71edo

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71edo

72edo →

Prime factorization	71 (prime)
Step size	16.9014 ¢
Fifth	42\71 (709.859 ¢)
Semitones (A1:m2)	10:3 (169 ¢ : 50.7 ¢)
Dual sharp fifth	42\71 (709.859 ¢)
Dual flat fifth	41\71 (692.958 ¢)
Dual major 2nd	12\71 (202.817 ¢)
Consistency limit	5
Distinct consistency limit	5

71 equal divisions of the octave (abbreviated 71edo or 71ed2), also called 71-tone equal temperament (71tet) or 71 equal temperament (71et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 71 equal parts of about 16.9 ¢ each. Each step represents a frequency ratio of 2^1/71, or the 71st root of 2.

Theory

71edo is a dual-fifth system, with the flat fifth (which is near the fifths of 26edo and 45edo) supporting flattone temperament, and the sharp fifth (which is near 22edo's fifth) supporting superpyth. Unlike small dual-fifth systems such as 18edo, both fifths are close approximations of 3/2.

Using the patent val, the equal temperament tempers out 20480/19683 and 393216/390625 in the 5-limit, 875/864, 1029/1024 and 4000/3969 in the 7-limit, 100/99 and 245/242 in the 11-limit, and 91/90 in the 13-limit. In the 13-limit it supplies the optimal patent val for the 29 & 71 and 34 & 37 temperaments.

Odd harmonics

Approximation of odd harmonics in 71edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+7.90	+2.42	-5.45	-1.09	+6.43	+4.54	-6.58	-3.55	+6.71	+2.46	-2.92
	Relative (%)	+46.8	+14.3	-32.2	-6.5	+38.0	+26.9	-38.9	-21.0	+39.7	+14.5	-17.3
Steps (reduced)		113 (42)	165 (23)	199 (57)	225 (12)	246 (33)	263 (50)	277 (64)	290 (6)	302 (18)	312 (28)	321 (37)

