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
| 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 | -43 | +22 | -42 | -30 | -33 | +36 | +9 | +41 | +1 | -36 | |
| Steps (reduced) |
125 (46) |
183 (25) |
222 (64) |
250 (13) |
273 (36) |
292 (55) |
309 (72) |
323 (7) |
336 (20) |
347 (31) |
357 (41) | |
79edo works well as a no-fives 17-limit tuning, or as a dual-5 17-limit tuning.
It tempers out 3125/3072 in the 5-limit, 4000/3969, 1728/1715 and 4375/4374 in the 7-limit, 99/98, 1331/1323, 243/242, 385/384 and 4000/3993 in the 11-limit, and 275/273, 169/168, 640/637, 1188/1183, 325/324, 351/350, 1575/1573, 2080/2079 and 2200/2197 in the 13-limit. It provides the optimal patent val for sentinel temperament.
79 is the 22nd prime EDO number.
79edo's fifth is slightly flat and very similar to the one of 12edo–the major third is too sharp to be the closest approximation of 5/4 so 79edo actually does not qualify as meantone in the patent val.
Decaononic whole tone
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 | Ups and downs notation | Approximate ratios |
|---|---|---|---|
| 0 | 0 | D | 1/1 |
| 1 | 15.1899 | ^D, E♭♭ | |
| 2 | 30.3797 | ^^D, v5E♭ | 55/54, 64/63, 66/65 |
| 3 | 45.5696 | ^3D, v4E♭ | 36/35, 40/39 |
| 4 | 60.7595 | ^4D, v3E♭ | 27/26, 80/77 |
| 5 | 75.9494 | ^5D, vvE♭ | 22/21 |
| 6 | 91.1392 | D♯, vE♭ | 55/52 |
| 7 | 106.329 | ^D♯, E♭ | 35/33 |
| 8 | 121.519 | ^^D♯, v5E | 77/72 |
| 9 | 136.709 | ^3D♯, v4E | 13/12, 27/25 |
| 10 | 151.899 | ^4D♯, v3E | 12/11, 35/32 |
| 11 | 167.089 | ^5D♯, vvE | 11/10 |
| 12 | 182.278 | D𝄪, vE | 10/9, 39/35, 72/65 |
| 13 | 197.468 | E | |
| 14 | 212.658 | ^E, F♭ | 44/39 |
| 15 | 227.848 | ^^E, v5F | 8/7 |
| 16 | 243.038 | ^3E, v4F | 15/13 |
| 17 | 258.228 | ^4E, v3F | |
| 18 | 273.418 | ^5E, vvF | |
| 19 | 288.608 | E♯, vF | 13/11 |
| 20 | 303.797 | F | |
| 21 | 318.987 | ^F, G♭♭ | 6/5, 65/54, 77/64 |
| 22 | 334.177 | ^^F, v5G♭ | 40/33, 63/52 |
| 23 | 349.367 | ^3F, v4G♭ | 11/9, 27/22 |
| 24 | 364.557 | ^4F, v3G♭ | 16/13, 26/21 |
| 25 | 379.747 | ^5F, vvG♭ | 81/65 |
| 26 | 394.937 | F♯, vG♭ | 44/35 |
| 27 | 410.127 | ^F♯, G♭ | 33/26, 80/63 |
| 28 | 425.316 | ^^F♯, v5G | 50/39 |
| 29 | 440.506 | ^3F♯, v4G | |
| 30 | 455.696 | ^4F♯, v3G | 13/10 |
| 31 | 470.886 | ^5F♯, vvG | 21/16, 72/55 |
| 32 | 486.076 | F𝄪, vG | 33/25 |
| 33 | 501.266 | G | 4/3 |
| 34 | 516.456 | ^G, A♭♭ | 27/20, 35/26 |
| 35 | 531.646 | ^^G, v5A♭ | 15/11 |
| 36 | 546.835 | ^3G, v4A♭ | 11/8, 48/35 |
| 37 | 562.025 | ^4G, v3A♭ | 18/13 |
| 38 | 577.215 | ^5G, vvA♭ | |
| 39 | 592.405 | G♯, vA♭ | 55/39 |
| 40 | 607.595 | ^G♯, A♭ | 78/55 |
| 41 | 622.785 | ^^G♯, v5A | 63/44 |
| 42 | 637.975 | ^3G♯, v4A | 13/9 |
| 43 | 653.165 | ^4G♯, v3A | 16/11, 35/24 |
| 44 | 668.354 | ^5G♯, vvA | 22/15, 81/55 |
| 45 | 683.544 | G𝄪, vA | 40/27, 52/35, 77/52 |
| 46 | 698.734 | A | 3/2 |
| 47 | 713.924 | ^A, B♭♭ | 50/33 |
| 48 | 729.114 | ^^A, v5B♭ | 32/21, 55/36 |
| 49 | 744.304 | ^3A, v4B♭ | 20/13 |
| 50 | 759.494 | ^4A, v3B♭ | |
| 51 | 774.684 | ^5A, vvB♭ | 39/25 |
| 52 | 789.873 | A♯, vB♭ | 52/33, 63/40 |
| 53 | 805.063 | ^A♯, B♭ | 35/22 |
| 54 | 820.253 | ^^A♯, v5B | 77/48 |
| 55 | 835.443 | ^3A♯, v4B | 13/8, 21/13, 81/50 |
| 56 | 850.633 | ^4A♯, v3B | 18/11, 44/27 |
| 57 | 865.823 | ^5A♯, vvB | 33/20 |
| 58 | 881.013 | A𝄪, vB | 5/3 |
| 59 | 896.203 | B | |
| 60 | 911.392 | ^B, C♭ | 22/13 |
| 61 | 926.582 | ^^B, v5C | |
| 62 | 941.772 | ^3B, v4C | |
| 63 | 956.962 | ^4B, v3C | 26/15 |
| 64 | 972.152 | ^5B, vvC | 7/4 |
| 65 | 987.342 | B♯, vC | 39/22 |
| 66 | 1002.53 | C | |
| 67 | 1017.72 | ^C, D♭♭ | 9/5, 65/36, 70/39 |
| 68 | 1032.91 | ^^C, v5D♭ | 20/11 |
| 69 | 1048.1 | ^3C, v4D♭ | 11/6, 64/35 |
| 70 | 1063.29 | ^4C, v3D♭ | 24/13, 50/27 |
| 71 | 1078.48 | ^5C, vvD♭ | |
| 72 | 1093.67 | C♯, vD♭ | 66/35 |
| 73 | 1108.86 | ^C♯, D♭ | |
| 74 | 1124.05 | ^^C♯, v5D | 21/11 |
| 75 | 1139.24 | ^3C♯, v4D | 52/27, 77/40 |
| 76 | 1154.43 | ^4C♯, v3D | 35/18, 39/20 |
| 77 | 1169.62 | ^5C♯, vvD | 63/32, 65/33 |
| 78 | 1184.81 | C𝄪, vD | |
| 79 | 1200 | D | 2/1 |
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)