78edo
| ← 77edo | 78edo | 79edo → |
78 equal divisions of the octave (abbreviated 78edo or 78ed2), also called 78-tone equal temperament (78tet) or 78 equal temperament (78et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 78 equal parts of about 15.4 ¢ each. Each step represents a frequency ratio of 21/78, or the 78th root of 2.
Theory
78edo is consistent to the 7-odd-limit, but the error of harmonic 3, inherited from 39edo, is quite large for the size of the system.
This tuning tempers out 2048/2025 in the 5-limit; 875/864 and 2401/2400 in the 7-limit; and 100/99, 385/384 and 1375/1372 in the 11-limit. It provides the optimal patent val for 11-limit keen temperament.
Much like 100bddd, the 78dd val can be used to construct an alternative to 22edo for pajara. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7 cents. The major third is 384.6 cents; less than two cents flat of just. The harmonic seventh is 984.6 cents, or about 15.8 cents sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16 cents} off. The 22-note 2mos generated in this way could be used to build straight-fretted guitars that would be tuned in tritones. The appeal of this scale is that it is less xenharmonic than 22edo is, for listeners accustomed to 12edo. In particular, the 163.6 cents "flat minor whole tone" of 22edo is now 169.2 cents, making it more clearly a whole tone (albeit noticeably flat), rather than a neutral second.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.74 | -1.70 | +0.40 | -3.91 | +2.53 | +5.63 | +4.04 | +2.74 | -5.21 | +6.14 | +2.49 |
| Relative (%) | +37.3 | -11.0 | +2.6 | -25.4 | +16.4 | +36.6 | +26.3 | +17.8 | -33.8 | +39.9 | +16.2 | |
| Steps (reduced) |
124 (46) |
181 (25) |
219 (63) |
247 (13) |
270 (36) |
289 (55) |
305 (71) |
319 (7) |
331 (19) |
343 (31) |
353 (41) | |
Subsets and supersets
Since 78 factors into 2 × 3 × 13, 78edo has subset edos 2, 3, 6, 13, 26, and 39. 156edo, which doubles it, is a notable tuning.
Intervals
| Steps | Cents | Approximate Ratios | Ups and Downs Notation (Dual Flat Fifth 45\78) |
Ups and Downs Notation (Dual Sharp Fifth 46\78) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 15.385 | ^D, vvE♭♭♭ | ^D, v3E♭ | |
| 2 | 30.769 | 49/48, 50/49, 56/55, 65/64, 66/65 | ^^D, vE♭♭♭ | ^^D, vvE♭ |
| 3 | 46.154 | 77/75 | D♯, E♭♭♭ | ^3D, vE♭ |
| 4 | 61.538 | 80/77 | ^D♯, vvE♭♭ | ^4D, E♭ |
| 5 | 76.923 | 22/21 | ^^D♯, vE♭♭ | ^5D, v9E |
| 6 | 92.308 | 55/52 | D𝄪, E♭♭ | ^6D, v8E |
| 7 | 107.692 | 16/15, 52/49 | ^D𝄪, vvE♭ | ^7D, v7E |
| 8 | 123.077 | 14/13, 15/14 | ^^D𝄪, vE♭ | ^8D, v6E |
| 9 | 138.462 | 13/12 | D♯𝄪, E♭ | ^9D, v5E |
| 10 | 153.846 | 12/11, 35/32 | ^D♯𝄪, vvE | D♯, v4E |
| 11 | 169.231 | 11/10 | ^^D♯𝄪, vE | ^D♯, v3E |
| 12 | 184.615 | 49/44 | E | ^^D♯, vvE |
| 13 | 200 | 28/25, 55/49 | ^E, vvF♭♭ | ^3D♯, vE |
| 14 | 215.385 | 25/22 | ^^E, vF♭♭ | E |
| 15 | 230.769 | 8/7, 55/48 | E♯, F♭♭ | ^E, v3F |
| 16 | 246.154 | 15/13, 52/45 | ^E♯, vvF♭ | ^^E, vvF |
