99edo: Difference between revisions
From Xenharmonic Reference
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{{Harmonics in ED|99}} | {{Harmonics in ED|99}} | ||
=== 7-prime-limited odd-limit analysis === | === 7-prime-limited odd-limit analysis === | ||
99edo is ''distinctly'' consistent and monotone (i.e. relative sizes of any two intervals are never conflated ''or'' reversed) | 99edo is ''distinctly'' consistent and monotone (i.e. relative sizes of any two intervals are never conflated ''or'' reversed) up to the 7-prime-limited 45-odd-limit, unlike all previous edos: | ||
36/35; 28/27; 16/15; 15/14; 27/25; 35/32; 10/9; 28/25; 9/8; 8/7; 7/6; 32/27; 25/21; 6/5; 56/45; 5/4; 32/25; 9/7; 35/27; 21/16; 4/3; 27/20; 48/35; 7/5; 45/32; 64/45; 10/7; 35/24; 40/27; 3/2; 32/21; 54/35; 14/9; 25/16; 8/5; 45/28; 5/3; 42/25; 27/16; 12/7; 7/4; 16/9; 25/14; 9/5; 64/35; 50/27; 28/15; 15/8; 27/14; 35/18; 2/1 | |||
The 7-prime-limited 49-odd-limit is where non-distinctness first shows up: namely, T49/48 = T50/49 (this is characteristic of all ennealimmal tunings). However 99edo remains monotone and consistent in these odd-limits: (todo) | The 7-prime-limited 49-odd-limit is where non-distinctness first shows up: namely, T49/48 = T50/49 (this is characteristic of all ennealimmal tunings). However 99edo remains monotone and consistent in these odd-limits: (todo) | ||
Revision as of 15:21, 8 April 2026
99edo is an equal tuning with steps of size 12.12... cents. It is arguably the edo below 100 that models 7-limit just intonation the most faithfully.
Theory
Prime approximations
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +1.1 | +1.6 | +0.9 | -5.9 | -4.2 | +4.1 | +5.5 | +2.0 | +0.7 | -5.6 |
| 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) | |
7-prime-limited odd-limit analysis
99edo is distinctly consistent and monotone (i.e. relative sizes of any two intervals are never conflated or reversed) up to the 7-prime-limited 45-odd-limit, unlike all previous edos:
36/35; 28/27; 16/15; 15/14; 27/25; 35/32; 10/9; 28/25; 9/8; 8/7; 7/6; 32/27; 25/21; 6/5; 56/45; 5/4; 32/25; 9/7; 35/27; 21/16; 4/3; 27/20; 48/35; 7/5; 45/32; 64/45; 10/7; 35/24; 40/27; 3/2; 32/21; 54/35; 14/9; 25/16; 8/5; 45/28; 5/3; 42/25; 27/16; 12/7; 7/4; 16/9; 25/14; 9/5; 64/35; 50/27; 28/15; 15/8; 27/14; 35/18; 2/1
The 7-prime-limited 49-odd-limit is where non-distinctness first shows up: namely, T49/48 = T50/49 (this is characteristic of all ennealimmal tunings). However 99edo remains monotone and consistent in these odd-limits: (todo)
Edostep interpretations
1\99 represents the following 7-limit ratios:
- 126/125
- 225/224
- 245/243
- 1029/1024
- 1728/1715
- 2048/2025
- 4000/3969
2\99 represents the following 7-limit ratios:
- 64/63
- 81/80
3\99 represents the following 7-limit ratios:
- 49/48
- 50/49
- 128/125
Temperaments
99edo notably supports
Derivation
Todo: Derive 99edo from
- Didacus
- Aberschismic
- Don't temper out 81/80, 64/63, or 128/125
- Ennealimmal
| View • Talk • EditEqual temperaments | |
|---|---|
| EDOs | |
| Macrotonal | 5 • 7 • 8 • 9 • 10 • 11 |
| 12-23 | 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 |
| 24-35 | 24 • 25 • 26 • 27 • 29 • 31 • 32 • 34 • 35 |
| 36-47 | 36 • 37 • 39 • 40 • 41 • 43 • 44 • 45 • 46 • 47 |
| 48-59 | 48 • 50 • 51 • 53 • 54 • 56 • 57 • 58 |
| 60-71 | 60 • 63 • 64 • 65 • 67 • 68 • 70 |
| 72-83 | 72 • 77 • 80 • 81 |
| 84-95 | 84 • 87 • 89 • 90 • 93 • 94 |
| Large EDOs | 99 • 104 • 106 • 111 • 118 • 130 • 140 • 152 • 159 • 171 • 217 • 224 • 239 • 270 • 306 • 311 • 612 • 665 |
| Nonoctave equal temperaments | |
| Tritave | 4 • 9 • 13 • 17 • 26 • 39 |
| Fifth | 8 • 9 • 11 • 20 |
| Other | |
