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

57 equal divisions of the octave (abbreviated 57edo or 57ed2), also called 57-tone equal temperament (57tet) or 57 equal temperament (57et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 57 equal parts of about 21.1 ¢ each. Each step represents a frequency ratio of 2^1/57, or the 57th root of 2.

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)

Approximation of odd harmonics in 57edo (continued)

Harmonic		25	27	29	31	33	35	37	39	41	43	45
Error	Absolute (¢)	+6.32	-0.60	+2.00	-8.19	+9.88	-7.77	+1.29	-5.64	-8.01	-6.25	-0.75
	Relative (%)	+30.0	-2.9	+9.5	-38.9	+47.0	-36.9	+6.1	-26.8	-38.0	-29.7	-3.6
Steps (reduced)		265 (37)	271 (43)	277 (49)	282 (54)	288 (3)	292 (7)	297 (12)	301 (16)	305 (20)	309 (24)	313 (28)

Subsets and supersets

57edo contains 3edo and 19edo as subsets. Tripling 57edo yields 171edo, which corrects the 3rd and 5th harmonics.

Intervals

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

* As a 2.3.5.7.11.13.17.19.23.29.31-subgroup temperament, in italics if inconsistent

Notation

Stein–Zimmermann–Gould notation

57edo can be notated using Stein–Zimmermann–Gould notation:

Step offset	0	1	2	3	4	5	6	7
Sharp symbol								
Flat symbol								

Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows.

Kite's ups and downs notation

Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.

Step offset	0	1	2	3	4	5	6	7
Sharp symbol
Flat symbol

Sagittal notation

This notation uses the same sagittal sequence as edos 50, 64, and 71b, and is a superset of the notation for 19edo.

Evo flavor

Revo flavor

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.

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)

Instruments

Since 57edo contains 19edo as a non-trivial subset, it would be possible to use three 19edo instruments tuned 1\57 apart from each other to play the full gamut of 57edo.

A Lumatone mapping for 57edo is available.

Music

Bryan Deister

• 57edo improv (2025)
• Prelude in 57edo (2025) (switches Lumatone mapping between sections)
  • Part 1 (short clip)
  • Part 2 (short clip)
  • Whole composition (but no view of Lumatone)
• 57edo groove (2025)