73edo: Difference between revisions
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== Theory == | == Theory == | ||
73edo has a very sharp tendency, with the approximations of [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]] all sharp. The equal temperament [[tempering out|tempers out]] [[78732/78125]] and [[262144/253125]] in the [[5-limit]]; [[126/125]] and [[245/243]] in the [[7-limit]]; [[176/175]], [[441/440]] and [[4000/3993]] in the [[11-limit]]; [[91/90]], [[169/168]], [[196/195]], [[325/324]], [[351/350]], and [[352/351]] in the [[13-limit]]. It provides the [[optimal patent val]] for the [[marrakesh]] temperament, though [[104edo]] and [[135edo]] tunes it better. | 73edo has a very sharp tendency, with the approximations of [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]] all sharp. | ||
The equal temperament [[tempering out|tempers out]] [[78732/78125]] and [[262144/253125]] in the [[5-limit]]; [[126/125]] and [[245/243]] in the [[7-limit]]; [[176/175]], [[441/440]] and [[4000/3993]] in the [[11-limit]]; [[91/90]], [[169/168]], [[196/195]], [[325/324]], [[351/350]], and [[352/351]] in the [[13-limit]]. It provides the [[optimal patent val]] for the [[marrakesh]] temperament, though [[104edo]] and [[135edo]] tunes it better. (''See [[regular temperament]] for more about what all this means and how to use it.'') | |||
73edo fits in [[mavila]] scale, by the 9;5 relation in the [[7L 2s|superdiatonic]] scheme. | 73edo fits in [[mavila]] scale, by the 9;5 relation in the [[7L 2s|superdiatonic]] scheme. | ||
Revision as of 07:27, 11 April 2025
| ← 72edo | 73edo | 74edo → |
73 equal divisions of the octave (abbreviated 73edo or 73ed2), also called 73-tone equal temperament (73tet) or 73 equal temperament (73et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 73 equal parts of about 16.4 ¢ each. Each step represents a frequency ratio of 21/73, or the 73rd root of 2.
Theory
73edo has a very sharp tendency, with the approximations of 3, 5, 7, 11 all sharp.
The equal temperament tempers out 78732/78125 and 262144/253125 in the 5-limit; 126/125 and 245/243 in the 7-limit; 176/175, 441/440 and 4000/3993 in the 11-limit; 91/90, 169/168, 196/195, 325/324, 351/350, and 352/351 in the 13-limit. It provides the optimal patent val for the marrakesh temperament, though 104edo and 135edo tunes it better. (See regular temperament for more about what all this means and how to use it.)
73edo fits in mavila scale, by the 9;5 relation in the superdiatonic scheme.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +4.89 | +8.21 | +1.04 | +7.59 | -2.17 | -6.33 | -1.62 | -3.62 | +6.04 | +5.65 |
| Relative (%) | +0.0 | +29.8 | +49.9 | +6.3 | +46.1 | -13.2 | -38.5 | -9.9 | -22.0 | +36.7 | +34.4 | |
