79edo
| ← 78edo | 79edo | 80edo → |
79 equal divisions of the octave (abbreviated 79edo or 79ed2), also called 79-tone equal temperament (79tet) or 79 equal temperament (79et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 79 equal parts of about 15.2 ¢ each. Each step represents a frequency ratio of 21/79, or the 79th root of 2.
Theory
79edo works well as a no-7 13- or 17-limit tuning. It is in fact consistent in the no-7 13-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, as 79edo misses 9/8 while having a near-perfect representation of 10/9 as 12\79. A maximal evenness variant of such a scale can be generated by naively stacking six 12edo diatonic majors and one Lydian tetrachord [clarification needed ]. Since the final tetrachord does not 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 [clarification needed ]. From a regular temperament theory perspective, these scales are a part of the bluebirds temperament.
Intervals
| Steps | Cents | Approximate Ratios | Ups and Downs Notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 15.19 | ^D, E♭♭ | |
| 2 | 30.38 | ^^D, v5E♭ | |
| 3 | 45.57 | 36/35, 38/37 | ^3D, v4E♭ |
| 4 | 60.759 | 29/28 | ^4D, v3E♭ |
| 5 | 75.949 | 23/22, 24/23 | ^5D, vvE♭ |
| 6 | 91.139 | D♯, vE♭ | |
| 7 | 106.329 | 17/16, 33/31 | ^D♯, E♭ |
| 8 | 121.519 | ^^D♯, v5E | |
| 9 | 136.709 | 13/12 | ^3D♯, v4E |
| 10 | 151.899 | 12/11, 35/32 | ^4D♯, v3E |
| 11 | 167.089 | 11/10 | ^5D♯, vvE |
| 12 | 182.278 | 10/9 | D𝄪, vE |
| 13 | 197.468 | E | |
| 14 | 212.658 | 26/23, 35/31 | ^E, F♭ |
| 15 | 227.848 | ^^E, v5F | |
| 16 | 243.038 | 23/20 | ^3E, v4F |
| 17 | 258.228 | 36/31 | ^4E, v3F |
| 18 | 273.418 | 34/29 | ^5E, vvF |
| 19 | 288.608 | 13/11 | E♯, vF |
| 20 | 303.797 | 31/26 | F |
| 21 | 318.987 | ^F, G♭♭ | |
| 22 | 334.177 | 17/14 | ^^F, v5G♭ |
| 23 | 349.367 | 11/9 | ^3F, v4G♭ |
| 24 | 364.557 | 21/17 | ^4F, v3G♭ |
| 25 | 379.747 | ^5F, vvG♭ | |
| 26 | 394.937 | 39/31 | F♯, vG♭ |
| 27 | 410.127 | 33/26 | ^F♯, G♭ |
| 28 | 425.316 | 23/18 | ^^F♯, v5G |
| 29 | 440.506 | 31/24 | ^3F♯, v4G |
| 30 | 455.696 | 13/10 | ^4F♯, v3G |
| 31 | 470.886 | 21/16, 38/29 | ^5F♯, vvG |
| 32 | 486.076 | F𝄪, vG | |
| 33 | 501.266 | 4/3 | G |
| 34 | 516.456 | 27/20, 31/23, 35/26 | ^G, A♭♭ |
| 35 | 531.646 | 19/14 | ^^G, v5A♭ |
| 36 | 546.835 | ^3G, v4A♭ | |
| 37 | 562.025 | 18/13, 29/21 | ^4G, v3A♭ |
| 38 | 577.215 | ^5G, vvA♭ | |
| 39 | 592.405 | 31/22 | G♯, vA♭ |
| 40 | 607.595 | ^G♯, A♭ | |
| 41 | 622.785 | 33/23 | ^^G♯, v5A |
| 42 | 637.975 | 13/9 | ^3G♯, v4A |
| 43 | 653.165 | 35/24 | ^4G♯, v3A |
| 44 | 668.354 | 28/19 | ^5G♯, vvA |
| 45 | 683.544 | G𝄪, vA | |
| 46 | 698.734 | 3/2 | A |
| 47 | 713.924 | ^A, B♭♭ | |
| 48 | 729.114 | 29/19, 32/21, 35/23 | ^^A, v5B♭ |
| 49 | 744.304 | 20/13 | ^3A, v4B♭ |
| 50 | 759.494 | 31/20 | ^4A, v3B♭ |
| 51 | 774.684 | 36/23 | ^5A, vvB♭ |
| 52 | 789.873 | A♯, vB♭ | |
| 53 | 805.063 | 35/22 | ^A♯, B♭ |
| 54 | 820.253 | ^^A♯, v5B | |
| 55 | 835.443 | 34/21 | ^3A♯, v4B |
| 56 | 850.633 | 18/11 | ^4A♯, v3B |
| 57 | 865.823 | 28/17, 33/20 | ^5A♯, vvB |
| 58 | 881.013 | A𝄪, vB | |
| 59 | 896.203 | B | |
| 60 | 911.392 | 22/13, 39/23 | ^B, C♭ |
| 61 | 926.582 | 29/17 | ^^B, v5C |
| 62 | 941.772 | 31/18 | ^3B, v4C |
| 63 | 956.962 | ^4B, v3C | |
| 64 | 972.152 | ^5B, vvC | |
| 65 | 987.342 | 23/13 | B♯, vC |
| 66 | 1002.532 | C | |
| 67 | 1017.722 | 9/5 | ^C, D♭♭ |
| 68 | 1032.911 | 20/11 | ^^C, v5D♭ |
| 69 | 1048.101 | 11/6 | ^3C, v4D♭ |
| 70 | 1063.291 | 24/13 | ^4C, v3D♭ |
| 71 | 1078.481 | ^5C, vvD♭ | |
| 72 | 1093.671 | 32/17 | C♯, vD♭ |
| 73 | 1108.861 | ^C♯, D♭ | |
| 74 | 1124.051 | 23/12 | ^^C♯, v5D |
| 75 | 1139.241 | ^3C♯, v4D | |
| 76 | 1154.43 | 35/18, 37/19, 39/20 | ^4C♯, v3D |
| 77 | 1169.62 | ^5C♯, vvD | |
| 78 | 1184.81 | C𝄪, vD | |
| 79 | 1200 | 2/1 | D |
Regular temperament properties
79edo supports the bluebirds temperament. It also supports oceanfront, meantone and sentinel.
Scales
Mos scales
- Bluebirds[6, 7, 13, 20, 33…]
- Meantone[7, 12, 19, 31…]
- Oceanfront[7, 12, 17, 22, 27…]
- Subsets of the above: see oceanfront scales
- Sentinel[5, 8, 11, 14, 17, 31…]
Others
- Meantone Minor Hexatonic: 13 7 13 13 20 13 ((13, 20, 33, 46, 66, 79)\79)
Music
- Gold (2022)
- Silence and Secrecy (Julian Malerman)