Subsets and supersets

71edo is the 20th prime edo, following 67edo and before 73edo. 142edo, which doubles it, provides correction for the harmonic 3.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation (Dual flat fifth 41\71)	Ups and downs notation (Dual sharp fifth 42\71)
0	0	1/1	D	D
1	16.9		^D, vE♭♭♭	^D, vvE♭
2	33.8		vD♯, E♭♭♭	^^D, vE♭
3	50.7	35/34	D♯, ^E♭♭♭	^3D, E♭
4	67.6	25/24, 26/25	^D♯, vE♭♭	^4D, ^E♭
5	84.5	21/20	vD𝄪, E♭♭	^5D, ^^E♭
6	101.4	17/16	D𝄪, ^E♭♭	v4D♯, ^3E♭
7	118.3	31/29	^D𝄪, vE♭	v3D♯, ^4E♭
8	135.2	13/12	vD♯𝄪, E♭	vvD♯, v5E
9	152.1	12/11, 35/32	D♯𝄪, ^E♭	vD♯, v4E
10	169	32/29	^D♯𝄪, vE	D♯, v3E
11	185.9	29/26	E	^D♯, vvE
12	202.8		^E, vF♭♭	^^D♯, vE
13	219.7	25/22	vE♯, F♭♭	E
14	236.6		E♯, ^F♭♭	^E, vvF
15	253.5	22/19, 29/25, 37/32	^E♯, vF♭	^^E, vF
16	270.4		vE𝄪, F♭	F
17	287.3	13/11	E𝄪, ^F♭	^F, vvG♭
18	304.2	25/21, 31/26, 37/31	^E𝄪, vF	^^F, vG♭
19	321.1		F	^3F, G♭
20	338	17/14, 28/23	^F, vG♭♭♭	^4F, ^G♭
21	354.9		vF♯, G♭♭♭	^5F, ^^G♭
22	371.8	26/21, 31/25	F♯, ^G♭♭♭	v4F♯, ^3G♭
23	388.7	5/4	^F♯, vG♭♭	v3F♯, ^4G♭
24	405.6	19/15, 24/19	vF𝄪, G♭♭	vvF♯, v5G
25	422.5	37/29	F𝄪, ^G♭♭	vF♯, v4G
26	439.4	31/24	^F𝄪, vG♭	F♯, v3G
27	456.3	13/10	vF♯𝄪, G♭	^F♯, vvG
28	473.2	21/16, 25/19	F♯𝄪, ^G♭	^^F♯, vG
29	490.1		^F♯𝄪, vG	G
30	507		G	^G, vvA♭
31	523.9	23/17	^G, vA♭♭♭	^^G, vA♭
32	540.8	26/19	vG♯, A♭♭♭	^3G, A♭
33	557.7	29/21	G♯, ^A♭♭♭	^4G, ^A♭
34	574.6	32/23	^G♯, vA♭♭	^5G, ^^A♭
35	591.5	31/22	vG𝄪, A♭♭	v4G♯, ^3A♭
36	608.5	37/26	G𝄪, ^A♭♭	v3G♯, ^4A♭
37	625.4	23/16	^G𝄪, vA♭	vvG♯, v5A
38	642.3	29/20	vG♯𝄪, A♭	vG♯, v4A
39	659.2	19/13	G♯𝄪, ^A♭	G♯, v3A
40	676.1	31/21, 34/23, 37/25	^G♯𝄪, vA	^G♯, vvA
41	693		A	^^G♯, vA
42	709.9		^A, vB♭♭♭	A
43	726.8	32/21, 35/23	vA♯, B♭♭♭	^A, vvB♭
44	743.7	20/13	A♯, ^B♭♭♭	^^A, vB♭
45	760.6	31/20	^A♯, vB♭♭	^3A, B♭
46	777.5		vA𝄪, B♭♭	^4A, ^B♭
47	794.4	19/12, 30/19	A𝄪, ^B♭♭	^5A, ^^B♭
48	811.3	8/5	^A𝄪, vB♭	v4A♯, ^3B♭
49	828.2	21/13	vA♯𝄪, B♭	v3A♯, ^4B♭
50	845.1	31/19	A♯𝄪, ^B♭	vvA♯, v5B
51	862	23/14, 28/17	^A♯𝄪, vB	vA♯, v4B
52	878.9		B	A♯, v3B
53	895.8		^B, vC♭♭	^A♯, vvB
54	912.7	22/13	vB♯, C♭♭	^^A♯, vB
55	929.6		B♯, ^C♭♭	B
56	946.5	19/11	^B♯, vC♭	^B, vvC
57	963.4		vB𝄪, C♭	^^B, vC
58	980.3	37/21	B𝄪, ^C♭	C
59	997.2		^B𝄪, vC	^C, vvD♭
60	1014.1		C	^^C, vD♭
61	1031	29/16	^C, vD♭♭♭	^3C, D♭
62	1047.9	11/6	vC♯, D♭♭♭	^4C, ^D♭
63	1064.8	24/13, 37/20	C♯, ^D♭♭♭	^5C, ^^D♭
64	1081.7		^C♯, vD♭♭	v4C♯, ^3D♭
65	1098.6	32/17	vC𝄪, D♭♭	v3C♯, ^4D♭
66	1115.5		C𝄪, ^D♭♭	vvC♯, v5D
67	1132.4	25/13	^C𝄪, vD♭	vC♯, v4D
68	1149.3		vC♯𝄪, D♭	C♯, v3D
69	1166.2		C♯𝄪, ^D♭	^C♯, vvD
70	1183.1		^C♯𝄪, vD	^^C♯, vD
71	1200	2/1	D	D

Notation

Sagittal notation

Best fifth notation

Evo flavor

Revo flavor

Evo-SZ 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.

Second-best fifth notation

This notation uses the same sagittal sequence as EDOs 50, 57, and 64.

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.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[113 -71⟩	[⟨71 113]]	−2.49	2.49	14.72
2.3.5	20480/19683, 393216/390625	[⟨71 113 165]]	−2.01	2.14	12.69
2.3.5.7	64/63, 245/243, 2200/2187	[⟨71 113 165 200]] (71d)	−2.53	2.06	12.19

Instruments

A Lumatone mapping for 71edo is available.

Music

Bryan Deister

• 71edo improv (2025)

Francium

• Dancing in the Mosh Pit (2023)

No Clue Music

• Spiraling - Randomness (2025)