96edo: Difference between revisions

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Readding this without the confusing language.
Adopt template: EDO intro; -"circulating temps" (it was becuz the context was entirely missing)
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{{Infobox ET}}
{{Infobox ET}}
{{Wikipedia|96 equal temperament}}
{{Wikipedia|96 equal temperament}}
The '''96 equal divisions of the octave''' ('''96edo'''), or the '''96-tone equal temperament''' ('''96tet'''), '''96 equal temperament''' ('''96et''') when viewed from a [[regular temperament]] perspective, divides the octave into 96 equal parts of exactly 12.5 [[cent]]s each.
{{EDO intro|96}}


== Theory ==
== Theory ==
Line 44: Line 44:
| 0.637
| 0.637
| 5.10
| 5.10
|}
== Circulating temperaments ==
{| class="wikitable center-all"
|+Circulating temperaments in 96edo
! Tones
! Pattern
! L:s
|-
| 5
| [[1L 4s]]
| 20:19
|-
| 6
| [[6edo]]
| equal
|-
| 7
| [[5L 2s]]
| 14:13
|-
| 8
| [[8edo]]
| equal
|-
| 9
| [[6L 3s]]
| 11:10
|-
| 10
| [[6L 4s]]
| 10:9
|-
| 11
| [[8L 3s]]
| 9:8
|-
| 12
| [[12edo]]
| equal
|-
| 13
| [[5L 8s]]
| 8:7
|-
| 14
| [[12L 2s]]
| rowspan="2" |7:6
|-
| 15
| [[6L 9s]]
|-
| 16
| [[16edo]]
| equal
|-
| 17
| [[11L 6s]]
| rowspan="3" |6:5
|-
| 18
| 6L 12s
|-
| 19
| 1L 18s
|-
| 20
| 16L 4s
| rowspan="4" |5:4
|-
| 21
| 12L 9s
|-
| 22
| 8L 14s
|-
| 23
| 4L 19s
|-
| 24
| [[24edo]]
| equal
|-
| 25
| 21L 4s
| rowspan="7" | 4:3
|-
| 26
| 18L 8s
|-
| 27
| 15L 12s
|-
| 28
| 12L 16s
|-
| 29
| 9L 20s
|-
| 30
| 6L 24s
|-
| 31
| 3L 28s
|-
| 32
| [[32edo]]
| equal
|-
| 33
| 30L 3s
| rowspan="15" | 3:2
|-
| 34
| 28L 6s
|-
| 35
| 26L 9s
|-
| 36
| 24L 12s
|-
| 37
| 22L 15s
|-
| 38
| 20L 18s
|-
| 39
| 18L 21s
|-
| 40
| 16L 24s
|-
| 41
| 14L 27s
|-
| 42
| 12L 30s
|-
| 43
| 10L 33s
|-
| 44
| 8L 36s
|-
| 45
| 6L 39s
|-
| 46
| 4L 42s
|-
| 47
| 2L 45s
|-
| 48
| [[48edo]]
| equal
|-
| 49
| 47L 2s
| rowspan="28" |2:1
|-
| 50
| 46L 4s
|-
| 51
| 45L 6s
|-
| 52
| 44L 8s
|-
| 53
| 43L 10s
|-
| 54
| 42L 12s
|-
| 55
| 41L 14s
|-
| 56
| 40L 16s
|-
| 57
| 39L 18s
|-
| 58
| 38L 20s
|-
| 59
| 37L 22s
|-
| 60
| 36L 24s
|-
| 61
| 35L 26s
|-
| 62
| 34L 28s
|-
| 63
| 33L 30s
|-
| 64
| 32L 32s
|-
| 65
| 31L 34s
|-
| 66
| 30L 36s
|-
| 67
| 29L 38s
|-
| 68
| 28L 40s
|-
| 69
| 27L 42s
|-
| 70
| 26L 44s
|-
| 71
| 25L 46s
|-
| 72
| 24L 48s
|-
| 73
| 23L 50s
|-
| 74
| 22L 52s
|-
| 75
| 21L 54s
|-
| 76
| 20L 56s
|}
|}


