99edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
The | {{ED intro}} The step size of this system is close to [[126/125]], the starling comma. | ||
== Theory == | == Theory == | ||
99edo is a very strong [[7-limit]] (and [[9-odd-limit]]) tuning. It [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and 1600000/1594323 ([[amity comma]]) in the [[5-limit]]; 2401/2400 ([[2401/2400|breedsma]]), 3136/3125 ([[hemimean comma]]), and 4375/4374 ([[4375/4374|ragisma]]) in the [[7-limit]], [[support]]ing [[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. | 99edo is a very strong [[7-limit]] (and [[9-odd-limit]]) tuning. It [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and 1600000/1594323 ([[amity comma]]) in the [[5-limit]]; 2401/2400 ([[2401/2400|breedsma]]), 3136/3125 ([[hemimean comma]]), and 4375/4374 ([[4375/4374|ragisma]]) in the [[7-limit]], [[support]]ing [[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 (1.1, 1.6, 0.9 cents) of its [[3/1|3]], [[5/1|5]], and [[7/1|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 {{val| 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. | 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 {{val| 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]]. | The same can be said of the mapping for [[13/1|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. | 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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=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|99}} | {{Harmonics in equal|99}} | ||
=== Octave stretch === | |||
99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|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. | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 99 factors into {{ | Since 99 factors into {{nowrap| 3<sup>2</sup> × 11 }}, 99edo has subset edos {{EDOs| 3, 9, 11, and 33 }}. | ||
== Intervals == | == Intervals == | ||
| Line 48: | Line 51: | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{ | | {{Monzo| 157 -99 }} | ||
| {{ | | {{Mapping| 99 157 }} | ||
| −0.339 | | −0.339 | ||
| 0.339 | | 0.339 | ||
| Line 56: | Line 59: | ||
| 2.3.5 | | 2.3.5 | ||
| 393216/390625, 1600000/1594323 | | 393216/390625, 1600000/1594323 | ||
| {{ | | {{Mapping| 99 157 230 }} | ||
| −0.451 | | −0.451 | ||
| 0.319 | | 0.319 | ||
| Line 63: | Line 66: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 3136/3125, 4375/4374 | | 2401/2400, 3136/3125, 4375/4374 | ||
| {{ | | {{Mapping| 99 157 230 278 }} | ||
| −0.416 | | −0.416 | ||
| 0.283 | | 0.283 | ||
| Line 70: | Line 73: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 243/242, 441/440, 896/891, 3136/3125 | | 243/242, 441/440, 896/891, 3136/3125 | ||
| {{ | | {{Mapping| 99 157 230 278 343 }} (99e) | ||
| −0.694 | | −0.694 | ||
| 0.612 | | 0.612 | ||
| Line 77: | Line 80: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 121/120, 176/175, 1375/1372, 2200/2187 | | 121/120, 176/175, 1375/1372, 2200/2187 | ||
| {{ | | {{Mapping| 99 157 230 278 342 }} (99) | ||
| +0.006 | | +0.006 | ||
| 0.881 | | 0.881 | ||
| Line 217: | Line 220: | ||
== See also == | == See also == | ||
* [[58edf]] – relative [[edf]] | |||
* [[157edt]] – relative [[edt]] | * [[157edt]] – relative [[edt]] | ||
* [[87edo]], [[94edo]], [[111edo]] – similarly sized edos all with consistency in higher harmonics. | * [[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. | * [[198edo]], the half-sized edo to reconcile the mappings of 11 and 13. | ||
Revision as of 15:14, 23 March 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 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 (1.1, 1.6, 0.9 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) | |
Octave stretch
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.
Subsets and supersets
Since 99 factors into 32 × 11, 99edo has subset edos 3, 9, 11, and 33.
Intervals
Approximation to JI
Zeta peak index
| Tuning | Strength | Octave (cents) | Integer limit | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per 8ve |
Step size (cents) |
Height | Integral | Gap | Size | Stretch | Consistent | Distinct | |
| Tempered | Pure | |||||||||
| 568zpi | 99.047335 | 12.115419 | 9.406495 | 7.543343 | 1.510412 | 18.536483 | 1199.426522 | −0.573478 | 12 | 12 |
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
Scales
Music
- 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
- Skip fretting system 99 6 11