| Steps (reduced) |
73 (0) |
116 (43) |
170 (24) |
205 (59) |
253 (34) |
270 (51) |
298 (6) |
310 (18) |
330 (38) |
355 (63) |
362 (70) | |
Subsets and supersets
73edo is the 21st prime edo, past 71edo and before 79edo.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 16.4 | ^D, v3E♭ | |
| 2 | 32.9 | ^^D, vvE♭ | |
| 3 | 49.3 | 36/35, 38/37 | ^3D, vE♭ |
| 4 | 65.8 | ^4D, E♭ | |
| 5 | 82.2 | 21/20, 22/21 | v4D♯, ^E♭ |
| 6 | 98.6 | 35/33 | v3D♯, ^^E♭ |
| 7 | 115.1 | 31/29 | vvD♯, ^3E♭ |
| 8 | 131.5 | 14/13, 27/25 | vD♯, ^4E♭ |
| 9 | 147.9 | 12/11, 37/34 | D♯, v4E |
| 10 | 164.4 | 11/10 | ^D♯, v3E |
| 11 | 180.8 | 10/9 | ^^D♯, vvE |
| 12 | 197.3 | ^3D♯, vE | |
| 13 | 213.7 | 26/23 | E |
| 14 | 230.1 | 8/7 | ^E, v3F |
| 15 | 246.6 | ^^E, vvF | |
| 16 | 263 | ^3E, vF | |
| 17 | 279.5 | F | |
| 18 | 295.9 | 19/16 | ^F, v3G♭ |
| 19 | 312.3 | 6/5 | ^^F, vvG♭ |
| 20 | 328.8 | 23/19, 29/24, 35/29 | ^3F, vG♭ |
| 21 | 345.2 | 11/9 | ^4F, G♭ |
| 22 | 361.6 | 16/13 | v4F♯, ^G♭ |
| 23 | 378.1 | v3F♯, ^^G♭ | |
| 24 | 394.5 | vvF♯, ^3G♭ | |
| 25 | 411 | vF♯, ^4G♭ | |
| 26 | 427.4 | F♯, v4G | |
| 27 | 443.8 | 31/24 | ^F♯, v3G |
| 28 | 460.3 | ^^F♯, vvG | |
| 29 | 476.7 | 29/22 | ^3F♯, vG |
| 30 | 493.2 | G | |
| 31 | 509.6 | ^G, v3A♭ | |
| 32 | 526 | 19/14, 23/17 | ^^G, vvA♭ |
| 33 | 542.5 | 26/19 | ^3G, vA♭ |
| 34 | 558.9 | 29/21 | ^4G, A♭ |
| 35 | 575.3 | v4G♯, ^A♭ | |
| 36 | 591.8 | 31/22 | v3G♯, ^^A♭ |
| 37 | 608.2 | 37/26 | vvG♯, ^3A♭ |
| 38 | 624.7 | vG♯, ^4A♭ | |
| 39 | 641.1 | 29/20 | G♯, v4A |
| 40 | 657.5 | 19/13 | ^G♯, v3A |
| 41 | 674 | 28/19, 31/21, 34/23 | ^^G♯, vvA |
| 42 | 690.4 | ^3G♯, vA | |
| 43 | 706.8 | A | |
| 44 | 723.3 | ^A, v3B♭ | |
| 45 | 739.7 | ^^A, vvB♭ | |
| 46 | 756.2 | 31/20 | ^3A, vB♭ |
| 47 | 772.6 | ^4A, B♭ | |
| 48 | 789 | v4A♯, ^B♭ | |
| 49 | 805.5 | 35/22 | v3A♯, ^^B♭ |
| 50 | 821.9 | 37/23 | vvA♯, ^3B♭ |
| 51 | 838.4 | 13/8 | vA♯, ^4B♭ |
| 52 | 854.8 | 18/11 | A♯, v4B |
| 53 | 871.2 | 38/23 | ^A♯, v3B |
| 54 | 887.7 | 5/3 | ^^A♯, vvB |
| 55 | 904.1 | 32/19 | ^3A♯, vB |
| 56 | 920.5 | B | |
| 57 | 937 | ^B, v3C | |
| 58 | 953.4 | ^^B, vvC | |
| 59 | 969.9 | 7/4 | ^3B, vC |
| 60 | 986.3 | 23/13 | C |
| 61 | 1002.7 | ^C, v3D♭ | |
| 62 | 1019.2 | 9/5 | ^^C, vvD♭ |
| 63 | 1035.6 | 20/11 | ^3C, vD♭ |
| 64 | 1052.1 | 11/6 | ^4C, D♭ |
| 65 | 1068.5 | 13/7 | v4C♯, ^D♭ |
| 66 | 1084.9 | v3C♯, ^^D♭ | |
| 67 | 1101.4 | vvC♯, ^3D♭ | |
| 68 | 1117.8 | 21/11 | vC♯, ^4D♭ |
| 69 | 1134.2 | C♯, v4D | |
| 70 | 1150.7 | 35/18, 37/19 | ^C♯, v3D |
| 71 | 1167.1 | ^^C♯, vvD | |
| 72 | 1183.6 | ^3C♯, vD | |
| 73 | 1200 | 2/1 | D |
Notation
Sagittal notation
This notation uses the same sagittal sequence as 80-EDO.
Evo flavor
Revo flavor
Scales
- Porky[7]: 10 10 10 13 10 10 10 ((10, 20, 30, 43, 53, 63, 73)\73)