Revision as of 08:47, 7 November 2023

← 95edo 96edo 97edo →
Prime factorization 25 × 3
Step size 12.5 ¢ 
Fifth 56\96 (700 ¢) (→ 7\12)
Semitones (A1:m2) 8:8 (100 ¢ : 100 ¢)
Consistency limit 5
Distinct consistency limit 5
English Wikipedia has an article on:

Template:EDO intro

Theory

As a 5-limit system, 96edo can be characterized by the fact that it tempers out both the Pythagorean comma, 531441/524288, Würschmidt's comma, 393216/390625, the unicorn comma, 1594323/1562500, and the kwazy comma, [-53 10 16⟩. It therefore has the same familiar 700-cent fifth as 12edo, and has a best major third of 387.5 cents, a bit over a cent sharp. There is therefore nothing to complain of with its representation of the 5-limit and it can be recommended as an approach to the Würschmidt family of temperaments. It also tempers out the unicorn comma, and serves a way of tuning temperaments in the unicorn family.

In the 7-limit, 96 has two possible mappings for 7/4, a sharp one of 975 cents from the patent val, and a flat one of 962.5 cents from 96d. Using the sharp mapping, 96 tempers out 225/224 and supports 7-limit würschmidt temperament, and using the flat mapping it tempers out 126/125 and supports worschmidt temperament. We can also dispense with 7 altogether, and use it as a no-sevens system, where it tempers out 243/242 in the 11-limit and 676/675 in the 13-limit. If we include 7, then the sharp mapping tempers out 99/98 and 176/175 in the 11-limit, and 169/168 in the 13-limit, and this provides the optimal patent val for the interpental temperament. With the flat 7 it tempers out 385/384 in the 11-limit and 196/195 and 364/363 in the 13-limit, and serves for the various temperaments of the unicorn family.

Prime harmonics

Approximation of prime harmonics in 96edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.96 +1.19 +6.17 -1.32 -3.03 -4.96 +2.49 -3.27 -4.58 +4.96
Relative (%) +0.0 -15.6 +9.5 +49.4 -10.5 -24.2 -39.6 +19.9 -26.2 -36.6 +39.7
Steps
(reduced) 		96
(0) 		152
(56) 		223
(31) 		270
(78) 		332
(44) 		355
(67) 		392
(8) 		408
(24) 		434
(50) 		466
(82) 		476
(92)

