56edo
(Redirected from 56 EDO)
Jump to navigation
Jump to search
56edo divides the octave into 56 parts of 21.429 cents each. 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.
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.
Intervals
The following table assumes the patent val ⟨56 89 130 157 194 207]. Other approaches are possible.
| # | Cents | Approximate Ratios |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 21.429 | 49/48, 64/63 |
| 2 | 42.857 | 28/27, 50/49, 81/80 |
| 3 | 64.286 | 25/24, 36/35, 33/32 |
| 4 | 85.714 | 21/20, 22/21 |
| 5 | 107.143 | 16/15 |
| 6 | 128.571 | 15/14, 13/12, 14/13 |
| 7 | 150.000 | 12/11 |
| 8 | 171.429 | 10/9, 11/10 |
| 9 | 192.857 | 28/25 |
| 10 | 214.286 | 9/8 |
| 11 | 235.714 | 8/7 |
| 12 | 257.143 | 7/6, 15/13 |
| 13 | 278.571 | 75/64, 13/11 |
| 14 | 300.000 | 25/21 |
| 15 | 321.429 | 6/5 |
| 16 | 342.857 | 11/9, 39/32 |
| 17 | 364.286 | 27/22, 16/13, 26/21 |
| 18 | 385.714 | 5/4 |
| 19 | 407.143 | 14/11 |
| 20 | 428.571 | 32/25, 33/26 |
| 21 | 450.000 | 9/7, 13/10 |
| 22 | 471.429 | 21/16 |
| 23 | 492.857 | 4/3 |
| 24 | 514.286 | |
| 25 | 535.714 | 27/20, 15/11 |
| 26 | 557.143 | 11/8 |
| 27 | 578.571 | 7/5 |
| 28 | 600.000 | 45/32, 64/45 |
| … | … | … |
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