66edo: Difference between revisions
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mentioned the over-5 properties since they seem to crop up |
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== Theory == | == Theory == | ||
The [[patent val]] of 66edo is [[contorted]] in the 5-limit, [[tempering out]] the same [[comma]]s ([[250/243]], [[2048/2025]], [[3125/3072]], etc.) as [[22edo]]. In the 7-limit it tempers out [[686/675]] and [[1029/1024]], in the 11-limit [[55/54]], [[100/99]] and [[121/120]], in the 13-limit [[91/90]], [[169/168]], [[196/195]] and in the 17-limit [[136/135]] and [[256/255]]. It provides the [[optimal patent val]] for the 11- and 13-limit [[ammonite]] temperament. | The [[patent val]] of 66edo is [[contorted]] in the 5-limit, [[tempering out]] the same [[comma]]s ([[250/243]], [[2048/2025]], [[3125/3072]], etc.) as [[22edo]]. In the 7-limit it tempers out [[686/675]] and [[1029/1024]], in the 11-limit [[55/54]], [[100/99]] and [[121/120]], in the 13-limit [[91/90]], [[169/168]], [[196/195]] and in the 17-limit [[136/135]] and [[256/255]]. It provides the [[optimal patent val]] for the 11- and 13-limit [[ammonite]] temperament. Otherwise, 66edo is not exceptional when it comes to approximating prime harmonics; however, it contains a quite accurate approximation to the 5:7:9:11:13 chord and can therefore be used for various [[over-5]] scales. | ||
The 66b val tempers out [[16875/16384]] in the 5-limit, [[126/125]], [[1728/1715]] and [[2401/2400]] in the 7-limit, [[99/98]] and [[385/384]] in the 11-limit, and [[105/104]], [[144/143]] and [[847/845]] in the 13-limit. | The 66b val tempers out [[16875/16384]] in the 5-limit, [[126/125]], [[1728/1715]] and [[2401/2400]] in the 7-limit, [[99/98]] and [[385/384]] in the 11-limit, and [[105/104]], [[144/143]] and [[847/845]] in the 13-limit. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 66 factors into {{factorization|66}}, 66edo has subset edos {{EDOs| 2, 3, 6, 11, 22, and 33 }}. [[198edo]], which triples it, corrects its approximation to many of the lower harmonics. | Since 66 factors into {{factorization|66}}, 66edo has subset edos {{EDOs| 2, 3, 6, 11, 22, and 33 }}. [[198edo]], which triples it, corrects its approximation to many of the lower harmonics. | ||
== Interval table == | == Interval table == | ||
Revision as of 12:26, 12 December 2024
| ← 65edo | 66edo | 67edo → |
Theory
The patent val of 66edo is contorted in the 5-limit, tempering out the same commas (250/243, 2048/2025, 3125/3072, etc.) as 22edo. In the 7-limit it tempers out 686/675 and 1029/1024, in the 11-limit 55/54, 100/99 and 121/120, in the 13-limit 91/90, 169/168, 196/195 and in the 17-limit 136/135 and 256/255. It provides the optimal patent val for the 11- and 13-limit ammonite temperament. Otherwise, 66edo is not exceptional when it comes to approximating prime harmonics; however, it contains a quite accurate approximation to the 5:7:9:11:13 chord and can therefore be used for various over-5 scales.
The 66b val tempers out 16875/16384 in the 5-limit, 126/125, 1728/1715 and 2401/2400 in the 7-limit, 99/98 and 385/384 in the 11-limit, and 105/104, 144/143 and 847/845 in the 13-limit.
109 steps of 66edo is extremely close to the acoustic pi with only +0.023 cents of error.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.14 | -4.50 | -5.19 | -3.91 | -5.86 | -4.16 | +2.64 | +4.14 | -6.60 | +1.95 | +8.09 |
| Relative (%) | +39.2 | -24.7 | -28.5 | -21.5 | -32.2 | -22.9 | +14.5 | +22.7 | -36.3 | +10.7 | +44.5 | |
| Steps (reduced) |
105 (39) |
153 (21) |
185 (53) |
209 (11) |
228 (30) |
244 (46) |
258 (60) |
270 (6) |
280 (16) |
290 (26) |
299 (35) | |
Subsets and supersets
Since 66 factors into 2 × 3 × 11, 66edo has subset edos 2, 3, 6, 11, 22, and 33. 198edo, which triples it, corrects its approximation to many of the lower harmonics.
