102edo
Jump to navigation
Jump to search
Prime factorization
2 × 3 × 17
Step size
11.7647¢
Fifth
60\102 (705.882¢) (→10\17)
Semitones (A1:m2)
12:6 (141.2¢ : 70.59¢)
Dual sharp fifth
60\102 (705.882¢) (→10\17)
Dual flat fifth
59\102 (694.118¢)
Dual major 2nd
17\102 (200¢) (→1\6)
Consistency limit
5
Distinct consistency limit
5
| ← 101edo | 102edo | 103edo → |
102 equal divisions of the octave (abbreviated 102edo or 102ed2), also called 102-tone equal temperament (102tet) or 102 equal temperament (102et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 102 equal parts of about 11.8 ¢ each. Each step represents a frequency ratio of 21/102, or the 102nd root of 2.
Theory
102edo is enfactored in the 5-limit, where it tempers out the same commas (2048/2025, 15625/15552, 20000/19683) as 34edo. In the 7-limit it tempers out 686/675 and 1029/1024; in the 11-limit 385/384, 441/440 and 4000/3993; in the 13-limit 91/90 and 169/168; in the 17-limit 136/135 and 154/153; and in the 19-limit 133/132 and 190/189. It is the optimal patent val for 13-limit echidnic temperament, and the rank-5 temperament tempering out 91/90.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.93 | +1.92 | -4.12 | -3.91 | +1.62 | -5.23 | +5.85 | +0.93 | -3.40 | -0.19 | -4.74 |
| Relative (%) | +33.4 | +16.3 | -35.0 | -33.2 | +13.8 | -44.5 | +49.7 | +7.9 | -28.9 | -1.6 | -40.3 | |
| Steps (reduced) |
162 (60) |
237 (33) |
286 (82) |
323 (17) |
353 (47) |
377 (71) |
399 (93) |
417 (9) |
433 (25) |
448 (40) |
461 (53) | |
Intervals
| Steps | Cents | Approximate Ratios | Ups and Downs Notation (Dual Flat Fifth 59\102) |
Ups and Downs Notation (Dual Sharp Fifth 60\102) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 11.765 | ^D, E♭♭♭ | ^D, v5E♭ | |
| 2 | 23.529 | ^^D, v4E♭♭ | ^^D, v4E♭ | |
| 3 | 35.294 | ^3D, v3E♭♭ | ^3D, v3E♭ | |
| 4 | 47.059 | 38/37, 39/38 | ^4D, vvE♭♭ | ^4D, vvE♭ |
| 5 | 58.824 | 30/29 | D♯, vE♭♭ | ^5D, vE♭ |
| 6 | 70.588 | 25/24 | ^D♯, E♭♭ | ^6D, E♭ |
| 7 | 82.353 | 21/20, 22/21, 43/41 | ^^D♯, v4E♭ | ^7D, v11E |
| 8 | 94.118 | 37/35 | ^3D♯, v3E♭ | ^8D, v10E |
| 9 | 105.882 | 17/16 | ^4D♯, vvE♭ | ^9D, v9E |
