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

It can be used to tune mothra temperament, and is an excellent tuning for the 2.5/3.7.11.13.17.19 just intonation subgroup. One way to describe 57 is that it has a 5-limit part consisting of three versions of 19, 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

Degree	Cents	Ups and downs notation (flat fifth 11\19)	Ups and downs notation (sharp fifth 34\57)
0	0.0000	D	D
1	21.0526	^D, ^E♭♭♭	^D, E♭
2	42.1053	vD♯, vE♭♭	^^D, ^E♭
3	63.1579	D♯, E♭♭	^3D, ^^E♭
4	84.2105	^D♯, ^E♭♭	^4D, ^3E♭
5	105.2632	vD𝄪, vE♭	^5D, ^4E♭
6	126.3158	D𝄪, E♭	v4D♯, v5E
7	147.3684	^D𝄪, ^E♭	v3D♯, v4E
8	168.42105	vD♯𝄪, vE	vvD♯, v3E
9	189.4737	E	vD♯, vvE
10	210.5263	^E, ^F♭♭	D♯, vE
11	231.57895	vE♯, vF♭	E
12	252.6316	E♯, F♭	F
13	273.6842	^E♯, ^F♭	^F, G♭
14	294.7368	vE𝄪, vF	^^F, ^G♭
15	315.7895	F	^3F, ^^G♭
16	336.8421	^F, ^G♭♭♭	^4F, ^3G♭
17	357.8947	vF♯, vG♭♭	^5F, ^4G♭
18	378.9474	F♯, G♭♭	v4F♯, v5G
19	400	^F♯, ^G♭♭	v3F♯, v4G
20	421.0526	vF𝄪, vG♭	vvF♯, v3G
21	442.1053	F𝄪, G♭	vF♯, vvG
22	463.1579	^F𝄪, ^G♭	F♯, vG
23	484.2105	vF♯𝄪, vG	G
24	505.2632	G	^G, A♭
25	526.3158	^G, ^A♭♭♭	^^G, ^A♭
26	547.3684	vG♯, vA♭♭	^3G, ^^A♭
27	568.42105	G♯, A♭♭	^4G, ^3A♭
28	589.4737	^G♯, ^A♭♭	^5G, ^4A♭
29	610.5263	vG𝄪, vA♭	v4G♯, v5A
30	631.57895	G𝄪, A♭	v3G♯, v4A
31	652.6316	^G𝄪, ^A♭	vvG♯, v3A
32	673.6842	vG♯𝄪, vA	vG♯, vvA
33	694.7368	A	G♯, vA
34	715.7895	^A, ^B♭♭♭	A
35	736.8421	vA♯, vB♭♭	^A, B♭
36	757.8947	A♯, B♭♭	^^A, ^B♭
37	778.9474	^A♯, ^B♭♭	^3A, ^^B♭
38	800	vA𝄪, vB♭	^4A, ^3B♭
39	821.0526	A𝄪, B♭	^5A, ^4B♭
40	842.1053	^A𝄪, ^B♭	v4A♯, v5B
41	863.1579	vA♯𝄪, vB	v3A♯, v4B
42	884.2105	B	vvA♯, v3B
43	905.2632	^B, ^C♭♭	vA♯, vvB
44	926.3158	vB♯, vC♭	A♯, vB
45	947.3684	B♯, C♭	B
46	968.42105	^B♯, ^C♭	C
47	989.4737	vB𝄪, vC	^C, D♭
48	1010.5263	C	^^C, ^D♭
49	1031.57895	^C, ^D♭♭♭	^3C, ^^D♭
50	1052.6316	vC♯, vD♭♭	^4C, ^3D♭
51	1073.6842	C♯, D♭♭	^5C, ^4D♭
52	1094.7368	^C♯, ^D♭♭	v4C♯, v5D
53	1115.7895	vC𝄪, vD♭	v3C♯, v4D
54	1136.8421	C𝄪, D♭	vvC♯, v3D
55	1157.8947	^C𝄪, ^D♭	vC♯, vvD
56	1178.9474	vC♯𝄪, vD	C♯, vD
57	1200	D	D

Scales of 57EDO

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)