101edo
| ← 100edo | 101edo | 102edo → |
101 equal divisions of the octave (abbreviated 101edo or 101ed2), also called 101-tone equal temperament (101tet) or 101 equal temperament (101et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 101 equal parts of about 11.9 ¢ each. Each step represents a frequency ratio of 21/101, or the 101st root of 2.
Theory
101edo is inconsistent in the 5-odd-limit, with harmonics 5 and 7 falling about halfway between its steps. As such, ⟨101 160 235 284] (patent val) and ⟨101 160 234 283] (101cd) are about as viable. Using the patent val, it tempers out 32805/32768 (schisma) and 51018336/48828125 in the 5-limit; 126/125 and 2430/2401 in the 7-limit. It can be used to tune the grackle temperament. The 101cd val provides an excellent tuning for witchcraft temperament, falling between the 13- and 15-odd-limit least squares tuning.
Odd harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.96 | +5.77 | +5.43 | -4.78 | +3.04 | +1.98 | -0.48 | +1.43 | +4.09 | -4.44 |
| Relative (%) | +0.0 | -8.1 | +48.5 | +45.7 | -40.3 | +25.6 | +16.6 | -4.1 | +12.0 | +34.4 | -37.4 | |
| Steps (reduced) |
101 (0) |
160 (59) |
235 (33) |
284 (82) |
349 (46) |
374 (71) |
413 (9) |
429 (25) |
457 (53) |
491 (87) |
500 (96) | |
Subsets and supersets
101edo is the 26th prime edo, following 97edo and before 103edo. 202edo, which doubles it, provides a good correction to the 5th, 7th, and 11th harmonics.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 11.9 | ^D, ^^E♭♭ | |
| 2 | 23.8 | ^^D, ^3E♭♭ | |
| 3 | 35.6 | ^3D, ^4E♭♭ | |
| 4 | 47.5 | 37/36, 38/37 | ^4D, v4E♭ |
| 5 | 59.4 | 29/28, 30/29 | v4D♯, v3E♭ |
| 6 | 71.3 | 24/23 | v3D♯, vvE♭ |
| 7 | 83.2 | 21/20, 43/41 | vvD♯, vE♭ |
| 8 | 95 | 19/18 | vD♯, E♭ |
| 9 | 106.9 | 17/16, 33/31 | D♯, ^E♭ |
| 10 | 118.8 | 15/14 | ^D♯, ^^E♭ |
| 11 | 130.7 | 14/13, 41/38 | ^^D♯, ^3E♭ |
| 12 | 142.6 | ^3D♯, ^4E♭ | |
| 13 | 154.5 | ^4D♯, v4E | |
| 14 | 166.3 | v4D𝄪, v3E | |
| 15 | 178.2 | 41/37 | v3D𝄪, vvE |
| 16 | 190.1 | 29/26 | vvD𝄪, vE |
| 17 | 202 | 9/8 | E |
| 18 | 213.9 | 26/23, 43/38 | ^E, ^^F♭ |
| 19 | 225.7 | 41/36 | ^^E, ^3F♭ |
| 20 | 237.6 | 31/27, 39/34 | ^3E, ^4F♭ |
| 21 | 249.5 | 15/13, 37/32 | ^4E, v4F |
| 22 | 261.4 | 43/37 | v4E♯, v3F |
| 23 | 273.3 | 34/29 | v3E♯, vvF |
| 24 | 285.1 | vvE♯, vF | |
| 25 | 297 | 19/16 | F |
| 26 | 308.9 | 43/36 | ^F, ^^G♭♭ |
| 27 | 320.8 | ^^F, ^3G♭♭ | |
| 28 | 332.7 | 23/19 | ^3F, ^4G♭♭ |
| 29 | 344.6 | 39/32 | ^4F, v4G♭ |
| 30 | 356.4 | 27/22 | v4F♯, v3G♭ |
| 31 | 368.3 | 26/21 | v3F♯, vvG♭ |
| 32 | 380.2 | vvF♯, vG♭ | |
| 33 | 392.1 | vF♯, G♭ | |
| 34 | 404 | 24/19 | F♯, ^G♭ |
| 35 | 415.8 | ^F♯, ^^G♭ | |
| 36 | 427.7 | 41/32 | ^^F♯, ^3G♭ |