| 17 | 261.538 | 7/6, 64/55, 65/56 | ^^E♯, vF♭ | ^3E, vF |
| 18 | 276.923 | 75/64 | E𝄪, F♭ | F |
| 19 | 292.308 | 13/11, 77/65 | ^E𝄪, vvF | ^F, v3G♭ |
| 20 | 307.692 | ^^E𝄪, vF | ^^F, vvG♭ | |
| 21 | 323.077 | 77/64 | F | ^3F, vG♭ |
| 22 | 338.462 | ^F, vvG♭♭♭ | ^4F, G♭ | |
| 23 | 353.846 | 16/13, 49/40, 60/49 | ^^F, vG♭♭♭ | ^5F, v9G |
| 24 | 369.231 | 26/21 | F♯, G♭♭♭ | ^6F, v8G |
| 25 | 384.615 | 5/4 | ^F♯, vvG♭♭ | ^7F, v7G |
| 26 | 400 | 44/35 | ^^F♯, vG♭♭ | ^8F, v6G |
| 27 | 415.385 | 14/11, 33/26 | F𝄪, G♭♭ | ^9F, v5G |
| 28 | 430.769 | 32/25, 77/60 | ^F𝄪, vvG♭ | F♯, v4G |
| 29 | 446.154 | ^^F𝄪, vG♭ | ^F♯, v3G | |
| 30 | 461.538 | 55/42, 64/49 | F♯𝄪, G♭ | ^^F♯, vvG |
| 31 | 476.923 | 21/16 | ^F♯𝄪, vvG | ^3F♯, vG |
| 32 | 492.308 | 4/3, 65/49 | ^^F♯𝄪, vG | G |
| 33 | 507.692 | 75/56 | G | ^G, v3A♭ |
| 34 | 523.077 | 65/48 | ^G, vvA♭♭♭ | ^^G, vvA♭ |
| 35 | 538.462 | 15/11 | ^^G, vA♭♭♭ | ^3G, vA♭ |
| 36 | 553.846 | 11/8 | G♯, A♭♭♭ | ^4G, A♭ |
| 37 | 569.231 | 18/13 | ^G♯, vvA♭♭ | ^5G, v9A |
| 38 | 584.615 | 7/5 | ^^G♯, vA♭♭ | ^6G, v8A |
| 39 | 600 | G𝄪, A♭♭ | ^7G, v7A | |
| 40 | 615.385 | 10/7 | ^G𝄪, vvA♭ | ^8G, v6A |
| 41 | 630.769 | 13/9, 75/52 | ^^G𝄪, vA♭ | ^9G, v5A |
| 42 | 646.154 | 16/11 | G♯𝄪, A♭ | G♯, v4A |
| 43 | 661.538 | 22/15 | ^G♯𝄪, vvA | ^G♯, v3A |
| 44 | 676.923 | 65/44, 77/52 | ^^G♯𝄪, vA | ^^G♯, vvA |
| 45 | 692.308 | A | ^3G♯, vA | |
| 46 | 707.692 | 3/2 | ^A, vvB♭♭♭ | A |
| 47 | 723.077 | 32/21 | ^^A, vB♭♭♭ | ^A, v3B♭ |
| 48 | 738.462 | 49/32, 75/49 | A♯, B♭♭♭ | ^^A, vvB♭ |
| 49 | 753.846 | 65/42 | ^A♯, vvB♭♭ | ^3A, vB♭ |
| 50 | 769.231 | 25/16 | ^^A♯, vB♭♭ | ^4A, B♭ |
| 51 | 784.615 | 11/7, 52/33 | A𝄪, B♭♭ | ^5A, v9B |
| 52 | 800 | 35/22 | ^A𝄪, vvB♭ | ^6A, v8B |
| 53 | 815.385 | 8/5, 77/48 | ^^A𝄪, vB♭ | ^7A, v7B |
| 54 | 830.769 | 21/13 | A♯𝄪, B♭ | ^8A, v6B |
| 55 | 846.154 | 13/8, 49/30, 80/49 | ^A♯𝄪, vvB | ^9A, v5B |
| 56 | 861.538 | ^^A♯𝄪, vB | A♯, v4B | |
| 57 | 876.923 | B | ^A♯, v3B | |
| 58 | 892.308 | ^B, vvC♭♭ | ^^A♯, vvB | |
| 59 | 907.692 | 22/13 | ^^B, vC♭♭ | ^3A♯, vB |
| 60 | 923.077 | 75/44 | B♯, C♭♭ | B |
| 61 | 938.462 | 12/7, 55/32 | ^B♯, vvC♭ | ^B, v3C |
| 62 | 953.846 | 26/15, 45/26 | ^^B♯, vC♭ | ^^B, vvC |
| 63 | 969.231 | 7/4 | B𝄪, C♭ | ^3B, vC |
| 64 | 984.615 | 44/25 | ^B𝄪, vvC | C |
| 65 | 1000 | 25/14 | ^^B𝄪, vC | ^C, v3D♭ |
| 66 | 1015.385 | C | ^^C, vvD♭ | |
| 67 | 1030.769 | 20/11 | ^C, vvD♭♭♭ | ^3C, vD♭ |
| 68 | 1046.154 | 11/6, 64/35 | ^^C, vD♭♭♭ | ^4C, D♭ |
| 69 | 1061.538 | 24/13 | C♯, D♭♭♭ | ^5C, v9D |
| 70 | 1076.923 | 13/7, 28/15 | ^C♯, vvD♭♭ | ^6C, v8D |
| 71 | 1092.308 | 15/8, 49/26 | ^^C♯, vD♭♭ | ^7C, v7D |
| 72 | 1107.692 | C𝄪, D♭♭ | ^8C, v6D | |
| 73 | 1123.077 | 21/11 | ^C𝄪, vvD♭ | ^9C, v5D |
| 74 | 1138.462 | 77/40 | ^^C𝄪, vD♭ | C♯, v4D |
| 75 | 1153.846 | C♯𝄪, D♭ | ^C♯, v3D | |
| 76 | 1169.231 | 49/25, 55/28, 65/33 | ^C♯𝄪, vvD | ^^C♯, vvD |
| 77 | 1184.615 | ^^C♯𝄪, vD | ^3C♯, vD | |
| 78 | 1200 | 2/1 | D | D |