99edo: Difference between revisions
ArrowHead294 (talk | contribs) m →Scales |
Tristanbay (talk | contribs) →Ups and downs notation: switched to SZ accidentals |
||
| Line 20: | Line 20: | ||
=== Ups and downs 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): | 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): | ||
{{Ups and downs sharpness}} | {{Ups and downs sharpness|99|true}} | ||
Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]: | Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]: | ||
Revision as of 06:17, 14 October 2025
| ← 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
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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol | 
|
 |
|
  |
|
|
|
|
 |
|
|
|
|
 |
|
|
|
|
 |
|
|
|
| Flat symbol | |
|
 |
|
|
|
|
 |
|
|
|
|
 |
|
|
|
|
 |
|
|
|
Another notation uses alternative ups and downs. Here, this can be done using sharps and flats with arrows, borrowed from extended Helmholtz–Ellis notation:
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol | |
 |
 |
 |
 |
|
 |
 |
 |
 |
|
 |
 |
| Flat symbol |  |
 |
 |
 |
|
 |
 |
 |
 |
|
 |
 |
| Step offset | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol |  |
 |
|
 |
 |
 |
 |
|
 |
 |
| Flat symbol |  |
 |
|
 |
 |
 |
 |
|
 |
 |
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