64edo

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64edo

65edo →

Prime factorization	26
Step size	18.75 ¢
Fifth	37\64 (693.75 ¢)
Semitones (A1:m2)	3:7 (56.25 ¢ : 131.3 ¢)
Dual sharp fifth	38\64 (712.5 ¢) (→ 19\32)
Dual flat fifth	37\64 (693.75 ¢)
Dual major 2nd	11\64 (206.25 ¢)
Consistency limit	3
Distinct consistency limit	3

Theory

64edo has two options of fifth almost equally far from just. The sharp fifth is inherited from 32edo and produces a hard superpythagorean scale, while the slightly more accurate flat fifth is within the meantone/flattone range. However bizarrely, the flat fifth does not support meantone or flattone in its patent val, and instead supports the obscure unnamed 7c & 12c (or 19 & 64) temperament which reaches 5/4 as a double-diminished fourth. In order to interpret it as flattone, the 64cd val must be used.

Still, the patent val tempers out 648/625 in the 5-limit and 225/224 in the 7-limit, plus 66/65, 121/120 and 441/440 in the 11-limit and 144/143 in the 13-limit. It provides the optimal patent val in the 7-, 11- and 13-limits for the 16&64 temperament.

64be val is a tuning for the beatles temperament and for the rank-3 temperaments heimlaug and vili in the 17-limit. 64bccc tunes dichotic, although that is an exotemperament. 64cdf is a tuning for vibhu.

64edo is a zeta valley EDO and is very bad at approximating JI for its size.

Odd harmonics

Approximation of odd harmonics in 64edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-8.21	+7.44	+6.17	+2.34	-7.57	+3.22	-0.77	+7.54	+2.49	-2.03	+9.23
	Relative (%)	-43.8	+39.7	+32.9	+12.5	-40.4	+17.2	-4.1	+40.2	+13.3	-10.8	+49.2
Steps (reduced)		101 (37)	149 (21)	180 (52)	203 (11)	221 (29)	237 (45)	250 (58)	262 (6)	272 (16)	281 (25)	290 (34)

Subsets and supersets

64edo is the 6th power of two edo, and it has subset edos 1, 2, 4, 8, 16, 32.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation (Dual flat fifth 37\64)	Ups and downs notation (Dual sharp fifth 38\64)
0	0	1/1	D	D
1	18.8		^D, E♭♭♭	^D, vE♭
2	37.5		vD♯, ^E♭♭♭	^^D, E♭
3	56.3	30/29, 31/30, 32/31	D♯, vE♭♭	^3D, ^E♭
4	75		^D♯, E♭♭	^4D, ^^E♭
5	93.8		vD𝄪, ^E♭♭	^5D, ^3E♭
6	112.5	16/15, 31/29	D𝄪, vE♭	v4D♯, ^4E♭
7	131.3	14/13	^D𝄪, E♭	v3D♯, v5E
8	150	12/11	vD♯𝄪, ^E♭	vvD♯, v4E
9	168.8	32/29	D♯𝄪, vE	vD♯, v3E
10	187.5	29/26	E	D♯, vvE
11	206.3		^E, F♭♭	^D♯, vE
12	225		vE♯, ^F♭♭	E
13	243.8	15/13, 23/20	E♯, vF♭	^E, vF
14	262.5		^E♯, F♭	F
15	281.3	20/17	vE𝄪, ^F♭	^F, vG♭
16	300	19/16	E𝄪, vF	^^F, G♭
17	318.8		F	^3F, ^G♭
18	337.5	17/14, 28/23	^F, G♭♭♭	^4F, ^^G♭
19	356.3	16/13	vF♯, ^G♭♭♭	^5F, ^3G♭
20	375		F♯, vG♭♭	v4F♯, ^4G♭
21	393.8		^F♯, G♭♭	v3F♯, v5G
22	412.5	19/15	vF𝄪, ^G♭♭	vvF♯, v4G
23	431.3		F𝄪, vG♭	vF♯, v3G
24	450		^F𝄪, G♭	F♯, vvG
25	468.8	21/16	vF♯𝄪, ^G♭	^F♯, vG
26	487.5		F♯𝄪, vG	G
27	506.2		G	^G, vA♭
28	525	19/14, 23/17	^G, A♭♭♭	^^G, A♭
29	543.8	26/19	vG♯, ^A♭♭♭	^3G, ^A♭
30	562.5	29/21	G♯, vA♭♭	^4G, ^^A♭
31	581.3	7/5	^G♯, A♭♭	^5G, ^3A♭
32	600		vG𝄪, ^A♭♭	v4G♯, ^4A♭
33	618.8	10/7	G𝄪, vA♭	v3G♯, v5A
34	637.5		^G𝄪, A♭	vvG♯, v4A
35	656.3	19/13	vG♯𝄪, ^A♭	vG♯, v3A
36	675	28/19, 31/21, 34/23	G♯𝄪, vA	G♯, vvA
37	693.8		A	^G♯, vA
38	712.5		^A, B♭♭♭	A
39	731.3	29/19, 32/21	vA♯, ^B♭♭♭	^A, vB♭
40	750		A♯, vB♭♭	^^A, B♭
41	768.8		^A♯, B♭♭	^3A, ^B♭
42	787.5	30/19	vA𝄪, ^B♭♭	^4A, ^^B♭
43	806.3		A𝄪, vB♭	^5A, ^3B♭
44	825		^A𝄪, B♭	v4A♯, ^4B♭
45	843.7	13/8, 31/19	vA♯𝄪, ^B♭	v3A♯, v5B
46	862.5	23/14, 28/17	A♯𝄪, vB	vvA♯, v4B
47	881.2		B	vA♯, v3B
48	900	32/19	^B, C♭♭	A♯, vvB
49	918.8	17/10	vB♯, ^C♭♭	^A♯, vB
50	937.5		B♯, vC♭	B
51	956.3	26/15	^B♯, C♭	^B, vC
52	975		vB𝄪, ^C♭	C
53	993.8		B𝄪, vC	^C, vD♭
54	1012.5		C	^^C, D♭
55	1031.3	29/16	^C, D♭♭♭	^3C, ^D♭
56	1050	11/6	vC♯, ^D♭♭♭	^4C, ^^D♭
57	1068.8	13/7	C♯, vD♭♭	^5C, ^3D♭
58	1087.5	15/8	^C♯, D♭♭	v4C♯, ^4D♭
59	1106.3		vC𝄪, ^D♭♭	v3C♯, v5D
60	1125		C𝄪, vD♭	vvC♯, v4D
61	1143.8	29/15, 31/16	^C𝄪, D♭	vC♯, v3D
62	1162.5		vC♯𝄪, ^D♭	C♯, vvD
63	1181.3		C♯𝄪, vD	^C♯, vD
64	1200	2/1	D	D