79edo
Theory
79edo works well as a no-7 13- or 17-limit tuning. It is in fact consistent in the no-7 15-odd-limit.
Using the patent val, it tempers out 3125/3072 in the 5-limit, 1728/1715, 4000/3969 and 4375/4374 in the 7-limit, 99/98, 243/242, 385/384, 1331/1323, and 4000/3993 in the 11-limit, and 169/168, 275/273, 325/324, 351/350, 640/637, 1188/1183, 1575/1573, 2080/2079, and 2200/2197 in the 13-limit. It provides the optimal patent val for the sentinel temperament.
The 79c val supports meantone with a tuning very close to 1/7-comma.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.22 | -6.57 | +3.33 | -6.44 | -4.48 | -5.08 | +5.40 | +1.37 | +6.28 | +0.11 | -5.49 |
| Relative (%) | -21.2 | -43.2 | +21.9 | -42.4 | -29.5 | -33.5 | +35.6 | +9.0 | +41.4 | +0.7 | -36.1 | |
| Steps (reduced) |
125 (46) |
183 (25) |
222 (64) |
250 (13) |
273 (36) |
292 (55) |
309 (72) |
323 (7) |
336 (20) |
347 (31) |
357 (41) | |
Subsets and supersets
79edo is the 22nd prime edo, past 73edo and before 83edo.
Miscellany
79edo adequately represents the Decaononic way of playing, where a tone is considered to be 10/9 instead of 9/8.
In 12edo and meantones close to it (used predominantly in Western music), when the difference between 10/9 and 9/8 is tempered out, what really happens is that only the 9/8 is used, and 10/9 is raised to be equal to 9/8. 79edo misses 9/8 while having a near-perfect representation of 10/9 as 12\79.
A maximum evenness variant of such scale can be generated by naively stacking 6 12edo diatonic majors and 1 Lydian tetrachord. Since the final tetrachord doesn't have a 2nd degree, this results in 6 II's stretched over 6+7/12 octaves, which is just enough to make the log2 of the number to be equal to 10/9. From a regular temperament theory perspective, these scales are a part of the bluebirds temperament.
Interval table
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 15.2 | ^D, E♭♭ | |
| 2 | 30.4 | ^^D, ^E♭♭ | |
| 3 | 45.6 | 36/35, 38/37 | ^3D, ^^E♭♭ |
| 4 | 60.8 | 29/28 | vvD♯, v3E♭ |
| 5 | 75.9 | 23/22, 24/23 | vD♯, vvE♭ |
| 6 | 91.1 | D♯, vE♭ | |
| 7 | 106.3 | 17/16, 33/31 | ^D♯, E♭ |
| 8 | 121.5 | ^^D♯, ^E♭ | |
| 9 | 136.7 | 13/12 | ^3D♯, ^^E♭ |
| 10 | 151.9 | 12/11, 35/32 | vvD𝄪, v3E |
| 11 | 167.1 | 11/10 | vD𝄪, vvE |
| 12 | 182.3 | 10/9 | D𝄪, vE |
| 13 | 197.5 | E | |
| 14 | 212.7 | 26/23, 35/31 | ^E, F♭ |
| 15 | 227.8 | ^^E, ^F♭ | |
| 16 | 243 | 23/20 | ^3E, ^^F♭ |
| 17 | 258.2 | 36/31 | vvE♯, v3F |
| 18 | 273.4 | 34/29 | vE♯, vvF |