Interval table

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 12.5 ^D, ^E♭♭
2 25 ^^D, ^^E♭♭
3 37.5 44/43 ^3D, ^3E♭♭
4 50 34/33, 37/36 ^4D, v4E♭
5 62.5 v3D♯, v3E♭
6 75 23/22, 24/23 vvD♯, vvE♭
7 87.5 20/19, 41/39 vD♯, vE♭
8 100 18/17 D♯, E♭
9 112.5 16/15 ^D♯, ^E♭
10 125 29/27, 43/40 ^^D♯, ^^E♭
11 137.5 13/12, 40/37 ^3D♯, ^3E♭
12 150 12/11 ^4D♯, v4E
13 162.5 11/10 v3D𝄪, v3E
14 175 21/19, 31/28 vvD𝄪, vvE
15 187.5 29/26 vD𝄪, vE
16 200 37/33 E
17 212.5 26/23, 35/31, 43/38 ^E, ^F♭
18 225 33/29, 41/36 ^^E, ^^F♭
19 237.5 39/34 ^3E, ^3F♭
20 250 15/13, 37/32 ^4E, v4F
21 262.5 43/37 v3E♯, v3F
22 275 27/23, 34/29 vvE♯, vvF
23 287.5 13/11 vE♯, vF
24 300 19/16, 25/21, 44/37 F
25 312.5 ^F, ^G♭♭
26 325 41/34 ^^F, ^^G♭♭
27 337.5 ^3F, ^3G♭♭
28 350 11/9, 38/31 ^4F, v4G♭
29 362.5 37/30 v3F♯, v3G♭
30 375 31/25, 36/29, 41/33 vvF♯, vvG♭
31 387.5 5/4 vF♯, vG♭
32 400 29/23, 34/27 F♯, G♭
33 412.5 33/26 ^F♯, ^G♭
34 425 23/18, 32/25 ^^F♯, ^^G♭
35 437.5 ^3F♯, ^3G♭
36 450 ^4F♯, v4G
37 462.5 17/13, 30/23 v3F𝄪, v3G
38 475 25/19 vvF𝄪, vvG
39 487.5 vF𝄪, vG
40 500 4/3 G
41 512.5 39/29, 43/32 ^G, ^A♭♭
42 525 23/17, 42/31 ^^G, ^^A♭♭
43 537.5 15/11 ^3G, ^3A♭♭
44 550 11/8 ^4G, v4A♭
45 562.5 18/13 v3G♯, v3A♭
46 575 vvG♯, vvA♭
47 587.5 vG♯, vA♭
48 600 41/29 G♯, A♭
49 612.5 37/26 ^G♯, ^A♭
50 625 33/23, 43/30 ^^G♯, ^^A♭
51 637.5 13/9 ^3G♯, ^3A♭
52 650 16/11 ^4G♯, v4A
53 662.5 22/15 v3G𝄪, v3A
54 675 31/21, 34/23 vvG𝄪, vvA
55 687.5 vG𝄪, vA
56 700 3/2 A
57 712.5 ^A, ^B♭♭
58 725 38/25, 41/27 ^^A, ^^B♭♭
59 737.5 23/15, 26/17 ^3A, ^3B♭♭
60 750 37/24 ^4A, v4B♭
61 762.5 v3A♯, v3B♭
62 775 25/16, 36/23 vvA♯, vvB♭
63 787.5 41/26 vA♯, vB♭
64 800 27/17 A♯, B♭
65 812.5 8/5 ^A♯, ^B♭
66 825 29/18, 37/23 ^^A♯, ^^B♭
67 837.5 ^3A♯, ^3B♭
68 850 18/11, 31/19 ^4A♯, v4B
69 862.5 v3A𝄪, v3B
70 875 vvA𝄪, vvB
71 887.5 vA𝄪, vB
72 900 32/19, 37/22, 42/25 B
73 912.5 22/13, 39/23 ^B, ^C♭
74 925 29/17, 41/24 ^^B, ^^C♭
75 937.5 43/25 ^3B, ^3C♭
76 950 26/15 ^4B, v4C
77 962.5 v3B♯, v3C
78 975 vvB♯, vvC
79 987.5 23/13 vB♯, vC
80 1000 41/23 C
81 1012.5 ^C, ^D♭♭
82 1025 38/21 ^^C, ^^D♭♭
83 1037.5 20/11 ^3C, ^3D♭♭
84 1050 11/6 ^4C, v4D♭
85 1062.5 24/13, 37/20 v3C♯, v3D♭
86 1075 vvC♯, vvD♭
87 1087.5 15/8 vC♯, vD♭
88 1100 17/9 C♯, D♭
89 1112.5 19/10 ^C♯, ^D♭
90 1125 23/12, 44/23 ^^C♯, ^^D♭
91 1137.5 ^3C♯, ^3D♭
92 1150 33/17 ^4C♯, v4D
93 1162.5 43/22 v3C𝄪, v3D
94 1175 vvC𝄪, vvD
95 1187.5 vC𝄪, vD
96 1200 2/1 D

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 3293216/390625, 531441/524288 [⟨96 152 223]] +0.240 0.732 5.86
2.3.5.11 243/242, 5632/5625, 131769/131072 [⟨96 152 223 332]] +0.276 0.637 5.10

History

96 equal divisions of the octave was first used by the Mexican composer and theorist Julián Carrillo. It has subsequently been used by a number of other composers.

Carrillo

Main article: Julián Carrillo

Other composers

Works for the Sauter's 1/16tone microtone piano by the composers Ernest Helmuth Flammer, Marc Kilchenmann, Bernfried E. G. Pröve, Martin Imholz, Franck Cristoph Yeznikian, Werner Grimmel, and Alain Bancquart, are recompilated on this CD: 'The Carrillo tone piano' .

Music

Julián Carrillo
Shahiin Mohajeri
Tony Salinas
Randy Wells
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