| 10 | 117.647 | D𝄪, vE♭ | ^10D, v8E | |
| 11 | 129.412 | 14/13, 41/38 | ^D𝄪, E♭ | ^11D, v7E |
| 12 | 141.176 | 38/35 | ^^D𝄪, v4E | D♯, v6E |
| 13 | 152.941 | 12/11, 35/32 | ^3D𝄪, v3E | ^D♯, v5E |
| 14 | 164.706 | 11/10 | ^4D𝄪, vvE | ^^D♯, v4E |
| 15 | 176.471 | 31/28, 41/37 | D♯𝄪, vE | ^3D♯, v3E |
| 16 | 188.235 | 39/35 | E | ^4D♯, vvE |
| 17 | 200 | ^E, F♭♭ | ^5D♯, vE | |
| 18 | 211.765 | 26/23, 35/31, 43/38 | ^^E, v4F♭ | E |
| 19 | 223.529 | 25/22, 33/29 | ^3E, v3F♭ | ^E, v5F |
| 20 | 235.294 | 39/34 | ^4E, vvF♭ | ^^E, v4F |
| 21 | 247.059 | E♯, vF♭ | ^3E, v3F | |
| 22 | 258.824 | 29/25, 43/37 | ^E♯, F♭ | ^4E, vvF |
| 23 | 270.588 | ^^E♯, v4F | ^5E, vF | |
| 24 | 282.353 | 20/17 | ^3E♯, v3F | F |
| 25 | 294.118 | ^4E♯, vvF | ^F, v5G♭ | |
| 26 | 305.882 | 31/26, 37/31 | E𝄪, vF | ^^F, v4G♭ |
| 27 | 317.647 | 6/5 | F | ^3F, v3G♭ |
| 28 | 329.412 | 23/19, 29/24 | ^F, G♭♭♭ | ^4F, vvG♭ |
| 29 | 341.176 | 28/23, 39/32 | ^^F, v4G♭♭ | ^5F, vG♭ |
| 30 | 352.941 | 38/31 | ^3F, v3G♭♭ | ^6F, G♭ |
| 31 | 364.706 | 21/17 | ^4F, vvG♭♭ | ^7F, v11G |
| 32 | 376.471 | 36/29 | F♯, vG♭♭ | ^8F, v10G |
| 33 | 388.235 | 5/4 | ^F♯, G♭♭ | ^9F, v9G |
| 34 | 400 | ^^F♯, v4G♭ | ^10F, v8G | |
| 35 | 411.765 | ^3F♯, v3G♭ | ^11F, v7G | |
| 36 | 423.529 | ^4F♯, vvG♭ | F♯, v6G | |
| 37 | 435.294 | F𝄪, vG♭ | ^F♯, v5G | |
| 38 | 447.059 | 22/17 | ^F𝄪, G♭ | ^^F♯, v4G |
| 39 | 458.824 | ^^F𝄪, v4G | ^3F♯, v3G | |
| 40 | 470.588 | 21/16 | ^3F𝄪, v3G | ^4F♯, vvG |
| 41 | 482.353 | 33/25, 37/28, 41/31 | ^4F𝄪, vvG | ^5F♯, vG |
| 42 | 494.118 | F♯𝄪, vG | G | |
| 43 | 505.882 | G | ^G, v5A♭ | |
| 44 | 517.647 | 31/23 | ^G, A♭♭♭ | ^^G, v4A♭ |
| 45 | 529.412 | 19/14 | ^^G, v4A♭♭ | ^3G, v3A♭ |
| 46 | 541.176 | 26/19 | ^3G, v3A♭♭ | ^4G, vvA♭ |
| 47 | 552.941 | 11/8 | ^4G, vvA♭♭ | ^5G, vA♭ |
| 48 | 564.706 | 43/31 | G♯, vA♭♭ | ^6G, A♭ |
| 49 | 576.471 | ^G♯, A♭♭ | ^7G, v11A | |
| 50 | 588.235 | ^^G♯, v4A♭ | ^8G, v10A | |
| 51 | 600 | ^3G♯, v3A♭ | ^9G, v9A | |
| 52 | 611.765 | 37/26 | ^4G♯, vvA♭ | ^10G, v8A |
| 53 | 623.529 | G𝄪, vA♭ | ^11G, v7A | |
| 54 | 635.294 | ^G𝄪, A♭ | G♯, v6A | |
| 55 | 647.059 | 16/11 | ^^G𝄪, v4A | ^G♯, v5A |
