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 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.

Using the full prime-limit patent val, the equal temperament tempers out 81/80, 1029/1024, and 3125/3072 in the 7-limit; and 99/98, 385/384, 441/440, and 625/616 in the 11-limit. 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. It can also be used to tune mothra as well as trismegistus.

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)

Subsets and supersets

57edo contains 3edo and 19edo as subsets.

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)