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

71edo is the 20th prime EDO. It 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 and archy. Unlike small dual-fifth systems such as 18edo, both fifths are close approximations of 3/2.

Theory

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)

It tempers out 20480/19683 and 393216/390625 in the 5-limit, 875/864, 4000/3969 and 1029/1024 in the 7-limit, 245/242 and 100/99 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.

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

Music

Francium

• Dancing in the Mosh Pit (2023)