| 56 | 658.824 | 19/13, 41/28 | ^3G𝄪, v3A | ^^G♯, v4A |
| 57 | 670.588 | 28/19 | ^4G𝄪, vvA | ^3G♯, v3A |
| 58 | 682.353 | G♯𝄪, vA | ^4G♯, vvA | |
| 59 | 694.118 | A | ^5G♯, vA | |
| 60 | 705.882 | ^A, B♭♭♭ | A | |
| 61 | 717.647 | ^^A, v4B♭♭ | ^A, v5B♭ | |
| 62 | 729.412 | 32/21 | ^3A, v3B♭♭ | ^^A, v4B♭ |
| 63 | 741.176 | 43/28 | ^4A, vvB♭♭ | ^3A, v3B♭ |
| 64 | 752.941 | 17/11 | A♯, vB♭♭ | ^4A, vvB♭ |
| 65 | 764.706 | ^A♯, B♭♭ | ^5A, vB♭ | |
| 66 | 776.471 | ^^A♯, v4B♭ | ^6A, B♭ | |
| 67 | 788.235 | 41/26 | ^3A♯, v3B♭ | ^7A, v11B |
| 68 | 800 | ^4A♯, vvB♭ | ^8A, v10B | |
| 69 | 811.765 | 8/5 | A𝄪, vB♭ | ^9A, v9B |
| 70 | 823.529 | 29/18, 37/23 | ^A𝄪, B♭ | ^10A, v8B |
| 71 | 835.294 | 34/21 | ^^A𝄪, v4B | ^11A, v7B |
| 72 | 847.059 | 31/19 | ^3A𝄪, v3B | A♯, v6B |
| 73 | 858.824 | 23/14 | ^4A𝄪, vvB | ^A♯, v5B |
| 74 | 870.588 | 38/23, 43/26 | A♯𝄪, vB | ^^A♯, v4B |
| 75 | 882.353 | 5/3 | B | ^3A♯, v3B |
| 76 | 894.118 | ^B, C♭♭ | ^4A♯, vvB | |
| 77 | 905.882 | ^^B, v4C♭ | ^5A♯, vB | |
| 78 | 917.647 | 17/10 | ^3B, v3C♭ | B |
| 79 | 929.412 | ^4B, vvC♭ | ^B, v5C | |
| 80 | 941.176 | B♯, vC♭ | ^^B, v4C | |
| 81 | 952.941 | ^B♯, C♭ | ^3B, v3C | |
| 82 | 964.706 | ^^B♯, v4C | ^4B, vvC | |
| 83 | 976.471 | 44/25 | ^3B♯, v3C | ^5B, vC |
| 84 | 988.235 | 23/13 | ^4B♯, vvC | C |
| 85 | 1000 | 41/23 | B𝄪, vC | ^C, v5D♭ |
| 86 | 1011.765 | C | ^^C, v4D♭ | |
| 87 | 1023.529 | ^C, D♭♭♭ | ^3C, v3D♭ | |
| 88 | 1035.294 | 20/11 | ^^C, v4D♭♭ | ^4C, vvD♭ |
| 89 | 1047.059 | 11/6 | ^3C, v3D♭♭ | ^5C, vD♭ |
| 90 | 1058.824 | 35/19 | ^4C, vvD♭♭ | ^6C, D♭ |
| 91 | 1070.588 | 13/7 | C♯, vD♭♭ | ^7C, v11D |
| 92 | 1082.353 | 43/23 | ^C♯, D♭♭ | ^8C, v10D |
| 93 | 1094.118 | 32/17 | ^^C♯, v4D♭ | ^9C, v9D |
| 94 | 1105.882 | ^3C♯, v3D♭ | ^10C, v8D | |
| 95 | 1117.647 | 21/11, 40/21 | ^4C♯, vvD♭ | ^11C, v7D |
| 96 | 1129.412 | C𝄪, vD♭ | C♯, v6D | |
| 97 | 1141.176 | 29/15 | ^C𝄪, D♭ | ^C♯, v5D |
| 98 | 1152.941 | 37/19 | ^^C𝄪, v4D | ^^C♯, v4D |
| 99 | 1164.706 | ^3C𝄪, v3D | ^3C♯, v3D | |
| 100 | 1176.471 | ^4C𝄪, vvD | ^4C♯, vvD | |
| 101 | 1188.235 | C♯𝄪, vD | ^5C♯, vD | |
| 102 | 1200 | 2/1 | D | D |