57edo

← 56edo

57edo

58edo →

Prime factorization	3 × 19
Step size	21.0526 ¢
Fifth	33\57 (694.737 ¢) (→ 11\19)
Semitones (A1:m2)	3:6 (63.16 ¢ : 126.3 ¢)
Dual sharp fifth	34\57 (715.789 ¢)
Dual flat fifth	33\57 (694.737 ¢) (→ 11\19)
Dual major 2nd	10\57 (210.526 ¢)
Consistency limit	7
Distinct consistency limit	7

Theory

57edo can be used to tune the mothra temperament, and is an excellent tuning for the 2.5/3.7.11.13.17.19 just intonation subgroup. One way to describe 57edo is that it has a 5-limit part consisting of three rings of 19edo, plus a no-threes no-fives part which is much more accurate. A good generator to exploit the 2.5/3.7.11.13.17.19 aspect of 57 is the approximate 11/8, which is 26\57. This gives the 19-limit 46 & 57 temperament heinz.

5-limit commas: 81/80, 3125/3072

7-limit commas: 81/80, 3125/3072, 1029/1024

11-limit commas: 99/98, 385/384, 441/440, 625/616

Odd harmonics

Approximation of odd harmonics in 57edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-7.22	-7.37	-0.40	+6.62	-3.95	+1.58	+6.47	+0.31	-2.78	-7.62	+3.30
	Relative (%)	-34.3	-35.0	-1.9	+31.4	-18.8	+7.5	+30.7	+1.5	-13.2	-36.2	+15.7
Steps (reduced)		90 (33)	132 (18)	160 (46)	181 (10)	197 (26)	211 (40)	223 (52)	233 (5)	242 (14)	250 (22)	258 (30)

Intervals

#	Cents	Ups and Downs Notation (Flat Fifth 11\19)	Ups and Downs Notation (Sharp Fifth 34\57)
0	0.00	D	D
1	21.05	^D, ^E♭♭♭	^D, E♭
2	42.11	vD♯, vE♭♭	^^D, ^E♭
3	63.16	D♯, E♭♭	^3D, ^^E♭
4	84.21	^D♯, ^E♭♭	^4D, ^3E♭
5	105.26	vD𝄪, vE♭	^5D, ^4E♭
6	126.32	D𝄪, E♭	v4D♯, v5E
7	147.37	^D𝄪, ^E♭	v3D♯, v4E
8	168.42	vD♯𝄪, vE	vvD♯, v3E
9	189.47	E	vD♯, vvE
10	210.53	^E, ^F♭♭	D♯, vE
11	231.58	vE♯, vF♭	E
12	252.63	E♯, F♭	F
13	273.68	^E♯, ^F♭	^F, G♭
14	294.74	vE𝄪, vF	^^F, ^G♭
15	315.79	F	^3F, ^^G♭
16	336.84	^F, ^G♭♭♭	^4F, ^3G♭
17	357.89	vF♯, vG♭♭	^5F, ^4G♭
18	378.95	F♯, G♭♭	v4F♯, v5G
19	400.00	^F♯, ^G♭♭	v3F♯, v4G
20	421.05	vF𝄪, vG♭	vvF♯, v3G
21	442.11	F𝄪, G♭	vF♯, vvG
22	463.16	^F𝄪, ^G♭	F♯, vG
23	484.21	vF♯𝄪, vG	G
24	505.26	G	^G, A♭
25	526.32	^G, ^A♭♭♭	^^G, ^A♭
26	547.37	vG♯, vA♭♭	^3G, ^^A♭
27	568.42	G♯, A♭♭	^4G, ^3A♭
28	589.47	^G♯, ^A♭♭	^5G, ^4A♭
29	610.53	vG𝄪, vA♭	v4G♯, v5A
30	631.58	G𝄪, A♭	v3G♯, v4A
31	652.63	^G𝄪, ^A♭	vvG♯, v3A
32	673.68	vG♯𝄪, vA	vG♯, vvA
33	694.74	A	G♯, vA
34	715.79	^A, ^B♭♭♭	A
35	736.84	vA♯, vB♭♭	^A, B♭
36	757.89	A♯, B♭♭	^^A, ^B♭
37	778.95	^A♯, ^B♭♭	^3A, ^^B♭
38	800.00	vA𝄪, vB♭	^4A, ^3B♭
39	821.05	A𝄪, B♭	^5A, ^4B♭
40	842.11	^A𝄪, ^B♭	v4A♯, v5B
41	863.16	vA♯𝄪, vB	v3A♯, v4B
42	884.21	B	vvA♯, v3B
43	905.26	^B, ^C♭♭	vA♯, vvB
44	926.32	vB♯, vC♭	A♯, vB
45	947.37	B♯, C♭	B
46	968.42	^B♯, ^C♭	C
47	989.47	vB𝄪, vC	^C, D♭
48	1010.53	C	^^C, ^D♭
49	1031.58	^C, ^D♭♭♭	^3C, ^^D♭
50	1052.63	vC♯, vD♭♭	^4C, ^3D♭
51	1073.68	C♯, D♭♭	^5C, ^4D♭
52	1094.74	^C♯, ^D♭♭	v4C♯, v5D
53	1115.79	vC𝄪, vD♭	v3C♯, v4D
54	1136.84	C𝄪, D♭	vvC♯, v3D
55	1157.89	^C𝄪, ^D♭	vC♯, vvD
56	1178.95	vC♯𝄪, vD	C♯, vD
57	1200.00	D	D

Scales

• 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 1 - 3mos of type 18L 21s (augene)