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.

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

Scales