
| -16
| -36
| -34
| +25
|-
! [[75edo]]
| 44\75
| 0
| +1
| +21
| -31
| +11
| +37
| -22
| +26
| +6
|-
! [[76edo]]
| N/A (4x 19edo)
|-
! [[77edo]]
| 45\77
| 0
| +1
| -8
| -26
| +35
| +32
| +7
| -3
| +18
|-
! [[78edo]]
| N/A (2x 39edo)
|-
! [[79edo]]
| 46\79
| 0
| +1
| +16
| +22
| -37
| -28
| -5
| -3
| -18
|-
! [[80edo]]
| 47\80
| 0
| +1
| +38
| +15
| +11
| +8
| -39
| -20
| +6
|-
! [[81edo]]
| 47\81
| 0
| +1
| +4
| +10
| -13
| +15
| +26
| +28
| -6
|-
! [[82edo]]
| N/A (2x 41edo)
|-
! [[83edo]]
| 49\83
| 0
| +1
| -13
| +20
| -6
| -31
| +12
| +36
| -33
|-
! [[84edo]]
| N/A (7x 12edo)
|-
! [[85edo]]
| N/A (5x 17edo)
|-
! [[86edo]]
| N/A (2x 43edo)
|-
! [[87edo]]
| N/A (3x 29edo)
|-
! [[88edo]]
| 51\88
| 0
| +1
| +4
| +29
| -32
| +34
| -24
| -22
| -6
|-
! [[89edo]]
| 52\89
| 0
| +1
| -8
| -26
| -42
| +32
| +7
| -3
| +30
|-
! [[90edo]]
| 53\90
| 0
| +1
| +43
| -19
| -23
| -9
| -44
| +14
| -11
|-
! [[91edo]]
| 53\91
| 0
| +1
| +16
| +34
| +42
| -40
| -5
| -3
| -30
|-
! [[92edo]]
| N/A (2x 46edo)
|-
! [[93edo]]
| N/A (3x 31edo)
|-
! [[94edo]]
| 55\94
| 0
| +1
| -8
| -14
| +23
| +20
| -46
| -3
| -35
|-
! [[95edo]]
| 56\95
| 0
| +1
| +26
| +37
| -6
| +47
| -27
| +14
| +45
|-
! [[96edo]]
| N/A (8x 12edo)
|-
! [[97edo]]
| 57\97
| 0
| +1
| -42
| +32
| +11
| +8
| -39
| -20
| +6
|-
! [[98edo]]
| 57\98
| 0
| +1
| +4
| -33
| -25
| +27
| -5
| -46
| +37
|-
! [[99edo]]
| 58\99
| 0
| +1
| -37
| -43
| -18
| -21
| +36
| -32
| -23
|}
== EDO information with fifthspans ==
=== [[5edo]] ===
{{Primes in edo_fs|columns=9|5}}
=== [[7edo]] ===
{{Primes in edo_fs|columns=9|7}}
=== [[8edo]] ===
{{Primes in edo_fs|columns=9|8}}
=== [[9edo]] ===
{{Primes in edo_fs|columns=9|9}}
=== [[11edo]] ===
{{Primes in edo_fs|columns=9|11}}
=== [[12edo]] ===
{{Primes in edo_fs|columns=9|12}}
=== [[13edo]] ===
{{Primes in edo_fs|columns=9|13}}
=== [[16edo]] ===
{{Primes in edo_fs|columns=9|16}}
=== [[17edo]] ===
{{Primes in edo_fs|columns=9|17}}
=== [[18edo]] ===
{{Primes in edo_fs|columns=9|18}}
=== [[19edo]] ===
{{Primes in edo_fs|columns=9|19}}
=== [[22edo]] ===
{{Primes in edo_fs|columns=9|22}}
=== [[23edo]] ===
{{Primes in edo_fs|columns=9|23}}
=== [[26edo]] ===
{{Primes in edo_fs|columns=9|26}}
=== [[27edo]] ===
{{Primes in edo_fs|columns=9|27}}
=== [[29edo]] ===
{{Primes in edo_fs|columns=9|29}}
=== [[31edo]] ===
{{Primes in edo_fs|columns=9|31}}
=== [[39edo]] ===
{{Primes in edo_fs|columns=9|39}}
=== [[41edo]] ===
{{Primes in edo_fs|columns=9|41}}
=== [[46edo]] ===
{{Primes in edo_fs|columns=9|46}}
=== [[53edo]] ===
{{Primes in edo_fs|columns=9|53}}
== Separate mod steps ==
Shown on the example of [[17edo]].
