99edo
| ← 98edo | 99edo | 100edo → |
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; 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.
Intervals
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.
What follows is a comparison of stretched- and compressed-octave 99edo tunings.
- Step size: 12.138 ¢, octave size: 1201.66 ¢
Stretching the octave of 99edo by around 1.5 ¢ results in improved primes 11, 13, 17, and 19, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.54 ¢. The tuning 567zpi does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.66 | +3.71 | +3.32 | +5.43 | +5.37 | +5.54 | +4.99 | -4.72 | -5.05 | -0.12 | -5.10 |
| Relative (%) | +13.7 | +30.6 | +27.4 | +44.7 | +44.3 | +45.6 | +41.1 | -38.9 | -41.6 | -1.0 | -42.0 | |
| Step | 99 | 157 | 198 | 230 | 256 | 278 | 297 | 313 | 328 | 342 | 354 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.98 | -4.94 | -3.00 | -5.49 | -1.20 | -3.05 | +0.45 | -3.39 | -2.89 | +1.54 | -2.59 | -3.44 |
| Relative (%) | +16.3 | -40.7 | -24.7 | -45.2 | -9.9 | -25.2 | +3.7 | -27.9 | -23.8 | +12.7 | -21.3 | -28.3 | |
| Step | 366 | 376 | 386 | 395 | 404 | 412 | 420 | 427 | 434 | 441 | 447 | 453 | |
- Step size: 12.123 ¢, octave size: 1200.18 ¢
Stretching the octave of 99edo by around a fifth of a cent results in improved primes 11 and 13, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.25 ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.18 | +1.36 | +0.35 | +1.98 | +1.53 | +1.37 | +0.53 | +2.71 | +2.15 | -5.25 | +1.71 |
| Relative (%) | +1.5 | +11.2 | +2.9 | +16.3 | +12.6 | +11.3 | +4.4 | +22.4 | +17.8 | -43.3 | +14.1 | |
| Step | 99 | 157 | 198 | 230 | 256 | 278 | 297 | 314 | 329 | 342 | 355 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.51 | +1.55 | +3.33 | +0.71 | +4.86 | +2.89 | -5.85 | +2.33 | +2.72 | -5.07 | +2.83 | +1.89 |
| Relative (%) | -29.0 | +12.7 | +27.5 | +5.8 | +40.1 | +23.8 | -48.3 | +19.2 | +22.5 | -41.9 | +23.3 | +15.6 | |
| Step | 366 | 377 | 387 | 396 | 405 | 413 | 420 | 428 | 435 | 441 | 448 | 454 | |
- 99edo
- Step size: 12.121 ¢, octave size: 1200.00 ¢
Pure-octaves 99edo approximates all harmonics up to 16 within 5.86 ¢.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +1.08 | +0.00 | +1.57 | +1.08 | +0.87 | +0.00 | +2.15 | +1.57 | -5.86 | +1.08 |
| Relative (%) | +0.0 | +8.9 | +0.0 | +12.9 | +8.9 | +7.2 | +0.0 | +17.7 | +12.9 | -48.4 | +8.9 | |
| Steps (reduced) |
99 (0) |
157 (58) |
198 (0) |
230 (32) |
256 (58) |
278 (80) |
297 (0) |
314 (17) |
329 (32) |
342 (45) |
355 (58) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.16 | +0.87 | +2.64 | +0.00 | +4.14 | +2.15 | +5.52 | +1.57 | +1.95 | -5.86 | +2.03 | +1.08 |
| Relative (%) | -34.4 | +7.2 | +21.8 | +0.0 | +34.1 | +17.7 | +45.5 | +12.9 | +16.1 | -48.4 | +16.7 | +8.9 | |
| Steps (reduced) |
366 (69) |
377 (80) |
387 (90) |
396 (0) |
405 (9) |
413 (17) |
421 (25) |
428 (32) |
435 (39) |
441 (45) |
448 (52) |
454 (58) | |
- Step size: 12.117 ¢, octave size: 1199.58 ¢
Compressing the octave of 99edo by around 0.6 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.71 ¢. Its 7-limit WE tuning and 7-limit TE tuning both do this. So does the tuning 256ed6 whose octave is identical within a thousandth of a cent.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.42 | +0.41 | -0.83 | +0.60 | -0.00 | -0.30 | -1.25 | +0.83 | +0.18 | +4.81 | -0.42 |
