99edo
99 equal divisions of the octave (abbreviated 99edo or 99ed2), also called 99-tone equal temperament (99tet) or 99 equal temperament (99et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 99 equal parts of about 12.1 ¢ each. Each step represents a frequency ratio of 21/99, or the 99th root of 2. The step size of this system is close to 144/143, the grossma.
Theory
99edo is a very strong 7-limit (and 9-odd-limit) tuning, with a sound defined by the slight sharpness (1.1, 1.6, 0.9 cents) of its 3, 5, and 7. As an equal temperament, it tempers out 393216/390625 (würschmidt comma) and 1600000/1594323 (amity comma) in the 5-limit; 5120/5103 (argent comma), 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.
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 ⟨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. 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. The same can be said of the mapping for 13, with the 99ef val tempering out 144/143, 196/195, 352/351 and 364/363, and its patent val tempering out 169/168, 351/350 and 352/351. Hence 99edo, in spite of the fact that it tunes 11 and 13 relatively badly, is an important 13-limit tuning in more than one way.
Being a zeta peak edo, 99edo is also a very strong no-11 no-13 system, where it is consistent to the 29-odd-limit with a sharp tendency. This favors the sharp mapping of 11 and 13, and allows these relatively weak approximations to somewhat blend with the rest for a full 29-limit (or 31-limit, using the sharp-tending 99efk val) temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +1.08 | +1.57 | +0.87 | -5.86 | -4.16 | +4.14 | +5.52 | +2.03 | +0.73 | -5.64 |
| Relative (%) | +0.0 | +8.9 | +12.9 | +7.2 | -48.4 | -34.4 | +34.1 | +45.5 | +16.7 | +6.0 | -46.5 | |
| Steps (reduced) |
99 (0) |
157 (58) |
230 (32) |
278 (80) |
342 (45) |
366 (69) |
405 (9) |
421 (25) |
448 (52) |
481 (85) |
490 (94) | |
Subsets and supersets
Since 99 factors into 32 × 11, 99edo has subset edos 3, 9, 11, and 33. Splitting 99edo's step in half yields 198edo, correcting prime 11, slightly improving prime 13, and aligning both 11 and 13 with the sharp tunings of the lower odd primes. Because of this, 198edo can be seen as a complex yet notable true full 13-limit tuning.
Intervals
Notation
Ups and downs notation
99edo can be notated with Kite's ups and downs notation. Note that quip (quintuple-up) is the same as quudsharp (quadruple-down sharp) and that quid (quintuple-down) is the same as quupflat (quadruple-up flat):
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |
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Another notation uses alternative ups and downs. Here, this can be done using sharps and flats with arrows, borrowed from extended Helmholtz–Ellis notation:
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| Step offset | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
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Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [157 -99⟩ | [⟨99 157]] | −0.339 | 0.339 | 2.80 |
| 2.3.5 | 393216/390625, 1600000/1594323 | [⟨99 157 230]] | −0.451 | 0.319 | 2.63 |
| 2.3.5.7 | 2401/2400, 3136/3125, 4375/4374 | [⟨99 157 230 278]] | −0.416 | 0.283 | 2.33 |
| 2.3.5.7.11 | 243/242, 441/440, 896/891, 3136/3125 | [⟨99 157 230 278 343]] (99e) | −0.694 | 0.612 | 5.05 |
| 2.3.5.7.11 | 121/120, 176/175, 1375/1372, 2200/2187 | [⟨99 157 230 278 342]] (99) | +0.006 | 0.881 | 7.27 |
- 99et is lower in relative error than any previous equal temperaments in the 7-limit. Not until 171 do we find a better equal temperament in terms of either absolute error or relative error.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | 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 (99) / paralytic (99e) / parkleismic (99) / paradigmic (99e) |
| 1 | 28\99 | 339.394 | 128/105 | Amity (99ef) / hitchcock (99) |
| 1 | 29\99 | 351.515 | 49/40 | Hemififths (99ef) |
| 1 | 32\99 | 387.879 | 5/4 | Würschmidt / whirrschmidt |
| 1 | 41\99 | 496.970 | 4/3 | Undecental |
| 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 (99) / ennealimnic (99ef) / ennealim (99e) / ennealiminal (99) |
| 11 | 41\99 (4\99) |
496.970 (48.485) |
4/3 (36/35) |
Hendecatonic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Octave stretch or compression
99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly compressing the octave is acceptable, using tunings such as 157edt or 256ed6. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice.
If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable, such as in 567zpi.
Scales
Instruments
Skip fretting
Skip fretting system 99 6 11 is a skip fretting system for 99edo. The frets correspond to 16.5edo (33ed4). All intervals are for 7-string guitar.
- Harmonics
1/1: string 2 open
2/1: string 5 fret 11
3/2: string 4 fret 6
5/4 is not easily accessible, but the next-best approximation is at string 5 open.
7/4: string 6 fret 6
11/8: string 5 fret 2
13/8: string 5 fret 6
Keyboards
Lumatone mappings for 99edo are now available.
Music
- microtonal improvisation in 99edo (2023)
- 99edo waltz (2025)
- Cloudtop Reverie (2021) – zeus[7] in 99edo tuning
- Nonaginta et Novem (archived 2010) SoundCloud | details | play
- Benny Smith-Palestrina in zeus7tri
See also
- 58edf – relative edf
- 157edt – relative edt
- 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