All fifthspans are given as positive values. (This can be transformed into mixed mode by subtracting <code>17</code> from all values <code> > 17/2 == 8.5</code>)
{| class="wikitable center-all"
! colspan="2" | <!-- empty cell -->
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
! prime 23
|- class="thick-border" style="border-top: 2px solid #777"
! rowspan="2" | Error
! absolute ([[cent|¢]])
| 0.0
| +3.9
| -33.4
| +19.4
| +13.4
| +6.5
| -34.3
| -15.2
| +7.0
|-
! [[Relative error|relative]] (%)
| 0
| +6
| -47
| +27
| +19
| +9
| -49
| -21
| +10
|- class="thick-border" style="border-top: 2px solid #777"
! rowspan="2" | [[patent val|Mapping]]
! value
| '''17'''
| 27
| 39
| 48
| 59
| 63
| 69
| 72
| 77
|-
! [[octave reduction|~ (mod 17)]]
| 0
| '''10'''
| 5
| 14
| 8
| 12
| '''1'''
| 4
| 9
|- class="thick-border" style="border-top: 2px solid #777"
! colspan="2" | [[Fifthspan]] (in steps)
| 0
| '''1'''
| 9
| 15
| 11
| 8
| '''12'''
| 14
| 6
|}
{| class="wikitable center-all"
|-
|+ Approximation of prime intervals in 12 EDO
|-
! colspan="2" | Prime interval
! style="min-width: 3em" | 2
! style="min-width: 3em" | 3
! style="min-width: 3em" | 5
! style="min-width: 3em" | 7
! style="min-width: 3em" | 11
! style="min-width: 3em" | 13
! style="min-width: 3em" | 17
! style="min-width: 3em" | 19
|-
! rowspan="2" | Error
! absolute ([[cent|¢]])
| 0.0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Relative error|relative]] (%)
| 0
| -2
| +14
| +31
| +49
| -41
| -5
| +2
|-
! rowspan="2" | Mapping
! [[patent val]] ''v''
| 12
| 19
| 28
| 34
| 42
| 44
| 49
| 51
|-
! [[Octave reduction|''v'' mod 12]]
| 0
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|}
----
{| class="wikitable center-all"
|-
|+ Approximation of prime intervals in 17 EDO
|-
! colspan="2" | Prime number
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
|-
! rowspan="2" | Error
! absolute ([[cent|¢]])
| +0.0
| +3.9
| −33.4
| +19.4
| +13.4
| +6.5
| −34.4
| −15.2
|-
! [[Relative error|relative]] (%)
| +0
| +6
| −47
| +27
| +19
| +9
| −49
| −21
|-
! rowspan="2" | Mapping
! [[patent val]] ''v''
| 17
| 27
| 39
| 48
| 59
| 63
| 69
| 72
|-
! ''v'' ([[octave-reduced|mod 17]])
| 0
| 10
| 5
| 14
| 8
| 12
| 1
| 4
|}
{| class="wikitable center-all"
|-
|+ Approximation of prime intervals in 17 EDO
|-
! colspan="2" | Prime number
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
|- class="thick-border" style="border-top: 2px solid #aaa"
! rowspan="2" | Error
! absolute ([[cent|¢]])
| +0.0
| +3.9
| −33.4
| +19.4
| +13.4
| +6.5
| −34.4
| −15.2
|-
! [[Relative error|relative]] (%)
| +0
| +6
| −47
| +27
| +19
| +9
| −49
| −21
|- class="thick-border" style="border-top: 2px solid #aaa"
! rowspan="2" | Mapping
! [[patent val]] ''v''
| 17
| 27
| 39
| 48
| 59
| 63
| 69
| 72
|-
! ''v'' ([[octave-reduced|mod 17]])
| 0
| 10
| 5
| 14
| 8
| 12
| 1
| 4
|- class="thick-border" style="border-top: 2px solid #aaa"
! colspan="2" | [[Fifthspan]]
| 0
| 1
| −8
| −2
| −6
| 8
| −5
| −3
|}
== Fifthspans of [[17edo]] steps ==
{| class="wikitable center-all"
|-
! step !! span
|-
| 0 || 0
|-
| 1 || 12
|-
| 2 || 7
|-
| 3 || 2
|-
| 4 || 14
|-
| 5 || 9
|-
| 6 || 4
|-
| 7 || 16
|-
| 8 || 11
|-
| 9 || 6
|-
| 10 || 1
|-
| 11 || 13
|-
| 12 || 8
|-
| 13 || 3
|-
| 14 || 15
|-
| 15 || 10
|-
| 16 || 5
|}
|}
Retrieved from "https://en.xen.wiki/w/User:Xenwolf/Fifthspan"