| Relative (%) | -3.4 | +3.4 | -6.9 | +4.9 | -0.0 | -2.5 | -10.3 | +6.8 | +1.5 | +39.7 | -3.5 | |
| Step | 99 | 157 | 198 | 230 | 256 | 278 | 297 | 314 | 329 | 343 | 355 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.71 | -0.72 | +1.01 | -1.67 | +2.43 | +0.41 | +3.74 | -0.24 | +0.11 | +4.40 | +0.14 | -0.84 |
| Relative (%) | -47.1 | -5.9 | +8.3 | -13.8 | +20.1 | +3.4 | +30.9 | -2.0 | +0.9 | +36.3 | +1.2 | -6.9 | |
| Step | 366 | 377 | 387 | 396 | 405 | 413 | 421 | 428 | 435 | 442 | 448 | 454 | |
- Step size: 12.115 ¢, octave size: 1199.39 ¢
Compressing the octave of 99edo by around 0.4 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.68 ¢. The tuning 568zpi does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.61 | +0.10 | -1.23 | +0.14 | -0.52 | -0.86 | -1.84 | +0.20 | -0.48 | +4.13 | -1.13 |
| Relative (%) | -5.1 | +0.8 | -10.2 | +1.1 | -4.3 | -7.1 | -15.2 | +1.7 | -4.0 | +34.1 | -9.3 | |
| Step | 99 | 157 | 198 | 230 | 256 | 278 | 297 | 314 | 329 | 343 | 355 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.68 | -1.47 | +0.24 | -2.46 | +1.62 | -0.42 | +2.90 | -1.09 | -0.76 | +3.51 | -0.75 | -1.75 |
| Relative (%) | +46.9 | -12.1 | +2.0 | -20.3 | +13.4 | -3.4 | +24.0 | -9.0 | -6.2 | +29.0 | -6.2 | -14.4 | |
| Step | 367 | 377 | 387 | 396 | 405 | 413 | 421 | 428 | 435 | 442 | 448 | 454 | |
- Step size: 12.114 ¢, octave size: 1199.32 ¢
Compressing the octave of 99edo by around 0.3 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.44 ¢. The tuning 157edt does this. So does 230ed5 whose octave is identical within a hundredth of a cent.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.68 | +0.00 | -1.36 | -0.01 | -0.68 | -1.03 | -2.03 | +0.00 | -0.69 | +3.91 | -1.36 |
| Relative (%) | -5.6 | +0.0 | -11.2 | -0.1 | -5.6 | -8.5 | -16.8 | +0.0 | -5.7 | +32.3 | -11.2 | |
| Steps (reduced) |
99 (99) |
157 (0) |
198 (41) |
230 (73) |
256 (99) |
278 (121) |
297 (140) |
314 (0) |
329 (15) |
343 (29) |
355 (41) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.44 | -1.71 | -0.01 | -2.71 | +1.36 | -0.68 | +2.63 | -1.37 | -1.03 | +3.23 | -1.04 | -2.03 |
| Relative (%) | +44.9 | -14.1 | -0.1 | -22.4 | +11.2 | -5.6 | +21.7 | -11.3 | -8.5 | +26.7 | -8.6 | -16.8 | |
| Steps (reduced) |
367 (53) |
377 (63) |
387 (73) |
396 (82) |
405 (91) |
413 (99) |
421 (107) |
428 (114) |
435 (121) |
442 (128) |
448 (134) |
454 (140) | |
- Step size: 12.103 ¢, octave size: 1199.16 ¢
Compressing the octave of 99edo by around 1 ¢ results in improved primes 11, 13, 17 and 19 but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.98 ¢. The tuning 58edf does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.84 | -1.84 | -3.67 | -2.70 | -3.67 | -4.28 | -5.51 | -3.67 | -4.53 | -0.10 | -5.51 |
| Relative (%) | -15.2 | -15.2 | -30.3 | -22.3 | -30.3 | -35.4 | -45.5 | -30.3 | -37.5 | -0.8 | -45.5 | |
| Steps (reduced) |
99 (41) |
157 (41) |
198 (24) |
230 (56) |
256 (24) |
278 (46) |
297 (7) |
314 (24) |
329 (39) |
343 (53) |
355 (7) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.15 | +5.98 | -4.53 | +4.76 | -3.37 | -5.51 | -2.29 | +5.73 | +5.98 | -1.94 | +5.83 | +4.76 |
| Relative (%) | +9.5 | +49.4 | -37.5 | +39.3 | -27.9 | -45.5 | -18.9 | +47.4 | +49.4 | -16.0 | +48.1 | +39.3 | |
| Steps (reduced) |
367 (19) |
378 (30) |
387 (39) |
397 (49) |
405 (57) |
413 (7) |
421 (15) |
429 (23) |
436 (30) |
442 (36) |
449 (43) |
455 (49) | |
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