| 37 | 439.6 | ^3F♯, ^4G♭ | |
| 38 | 451.5 | ^4F♯, v4G | |
| 39 | 463.4 | 17/13 | v4F𝄪, v3G |
| 40 | 475.2 | v3F𝄪, vvG | |
| 41 | 487.1 | 45/34 | vvF𝄪, vG |
| 42 | 499 | 4/3 | G |
| 43 | 510.9 | 39/29, 43/32 | ^G, ^^A♭♭ |
| 44 | 522.8 | 23/17 | ^^G, ^3A♭♭ |
| 45 | 534.7 | ^3G, ^4A♭♭ | |
| 46 | 546.5 | 37/27 | ^4G, v4A♭ |
| 47 | 558.4 | 29/21, 40/29 | v4G♯, v3A♭ |
| 48 | 570.3 | 32/23 | v3G♯, vvA♭ |
| 49 | 582.2 | 7/5 | vvG♯, vA♭ |
| 50 | 594.1 | 31/22, 38/27 | vG♯, A♭ |
| 51 | 605.9 | 27/19, 44/31 | G♯, ^A♭ |
| 52 | 617.8 | 10/7 | ^G♯, ^^A♭ |
| 53 | 629.7 | 23/16 | ^^G♯, ^3A♭ |
| 54 | 641.6 | 29/20, 42/29 | ^3G♯, ^4A♭ |
| 55 | 653.5 | ^4G♯, v4A | |
| 56 | 665.3 | v4G𝄪, v3A | |
| 57 | 677.2 | 34/23 | v3G𝄪, vvA |
| 58 | 689.1 | vvG𝄪, vA | |
| 59 | 701 | 3/2 | A |
| 60 | 712.9 | ^A, ^^B♭♭ | |
| 61 | 724.8 | 41/27 | ^^A, ^3B♭♭ |
| 62 | 736.6 | 26/17 | ^3A, ^4B♭♭ |
| 63 | 748.5 | 37/24 | ^4A, v4B♭ |
| 64 | 760.4 | 45/29 | v4A♯, v3B♭ |
| 65 | 772.3 | v3A♯, vvB♭ | |
| 66 | 784.2 | vvA♯, vB♭ | |
| 67 | 796 | 19/12 | vA♯, B♭ |
| 68 | 807.9 | 43/27 | A♯, ^B♭ |
| 69 | 819.8 | 45/28 | ^A♯, ^^B♭ |
| 70 | 831.7 | 21/13 | ^^A♯, ^3B♭ |
| 71 | 843.6 | 44/27 | ^3A♯, ^4B♭ |
| 72 | 855.4 | ^4A♯, v4B | |
| 73 | 867.3 | 38/23 | v4A𝄪, v3B |
| 74 | 879.2 | v3A𝄪, vvB | |
| 75 | 891.1 | vvA𝄪, vB | |
| 76 | 903 | 32/19 | B |
| 77 | 914.9 | 39/23 | ^B, ^^C♭ |
| 78 | 926.7 | 29/17, 41/24 | ^^B, ^3C♭ |
| 79 | 938.6 | ^3B, ^4C♭ | |
| 80 | 950.5 | 26/15, 45/26 | ^4B, v4C |
| 81 | 962.4 | v4B♯, v3C | |
| 82 | 974.3 | v3B♯, vvC | |
| 83 | 986.1 | 23/13 | vvB♯, vC |
| 84 | 998 | 16/9 | C |
| 85 | 1009.9 | 43/24 | ^C, ^^D♭♭ |
| 86 | 1021.8 | ^^C, ^3D♭♭ | |
| 87 | 1033.7 | ^3C, ^4D♭♭ | |
| 88 | 1045.5 | ^4C, v4D♭ | |
| 89 | 1057.4 | v4C♯, v3D♭ | |
| 90 | 1069.3 | 13/7 | v3C♯, vvD♭ |
| 91 | 1081.2 | 28/15, 43/23 | vvC♯, vD♭ |
| 92 | 1093.1 | 32/17 | vC♯, D♭ |
| 93 | 1105 | 36/19 | C♯, ^D♭ |
| 94 | 1116.8 | 40/21 | ^C♯, ^^D♭ |
| 95 | 1128.7 | 23/12 | ^^C♯, ^3D♭ |
| 96 | 1140.6 | 29/15 | ^3C♯, ^4D♭ |
| 97 | 1152.5 | 37/19 | ^4C♯, v4D |
| 98 | 1164.4 | 45/23 | v4C𝄪, v3D |
| 99 | 1176.2 | v3C𝄪, vvD | |
| 100 | 1188.1 | vvC𝄪, vD | |
| 101 | 1200 | 2/1 | D |
Scales
Mos scales
- 3L 2s: 25 13 25 25 13 ((25 38 63 88 101)\101)[clarification needed]
- Grackle[7] 5L 2s: 17 17 8 17 17 17 8 ((17 34 42 59 76 93)\101)
- Grumpy Octatonic 7L 1s: 13 13 13 13 13 13 13 10 ((13 26 39 52 65 78 91 101)\101)
- Superdiatonic 1/13-tone 13;5 relation: 13 13 13 5 13 13 13 13 5 ((13 26 39 44 57 70 83 96 101)\101)
- Sensi[11] 8L 3s: 10 10 7 10 10 10 7 10 10 10 7 ((10 20 27 37 47 57 64 74 84 94)\101)[clarification needed]
- Anti-Ketradektriatoh 3L 11s: 7 7 7 8 7 7 7 7 8 7 7 7 7 8 ((7 14 22 29 36 43 50 58 65 72 79 86 93 101)\101)