Interval table
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 38\66) |
Ups and downs notation (Dual sharp fifth 39\66) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 18.2 | ^D, vE♭♭♭♭ | ^D, vvE♭ | |
| 2 | 36.4 | D♯, E♭♭♭♭ | ^^D, vE♭ | |
| 3 | 54.5 | 31/30, 32/31, 33/32, 34/33 | ^D♯, vE♭♭♭ | ^3D, E♭ |
| 4 | 72.7 | 24/23 | D𝄪, E♭♭♭ | ^4D, ^E♭ |
| 5 | 90.9 | 20/19 | ^D𝄪, vE♭♭ | v4D♯, ^^E♭ |
| 6 | 109.1 | 16/15, 33/31 | D♯𝄪, E♭♭ | v3D♯, ^3E♭ |
| 7 | 127.3 | 14/13 | ^D♯𝄪, vE♭ | vvD♯, ^4E♭ |
| 8 | 145.5 | D𝄪𝄪, E♭ | vD♯, v4E | |
| 9 | 163.6 | 11/10, 34/31 | ^D𝄪𝄪, vE | D♯, v3E |
| 10 | 181.8 | E | ^D♯, vvE | |
| 11 | 200 | 28/25 | ^E, vF♭♭♭ | ^^D♯, vE |
| 12 | 218.2 | 17/15, 25/22 | E♯, F♭♭♭ | E |
| 13 | 236.4 | ^E♯, vF♭♭ | ^E, vvF | |
| 14 | 254.5 | 22/19 | E𝄪, F♭♭ | ^^E, vF |
| 15 | 272.7 | 34/29 | ^E𝄪, vF♭ | F |
| 16 | 290.9 | 13/11 | E♯𝄪, F♭ | ^F, vvG♭ |
| 17 | 309.1 | ^E♯𝄪, vF | ^^F, vG♭ | |
| 18 | 327.3 | 29/24 | F | ^3F, G♭ |
| 19 | 345.5 | ^F, vG♭♭♭♭ | ^4F, ^G♭ | |
| 20 | 363.6 | 21/17 | F♯, G♭♭♭♭ | v4F♯, ^^G♭ |
| 21 | 381.8 | ^F♯, vG♭♭♭ | v3F♯, ^3G♭ | |
| 22 | 400 | 29/23 | F𝄪, G♭♭♭ | vvF♯, ^4G♭ |
| 23 | 418.2 | 14/11 | ^F𝄪, vG♭♭ | vF♯, v4G |
| 24 | 436.4 | F♯𝄪, G♭♭ | F♯, v3G | |
| 25 | 454.5 | 13/10 | ^F♯𝄪, vG♭ | ^F♯, vvG |
| 26 | 472.7 | 21/16, 25/19 | F𝄪𝄪, G♭ | ^^F♯, vG |
| 27 | 490.9 | ^F𝄪𝄪, vG | G | |
| 28 | 509.1 | G | ^G, vvA♭ | |
| 29 | 527.3 | 19/14, 23/17 | ^G, vA♭♭♭♭ | ^^G, vA♭ |
| 30 | 545.5 | 26/19 | G♯, A♭♭♭♭ | ^3G, A♭ |
| 31 | 563.6 | ^G♯, vA♭♭♭ | ^4G, ^A♭ | |
| 32 | 581.8 | 7/5 | G𝄪, A♭♭♭ | v4G♯, ^^A♭ |
| 33 | 600 | 17/12, 24/17 | ^G𝄪, vA♭♭ | v3G♯, ^3A♭ |
| 34 | 618.2 | 10/7 | G♯𝄪, A♭♭ | vvG♯, ^4A♭ |
| 35 | 636.4 | ^G♯𝄪, vA♭ | vG♯, v4A | |
| 36 | 654.5 | 19/13 | G𝄪𝄪, A♭ | G♯, v3A |
| 37 | 672.7 | 28/19, 31/21, 34/23 | ^G𝄪𝄪, vA | ^G♯, vvA |
| 38 | 690.9 | A | ^^G♯, vA | |
| 39 | 709.1 | ^A, vB♭♭♭♭ | A | |
| 40 | 727.3 | 32/21 | A♯, B♭♭♭♭ | ^A, vvB♭ |
| 41 | 745.5 | 20/13 | ^A♯, vB♭♭♭ | ^^A, vB♭ |
| 42 | 763.6 | A𝄪, B♭♭♭ | ^3A, B♭ | |
| 43 | 781.8 | 11/7 | ^A𝄪, vB♭♭ | ^4A, ^B♭ |
| 44 | 800 | 35/22 | A♯𝄪, B♭♭ | v4A♯, ^^B♭ |
| 45 | 818.2 | ^A♯𝄪, vB♭ | v3A♯, ^3B♭ | |
| 46 | 836.4 | 34/21 | A𝄪𝄪, B♭ | vvA♯, ^4B♭ |
| 47 | 854.5 | ^A𝄪𝄪, vB | vA♯, v4B | |
| 48 | 872.7 | B | A♯, v3B | |
| 49 | 890.9 | ^B, vC♭♭♭ | ^A♯, vvB | |
| 50 | 909.1 | 22/13 | B♯, C♭♭♭ | ^^A♯, vB |
| 51 | 927.3 | 29/17 | ^B♯, vC♭♭ | B |
| 52 | 945.5 | 19/11 | B𝄪, C♭♭ | ^B, vvC |
| 53 | 963.6 | ^B𝄪, vC♭ | ^^B, vC | |
| 54 | 981.8 | 30/17 | B♯𝄪, C♭ | C |
| 55 | 1000 | 25/14 | ^B♯𝄪, vC | ^C, vvD♭ |
| 56 | 1018.2 | C | ^^C, vD♭ | |
| 57 | 1036.4 | 20/11, 31/17 | ^C, vD♭♭♭♭ | ^3C, D♭ |
| 58 | 1054.5 | 35/19 | C♯, D♭♭♭♭ | ^4C, ^D♭ |
| 59 | 1072.7 | 13/7 | ^C♯, vD♭♭♭ | v4C♯, ^^D♭ |
| 60 | 1090.9 | 15/8 | C𝄪, D♭♭♭ | v3C♯, ^3D♭ |
| 61 | 1109.1 | 19/10 | ^C𝄪, vD♭♭ | vvC♯, ^4D♭ |
| 62 | 1127.3 | 23/12 | C♯𝄪, D♭♭ | vC♯, v4D |
| 63 | 1145.5 | 31/16, 33/17 | ^C♯𝄪, vD♭ | C♯, v3D |
| 64 | 1163.6 | C𝄪𝄪, D♭ | ^C♯, vvD | |
| 65 | 1181.8 | ^C𝄪𝄪, vD | ^^C♯, vD | |
| 66 | 1200 | 2/1 | D | D |