99edo
99edo is the equal division of the octave into 99 parts of 12.1212 cents each.
Theory
99edo is a very strong 7-limit (and 9-odd-limit) tuning. It tempers out 393216/390625 (würschmidt comma) and 1600000/1594323 (amity comma) in the 5-limit; 2401/2400 (breedsma), 3136/3125 (hemimean comma), and 4375/4374 (ragisma) in the 7-limit, supporting hemififths, amity, parakleismic, hemiwürschmidt and ennealimmal temperaments, and is pretty well a perfect tuning for hendecatonic temperament. It has a sound defined by the slight sharpness (1.075, 1.565, 0.871 cents) of its 3, 5, and 7.
Extending it to the 11-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the patent val, 99edo is the optimal patent val for the rank-4 temperament tempering out 121/120; zeus, the rank-3 temperament tempering out 121/120 and 176/175; hemiwür, one of the rank-2 11-limit extensions of hemiwürschmidt; and hitchcock (an 11-limit amity extension), the rank-2 temperament which also tempers out 2200/2187. Using the ⟨99 157 230 278 343] (99e) val, it tempers out 243/242, 441/440, 540/539 and 896/891, and is an excellent tuning for the 11-limit version of hemififths temperament. Hence 99 equal divisions, in spite of the fact that it tunes 11 relatively badly, is an important 11-limit tuning in more than one way.
The same can be said of the mapping for 13, with its patent val tempering out 169/168, 351/350 and 352/351, and the 99ef val tempering out 144/143, 196/195, 352/351 and 364/363.
Skipping 11 and 13, it is a very strong system in the 2.3.5.7.17.19.23.29 subgroup.
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Intervals
Temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperament |
|---|---|---|---|---|
| 1 | 2\99 | 24.242 | 686/675, 99/98 | Sengagen (99e) / Sengage (99ef) |
| 1 | 7\99 | 84.848 | 21/20 | Amicable |
| 1 | 16\99 | 193.939 | 28/25 | Hemiwürschmidt (99e) / Hemithir (99ef) / Hemiwur (99f) |
| 1 | 19\99 | 230.303 | 8/7 | Gamera |
| 1 | 20\99 | 242.424 | 147/128 | Septiquarter |
| 1 | 25\99 | 303.030 | 25/21 | Quinmite |
| 1 | 26\99 | 315.152 | 6/5 | Parakleismic / Parkleismic / Paradigmic (99e) |
| 1 | 28\99 | 339.394 | 128/105 | Amity (99ef) / Hitchcock |
| 1 | 29\99 | 351.515 | 49/40 | Hemififths (99ef) |
| 1 | 32\99 | 387.879 | 5/4 | Würschmidt / Whirrschmidt |
| 1 | 37\99 | 448.485 | 35/27 | Semidimfourth |
| 3 | 5\99 | 60.606 | 28/27 | Chromat |
| 3 | 13\99 | 157.576 | 35/32 | Nessafof |
| 3 | 41\99 (8\99) |
496.970 (96.970) |
4/3 (18/17~19/18) |
Misty |
| 9 | 4\99 | 48.485 | 36/35 | Ennealimmal (99e) / ennealimmia / ennealimnic (99ef) / ennealim (99e) / ennealiminal |
| 11 | 41\99 (4\99) |
496.970 (48.485) |
4/3 (36/35) |
Hendecatonic |
Scales
Since 99edo has a step of 12.1212 cents, it also allows one to use its MOS scales as circulating temperaments[clarification needed].
| Tones | Pattern | L:s |
|---|---|---|
| 5 | 4L 1s | 20:19 |
| 6 | 3L 3s | 17:16 |
| 7 | 1L 6s | 15:14 |
| 8 | 3L 5s | 13:12 |
| 9 | 9edo | equal |
| 10 | 9L 1s | 10:9 |
| 11 | 11edo | equal |
| 12 | 3L 9s | 9:8 |
| 13 | 8L 5s | 8:7 |
| 14 | 1L 13s | |
| 15 | 9L 6s | 7:6 |
| 16 | 3L 13s | |
| 17 | 14L 3s | 6:5 |
| 18 | 9L 9s | |
| 19 | 4L 15s | |
| 20 | 19L 1s | 5:4 |
| 21 | 15L 6s | |
| 22 | 11L 11s | |
| 23 | 7L 16s | |
| 24 | 3L 21s | |
| 25 | 24L 1s | 4:3 |
| 26 | 21L 5s | |
| 27 | 18L 9s | |
| 28 | 15L 13s | |
| 29 | 12L 17s | |
| 30 | 9L 21s | |
| 31 | 6L 25s | |
| 32 | 3L 29s | |
| 33 | 33edo | equal |
| 34 | 31L 3s | 3:2 |
| 35 | 29L 6s | |
| 36 | 27L 9s | |
| 37 | 25L 12s | |
| 38 | 23L 15s | |
| 39 | 21L 18s | |
| 40 | 19L 21s | |
| 41 | 17L 24s | |
| 42 | 15L 27s | |
| 43 | 13L 30s | |
| 44 | 11L 33s | |
| 45 | 9L 36s | |
| 46 | 7L 39s | |
| 47 | 5L 42s | |
| 48 | 3L 45s | |
| 49 | 1L 48s | |
| 50 | 49L 1s | 2:1 |
| 51 | 48L 3s | |
| 52 | 47L 5s | |
| 53 | 46L 7s | |
| 54 | 45L 9s | |
| 55 | 44L 11s | |
| 56 | 43L 13s | |
| 57 | 42L 15s | |
| 58 | 41L 17s | |
| 59 | 40L 19s | |
| 60 | 39L 21s | |
| 61 | 38L 23s | |
| 62 | 37L 25s | |
| 63 | 36L 27s | |
| 64 | 35L 29s | |
| 65 | 34L 31s | |
| 66 | 33L 33s | |
| 67 | 32L 35s | |
| 68 | 31L 37s | |
| 69 | 30L 39s | |
| 70 | 29L 41s | |
| 71 | 28L 43s | |
| 72 | 27L 45s | |
| 73 | 26L 47s | |
| 74 | 25L 49s | |
| 75 | 24L 51s | |
| 76 | 23L 53s | |
| 77 | 22L 55s | |
| 78 | 21L 57s | |
| 79 | 20L 59s |
Music
- Nonaginta et Novem play by Gene Ward Smith
- Benny Smith-Palestrina in zeus7tri
See also
- 157edt – relative EDT
- 58edf – relative EDF
- 87edo, 94edo, 111edo – similarly sized edos all with consistency in higher harmonics.
- 198edo, the half-sized edo to reconcile the mappings of 11 and 13.
- 105edo, a similarly sized edo that supports meantone, septimal meantone, undecimal meantone and grosstone