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

64 equal divisions of the octave (abbreviated 64edo or 64ed2), also called 64-tone equal temperament (64tet) or 64 equal temperament (64et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 64 equal parts of about 18.8 ¢ each. Each step represents a frequency ratio of 2^1/64, or the 64th root of 2.

Theory

64edo is a zeta valley edo and is very bad at approximating JI for its size. It has two options of fifth almost equally far from just. The sharp fifth from the 64b val is inherited from 32edo and produces a hard superpythagorean scale, while the slightly more accurate flat fifth from the patent val 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.

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)

Octave stretch

64edo's approximations of 3/1, 5/1, 7/1, 11/1 and 17/1 are improved by 180ed7, a compressed-octave version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.

149ed5 can also be used: it is similar to 180ed7 but both the improvements and shortcomings are amplified. Most notably its 2/1 isn’t as accurate as 180ed7's.

If one prefers a stretched-octave, 64edo's approximations of 3/1, 5/1, 11/1 and 17/1 are improved by 221ed11, a stretched version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.

64ed257/128 can also be used: it is similar to 221ed11 but both the improvements and shortcomings are amplified. Most notably its 2/1 is not as accurate as 221ed11's.

There are also some nearby zeta peak index (ZPI) tunings which can be used to improve 64edo's JI approximations: 326zpi, 327zpi, 328zpi and 329zpi. The main Zeta peak index page details all four tunings.

Subsets and supersets

64edo is the 6th power of two edo, and it has subset edos 2, 4, 8, 16, 32. 128edo, which doubles it, corrects its approximation to many of the lower harmonics.

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

Notation

Sagittal notation

Best fifth notation

This notation uses the same sagittal sequence as EDOs 50, 57, and 71b.

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.

Second-best fifth notation

This notation is a superset of the notation for 32-EDO.

Evo flavor

Revo flavor

Evo-SZ flavor