64edo

← 63edo

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 has two options of fifth equally far from just. The sharp fifth is inherited from 32edo and produces a hard superpythagorean scale, while the flat fifth is within the meantone/flattone range, supporting flattone temperament.

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, which would perhaps be of more interest if it was lower in badness.

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 (%)	-44	+40	+33	+12	-40	+17	-4	+40	+13	-11	+49
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	Ups and downs notation (dual flat fifth 37\64)	Ups and downs notation (dual sharp fifth 38\64)	Approximate ratios
0	0	D	D	1/1
1	18.75	↑D, E♭♭♭	↑D, ↓E♭	78/77
2	37.5	↑↑D, ↓↓E♭♭	↑↑D, E♭	45/44, 50/49, 56/55
3	56.25	D♯, ↓E♭♭	↑3D, ↓9E
4	75	↑D♯, E♭♭	↑4D, ↓8E	22/21
5	93.75	↑↑D♯, ↓↓E♭	↑5D, ↓7E	55/52
6	112.5	D𝄪, ↓E♭	↑6D, ↓6E	15/14, 16/15
7	131.25	↑D𝄪, E♭	↑7D, ↓5E	14/13
8	150	↑↑D𝄪, ↓↓E	↑8D, ↓4E	12/11
9	168.75	D♯𝄪, ↓E	↑9D, ↓3E
10	187.5	E	D♯, ↓↓E
11	206.25	↑E, F♭♭	↑D♯, ↓E	44/39
12	225	↑↑E, ↓↓F♭	E	8/7
13	243.75	E♯, ↓F♭	↑E, ↓F	15/13
14	262.5	↑E♯, F♭	F	64/55, 65/56
15	281.25	↑↑E♯, ↓↓F	↑F, ↓G♭	75/64
16	300	E𝄪, ↓F	↑↑F, G♭
17	318.75	F	↑3F, ↓9G	77/64
18	337.5	↑F, G♭♭♭	↑4F, ↓8G	39/32
19	356.25	↑↑F, ↓↓G♭♭	↑5F, ↓7G	16/13, 49/40
20	375	F♯, ↓G♭♭	↑6F, ↓6G	26/21
21	393.75	↑F♯, G♭♭	↑7F, ↓5G	5/4
22	412.5	↑↑F♯, ↓↓G♭	↑8F, ↓4G
23	431.25	F𝄪, ↓G♭	↑9F, ↓3G	77/60
24	450	↑F𝄪, G♭	F♯, ↓↓G	13/10
25	468.75	↑↑F𝄪, ↓↓G	↑F♯, ↓G	21/16, 55/42
26	487.5	F♯𝄪, ↓G	G	65/49
27	506.25	G	↑G, ↓A♭	75/56
28	525	↑G, A♭♭♭	↑↑G, A♭
29	543.75	↑↑G, ↓↓A♭♭	↑3G, ↓9A	15/11
30	562.5	G♯, ↓A♭♭	↑4G, ↓8A
31	581.25	↑G♯, A♭♭	↑5G, ↓7A	7/5
32	600	↑↑G♯, ↓↓A♭	↑6G, ↓6A	55/39, 78/55
33	618.75	G𝄪, ↓A♭	↑7G, ↓5A	10/7, 63/44
34	637.5	↑G𝄪, A♭	↑8G, ↓4A	75/52
35	656.25	↑↑G𝄪, ↓↓A	↑9G, ↓3A	22/15
36	675	G♯𝄪, ↓A	G♯, ↓↓A	77/52
37	693.75	A	↑G♯, ↓A
38	712.5	↑A, B♭♭♭	A
39	731.25	↑↑A, ↓↓B♭♭	↑A, ↓B♭	32/21, 75/49
40	750	A♯, ↓B♭♭	↑↑A, B♭	20/13
41	768.75	↑A♯, B♭♭	↑3A, ↓9B
42	787.5	↑↑A♯, ↓↓B♭	↑4A, ↓8B
43	806.25	A𝄪, ↓B♭	↑5A, ↓7B	8/5
44	825	↑A𝄪, B♭	↑6A, ↓6B	21/13, 77/48
45	843.75	↑↑A𝄪, ↓↓B	↑7A, ↓5B	13/8, 80/49
46	862.5	A♯𝄪, ↓B	↑8A, ↓4B	64/39
47	881.25	B	↑9A, ↓3B
48	900	↑B, C♭♭	A♯, ↓↓B
49	918.75	↑↑B, ↓↓C♭	↑A♯, ↓B
50	937.5	B♯, ↓C♭	B	55/32
51	956.25	↑B♯, C♭	↑B, ↓C	26/15
52	975	↑↑B♯, ↓↓C	C	7/4
53	993.75	B𝄪, ↓C	↑C, ↓D♭	39/22
54	1012.5	C	↑↑C, D♭
55	1031.25	↑C, D♭♭♭	↑3C, ↓9D
56	1050	↑↑C, ↓↓D♭♭	↑4C, ↓8D	11/6
57	1068.75	C♯, ↓D♭♭	↑5C, ↓7D	13/7
58	1087.5	↑C♯, D♭♭	↑6C, ↓6D	15/8, 28/15
59	1106.25	↑↑C♯, ↓↓D♭	↑7C, ↓5D
60	1125	C𝄪, ↓D♭	↑8C, ↓4D	21/11
61	1143.75	↑C𝄪, D♭	↑9C, ↓3D
62	1162.5	↑↑C𝄪, ↓↓D	C♯, ↓↓D	49/25, 55/28
63	1181.25	C♯𝄪, ↓D	↑C♯, ↓D	77/39
64	1200	D	D	2/1