99edo: Difference between revisions
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== Music == | == Music == | ||
Revision as of 13:56, 30 May 2023
| ← 98edo | 99edo | 100edo → |
The 99 equal divisions of the octave (99edo), or the 99(-tone) equal temperament (99tet, 99et) when viewed from a regular temperament perspective, is the equal division of the octave into 99 parts of about 12.1 cents each, a size close to 126/125, the starling comma.
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.
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) | |
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 (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 (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 |
Scales
Music
- Nonaginta et Novem play
- 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