# 56edo

(Redirected from 56 EDO)

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.

Approximation of prime harmonics in 56edo
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