| 19 | 288.6 | 13/11 | E♯, vF |
| 20 | 303.8 | 31/26 | F |
| 21 | 319 | ^F, G♭♭ | |
| 22 | 334.2 | 17/14 | ^^F, ^G♭♭ |
| 23 | 349.4 | 11/9 | ^3F, ^^G♭♭ |
| 24 | 364.6 | 21/17 | vvF♯, v3G♭ |
| 25 | 379.7 | vF♯, vvG♭ | |
| 26 | 394.9 | 39/31 | F♯, vG♭ |
| 27 | 410.1 | 33/26 | ^F♯, G♭ |
| 28 | 425.3 | 23/18 | ^^F♯, ^G♭ |
| 29 | 440.5 | 31/24 | ^3F♯, ^^G♭ |
| 30 | 455.7 | 13/10 | vvF𝄪, v3G |
| 31 | 470.9 | 21/16, 38/29 | vF𝄪, vvG |
| 32 | 486.1 | F𝄪, vG | |
| 33 | 501.3 | 4/3 | G |
| 34 | 516.5 | 27/20, 31/23, 35/26 | ^G, A♭♭ |
| 35 | 531.6 | 19/14 | ^^G, ^A♭♭ |
| 36 | 546.8 | ^3G, ^^A♭♭ | |
| 37 | 562 | 18/13, 29/21 | vvG♯, v3A♭ |
| 38 | 577.2 | vG♯, vvA♭ | |
| 39 | 592.4 | 31/22 | G♯, vA♭ |
| 40 | 607.6 | ^G♯, A♭ | |
| 41 | 622.8 | 33/23 | ^^G♯, ^A♭ |
| 42 | 638 | 13/9 | ^3G♯, ^^A♭ |
| 43 | 653.2 | 35/24 | vvG𝄪, v3A |
| 44 | 668.4 | 28/19 | vG𝄪, vvA |
| 45 | 683.5 | G𝄪, vA | |
| 46 | 698.7 | 3/2 | A |
| 47 | 713.9 | ^A, B♭♭ | |
| 48 | 729.1 | 29/19, 32/21, 35/23 | ^^A, ^B♭♭ |
| 49 | 744.3 | 20/13 | ^3A, ^^B♭♭ |
| 50 | 759.5 | 31/20 | vvA♯, v3B♭ |
| 51 | 774.7 | 36/23 | vA♯, vvB♭ |
| 52 | 789.9 | A♯, vB♭ | |
| 53 | 805.1 | 35/22 | ^A♯, B♭ |
| 54 | 820.3 | ^^A♯, ^B♭ | |
| 55 | 835.4 | 34/21 | ^3A♯, ^^B♭ |
| 56 | 850.6 | 18/11 | vvA𝄪, v3B |
| 57 | 865.8 | 28/17, 33/20 | vA𝄪, vvB |
| 58 | 881 | A𝄪, vB | |
| 59 | 896.2 | B | |
| 60 | 911.4 | 22/13, 39/23 | ^B, C♭ |
| 61 | 926.6 | 29/17 | ^^B, ^C♭ |
| 62 | 941.8 | 31/18 | ^3B, ^^C♭ |
| 63 | 957 | vvB♯, v3C | |
| 64 | 972.2 | vB♯, vvC | |
| 65 | 987.3 | 23/13 | B♯, vC |
| 66 | 1002.5 | C | |
| 67 | 1017.7 | 9/5 | ^C, D♭♭ |
| 68 | 1032.9 | 20/11 | ^^C, ^D♭♭ |
| 69 | 1048.1 | 11/6 | ^3C, ^^D♭♭ |
| 70 | 1063.3 | 24/13 | vvC♯, v3D♭ |
| 71 | 1078.5 | vC♯, vvD♭ | |
| 72 | 1093.7 | 32/17 | C♯, vD♭ |
| 73 | 1108.9 | ^C♯, D♭ | |
| 74 | 1124.1 | 23/12 | ^^C♯, ^D♭ |
| 75 | 1139.2 | ^3C♯, ^^D♭ | |
| 76 | 1154.4 | 35/18, 37/19, 39/20 | vvC𝄪, v3D |
| 77 | 1169.6 | vC𝄪, vvD | |
| 78 | 1184.8 | C𝄪, vD | |
| 79 | 1200 | 2/1 | D |
Regular temperament properties
79edo supports the bluebirds temperament. It also supports oceanfront, meantone and sentinel.
Scales
Bluebirds
- Bluebirds[6, 7, 13, 33...] MOSes
Meantone
- Meantone[7, 12, 19, 31...] MOSes
- Meantone Minor Hexatonic
- 197.802
- 303.297
- 501.099
- 698.901
- 1002.198
- 1200.000
Oceanfront
- Oceanfront[7, 12, 17, 22, 27...] MOSes
- Subsets of the above: see oceanfront scales
Sentinel
- Sentinel[5, 8, 11, 14, 17, 31...] MOSes
Music
- Silence and Secrecy (Julian Malerman)