56edo
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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.
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
---|---|---|---|---|---|---|---|---|---|---|---|
Error | absolute (¢) | +0.00 | +5.19 | -0.60 | -4.54 | +5.82 | -4.81 | +2.19 | +2.49 | -6.85 | -1.01 |
relative (%) | +0 | +24 | -3 | -21 | +27 | -22 | +10 | +12 | -32 | -5 | |
Steps (reduced) |
56 (0) |
89 (33) |
130 (18) |
157 (45) |
194 (26) |
207 (39) |
229 (5) |
238 (14) |
253 (29) |
272 (48) |
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