96edo

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← 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:

96 equal divisions of the octave (abbreviated 96edo or 96ed2), also called 96-tone equal temperament (96tet) or 96 equal temperament (96et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 96 equal parts of exactly 12.5 ¢ each. Each step represents a frequency ratio of 21/96, or the 96th root of 2.

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. It supports sitcom temperament.

One notable benefit of 96edo's representation of the 5-limit is that its dramatic narrowing of 81/80 allows for a less dissonant ~40/27 wolf fifth. This allows for the potential of a 12-note subset of 96edo being seen as a well temperament, and as part of an equal temperament, this scale could be rotated around during the run-time of a piece of music.

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)

As a tuning standard

A step of 96edo is known as a triamu (third MIDI-resolution unit, 3mu, 23 = 8 equal divisions of the 12edo semitone). The internal data structure of the 3mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and three other bits which are not used.

Subsets and supersets

Since 96 factors into 25 × 3, 96edo has subset edos 2, 3, 4, 6, 8, 12, 16, 24, 32, and 48.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 12.5 ^D, v7E♭
2 25 ^^D, v6E♭
3 37.5 44/43 ^3D, v5E♭
4 50 34/33, 37/36 ^4D, v4E♭
5 62.5 ^5D, v3E♭
6 75 23/22, 24/23 ^6D, vvE♭
7 87.5 20/19, 41/39 ^7D, vE♭
8 100 18/17 D♯, E♭
9 112.5 16/15 ^D♯, v7E
10 125 29/27, 43/40 ^^D♯, v6E
11 137.5 13/12, 40/37 ^3D♯, v5E
12 150 12/11 ^4D♯, v4E
13 162.5 11/10 ^5D♯, v3E
14 175 21/19, 31/28 ^6D♯, vvE
15 187.5 29/26 ^7D♯, vE
16 200 37/33 E
17 212.5 26/23, 35/31, 43/38 ^E, v7F
18 225 33/29, 41/36 ^^E, v6F
19 237.5 39/34 ^3E, v5F
20 250 15/13, 37/32 ^4E, v4F
21 262.5 43/37 ^5E, v3F
22 275 27/23, 34/29 ^6E, vvF
23 287.5 13/11 ^7E, vF
24 300 19/16, 25/21, 44/37 F
25 312.5 ^F, v7G♭
26 325 41/34 ^^F, v6G♭
27 337.5 ^3F, v5G♭
28 350 11/9, 38/31 ^4F, v4G♭
29 362.5 37/30 ^5F, v3G♭
30 375 31/25, 36/29, 41/33 ^6F, vvG♭
31 387.5 5/4 ^7F, vG♭
32 400 29/23, 34/27 F♯, G♭
33 412.5 33/26 ^F♯, v7G
34 425 23/18, 32/25 ^^F♯, v6G
35 437.5 ^3F♯, v5G
36 450 ^4F♯, v4G
37 462.5 17/13, 30/23 ^5F♯, v3G
38 475 25/19 ^6F♯, vvG
39 487.5 ^7F♯, vG
40 500 4/3 G
41 512.5 39/29, 43/32 ^G, v7A♭
42 525 23/17, 42/31 ^^G, v6A♭
43 537.5 15/11 ^3G, v5A♭
44 550 11/8 ^4G, v4A♭
45 562.5 18/13 ^5G, v3A♭
46 575 ^6G, vvA♭
47 587.5 ^7G, vA♭
48 600 41/29 G♯, A♭
49 612.5 37/26 ^G♯, v7A
50 625 33/23, 43/30 ^^G♯, v6A
51 637.5 13/9 ^3G♯, v5A
52 650 16/11 ^4G♯, v4A
53 662.5 22/15 ^5G♯, v3A
54 675 31/21, 34/23 ^6G♯, vvA
55 687.5 ^7G♯, vA
56 700 3/2 A
57 712.5 ^A, v7B♭
58 725 38/25, 41/27 ^^A, v6B♭
59 737.5 23/15, 26/17 ^3A, v5B♭
60 750 37/24 ^4A, v4B♭
61 762.5 ^5A, v3B♭
62 775 25/16, 36/23 ^6A, vvB♭
63 787.5 41/26 ^7A, vB♭
64 800 27/17 A♯, B♭
65 812.5 8/5 ^A♯, v7B
66 825 29/18, 37/23 ^^A♯, v6B
67 837.5 ^3A♯, v5B
68 850 18/11, 31/19 ^4A♯, v4B
69 862.5 ^5A♯, v3B
70 875 ^6A♯, vvB
71 887.5 ^7A♯, vB
72 900 32/19, 37/22, 42/25 B
73 912.5 22/13, 39/23 ^B, v7C
74 925 29/17, 41/24 ^^B, v6C
75 937.5 43/25 ^3B, v5C
76 950 26/15 ^4B, v4C
77 962.5 ^5B, v3C
78 975 ^6B, vvC
79 987.5 23/13 ^7B, vC
80 1000 41/23 C
81 1012.5 ^C, v7D♭
82 1025 38/21 ^^C, v6D♭
83 1037.5 20/11 ^3C, v5D♭
84 1050 11/6 ^4C, v4D♭
85 1062.5 24/13, 37/20 ^5C, v3D♭
86 1075 ^6C, vvD♭
87 1087.5 15/8 ^7C, vD♭
88 1100 17/9 C♯, D♭
89 1112.5 19/10 ^C♯, v7D
90 1125 23/12, 44/23 ^^C♯, v6D
91 1137.5 ^3C♯, v5D
92 1150 33/17 ^4C♯, v4D
93 1162.5 43/22 ^5C♯, v3D
94 1175 ^6C♯, vvD
95 1187.5 ^7C♯, vD
96 1200 2/1 D

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3.5 393216/390625, 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

Rank-2 temperaments

Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperament
1 5\96 62.5 28/27 Unicorn / camahueto (96d)
1 21\96 262.5 64/55 Spog
1 29\96 362.5 16/13 Submajor / interpental (96)
1 31\96 387.5 5/4 Würschmidt (96) / worschmidt (96d)
2 5\96 62.5 28/27 Monocerus (96d)
2 13\96 162.5 11/10 Gwazy (96) / bisupermajor (96d)
2 25\96 312.5 6/5 Vines (96d)
12 31\96
(1\96) 		387.5
(12.5) 		5/4
(126/125) 		Compton (7-limit, 96)
24 31\96
(1\96) 		387.5
(12.5) 		5/4
(245/243) 		Hours (96d)

Scales

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

See also

External links

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