56edo
| ← 55edo | 56edo | 57edo → |
56 equal divisions of the octave (56edo), or 56-tone equal temperament (56tet), 56 equal temperament (56et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 56 equal parts of about 21.4 ¢ each, a size close to the syntonic comma 81/80.
Theory
It shares it's near perfect major third with 28edo, which it doubles, while also adding a superpythagorean 5th that is a convergent towards the bronze metallic mean, following 17edo and preceding 185edo. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third 5/4 and the Pythagorean major third 81/64.
56edo can be used to tune hemithirds, superkleismic, sycamore and keen temperaments, and using ⟨56 89 130 158] (56d) as the equal temperament val, for pajara. It provides the optimal patent val for 7-, 11- and 13-limit sycamore, and the 11-limit 56d val is close to the POTE tuning for 11-limit pajara. 56edo can be used to tune Barium temperament which sets 56 syntonci commas to the octave.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +5.19 | -0.60 | -4.54 | +5.82 | -4.81 | +2.19 | +2.49 | -6.85 | -1.01 | -9.32 |
| Relative (%) | +0.0 | +24.2 | -2.8 | -21.2 | +27.2 | -22.5 | +10.2 | +11.6 | -31.9 | -4.7 | -43.5 | |
| Steps (reduced) |
56 (0) |
89 (33) |
130 (18) |
157 (45) |
194 (26) |
207 (39) |
229 (5) |
238 (14) |
253 (29) |
272 (48) |
277 (53) | |
Intervals
The following table assumes the patent val ⟨56 89 130 157 194 207]. Other approaches are possible.
| # | Cents | Approximate Ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0.000 | 1/1 | D |
| 1 | 21.429 | 49/48, 64/63 | ^D, vvE♭ |
| 2 | 42.857 | 28/27, 50/49, 81/80 | ^^D, vE♭ |
| 3 | 64.286 | 25/24, 36/35, 33/32 | ^3D, E♭ |
| 4 | 85.714 | 21/20, 22/21 | v3D♯, ^E♭ |
| 5 | 107.143 | 16/15 | vvD♯, ^^E♭ |
| 6 | 128.571 | 15/14, 13/12, 14/13 | vD♯, ^3E♭ |
| 7 | 150.000 | 12/11 | D♯, v3E |
| 8 | 171.429 | 10/9, 11/10 | ^D♯, vvE |
| 9 | 192.857 | 28/25 | ^^D♯, vE |
| 10 | 214.286 | 9/8 | E |
| 11 | 235.714 | 8/7 | ^E, vvF |
| 12 | 257.143 | 7/6, 15/13 | step=12}] |
| 13 | 278.571 | 75/64, 13/11 | F |
| 14 | 300.000 | 25/21 | ^F, vvG♭ |
| 15 | 321.429 | 6/5 | ^^F, vG♭ |
| 16 | 342.857 | 11/9, 39/32 | ^3F, G♭ |
| 17 | 364.286 | 27/22, 16/13, 26/21 | v3F♯, ^G♭ |
| 18 | 385.714 | 5/4 | vvF♯, ^^G♭ |
| 19 | 407.143 | 14/11 | vF♯, ^3G♭ |
| 20 | 428.571 | 32/25, 33/26 | F♯, v3G |
| 21 | 450.000 | 9/7, 13/10 | ^F♯, vvG |
| 22 | 471.429 | 21/16 | ^^F♯, vG |
| 23 | 492.857 | 4/3 | G |
| 24 | 514.286 | ^G, vvA♭ | |
| 25 | 535.714 | 27/20, 15/11 | |
| 26 | 557.143 | 11/8 | ^^G, vA♭ |
| 27 | 578.571 | 7/5 | ^3G, A♭ |
| 28 | 600.000 | 45/32, 64/45 | v3G♯, ^A♭ |
| … | … | … | … |
Commas
- 5-limit commas: 2048/2025, [-5 -10 9⟩;
- 7-limit commas: 686/675, 875/864, 1029/1024
- 11-limit commas: 100/99, 245/242, 385/384, 686/675