96edo
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 cents each.
| ← 95edo | 96edo | 97edo → |
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
| 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
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' .