56edo

← 55edo

56edo

57edo →

Prime factorization	23 × 7
Step size	21.4286 ¢
Fifth	33\56 (707.143 ¢)
Semitones (A1:m2)	7:3 (150 ¢ : 64.29 ¢)
Consistency limit	7
Distinct consistency limit	7

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 ha a near perfect 5/4 and a step size very close to the syntonic comma, 56edo contains very accurate approximations of both 5/4 and 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.

Approximation of prime harmonics in 56edo

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
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

• Supra7
• Supra12

See also

Lumatone mapping for 56edo