99edo: Difference between revisions
From Xenharmonic Reference
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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 | 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 the 7-prime-limited 125-odd-limit | 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 the 7-prime-limited 125-odd-limit: | ||
126/125; 81/80; 64/63; 50/49; 49/48; 128/125; 36/35; 28/27; 16/15; 15/14; 27/25; 35/32; 54/49; 10/9; 125/112; 28/25; 9/8; 8/7; 125/108; 7/6; 75/64; 32/27; 25/21; 6/5; 98/81; 128/105; 49/40; 100/81; 56/45; 5/4; 63/50; 81/64; 80/63; 32/25; 9/7; 162/125; 35/27; 125/96; 64/49; 98/75; 21/16; 4/3; 75/56; 168/125; 27/20; 49/36; 48/35; 112/81; 7/5; 45/32; 64/45; 10/7; 81/56; 35/24; 72/49; 40/27; 125/84; 112/75; 3/2; 32/21; 75/49; 49/32; 192/125; 54/35; 125/81; 14/9; 25/16; 63/40; 128/81; 100/63; 8/5; 45/28; 81/50; 80/49; 105/64; 81/49; 5/3; 42/25; 27/16; 128/75; 12/7; 216/125; 7/4; 16/9; 25/14; 224/125; 9/5; 49/27; 64/35; 50/27; 28/15; 15/8; 27/14; 35/18; 125/64; 96/49; 49/25; 63/32; 160/81; 125/63; 2/1 | |||
=== Edostep interpretations === | === Edostep interpretations === | ||
Revision as of 16:15, 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 the 7-prime-limited 125-odd-limit:
126/125; 81/80; 64/63; 50/49; 49/48; 128/125; 36/35; 28/27; 16/15; 15/14; 27/25; 35/32; 54/49; 10/9; 125/112; 28/25; 9/8; 8/7; 125/108; 7/6; 75/64; 32/27; 25/21; 6/5; 98/81; 128/105; 49/40; 100/81; 56/45; 5/4; 63/50; 81/64; 80/63; 32/25; 9/7; 162/125; 35/27; 125/96; 64/49; 98/75; 21/16; 4/3; 75/56; 168/125; 27/20; 49/36; 48/35; 112/81; 7/5; 45/32; 64/45; 10/7; 81/56; 35/24; 72/49; 40/27; 125/84; 112/75; 3/2; 32/21; 75/49; 49/32; 192/125; 54/35; 125/81; 14/9; 25/16; 63/40; 128/81; 100/63; 8/5; 45/28; 81/50; 80/49; 105/64; 81/49; 5/3; 42/25; 27/16; 128/75; 12/7; 216/125; 7/4; 16/9; 25/14; 224/125; 9/5; 49/27; 64/35; 50/27; 28/15; 15/8; 27/14; 35/18; 125/64; 96/49; 49/25; 63/32; 160/81; 125/63; 2/1
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
