66edo

← 65edo

66edo

67edo →

Prime factorization	2 × 3 × 11
Step size	18.1818¢
Fifth	39\66 (709.091¢) (→13\22)
Semitones (A1:m2)	9:3 (163.6¢ : 54.55¢)
Dual sharp fifth	39\66 (709.091¢) (→13\22)
Dual flat fifth	38\66 (690.909¢) (→19\33)
Dual major 2nd	11\66 (200¢) (→1\6)
Consistency limit	3
Distinct consistency limit	3

66 equal divisions of the octave (abbreviated 66edo or 66ed2), also called 66-tone equal temperament (66tet) or 66 equal temperament (66et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 66 equal parts of about 18.2 ¢ each. Each step represents a frequency ratio of 2^1/66, or the 66th root of 2.

Theory

Approximation of odd harmonics in 66edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	+7.14	-4.50	-5.19	-3.91	-5.86	-4.16	+2.64	+4.14	-6.60	+1.95	+8.09
	relative (%)	+39	-25	-29	-22	-32	-23	+15	+23	-36	+11	+44
Steps (reduced)		105 (39)	153 (21)	185 (53)	209 (11)	228 (30)	244 (46)	258 (60)	270 (6)	280 (16)	290 (26)	299 (35)

The patent val is contorted in the 5-limit, tempering out the same commas 250/243, 2048/2025 and 3125/3072 as 22edo. In the 7-limit it tempers out 686/675 and 1029/1024, in the 11-limit 55/54, 100/99 and 121/120, in the 13-limit 91/90, 169/168, 196/195 and in the 17-limit 136/135 and 256/255. It provides the optimal patent val for 11- and 13-limit ammonite temperament.

The 66b val tempers out 16875/16384 in the 5-limit, 126/125, 1728/1715 and 2401/2400 in the 7-limit, 99/98 and 385/384 in the 11-limit, and 105/104, 144/143 and 847/845 in the 13-limit.

109 steps of 66edo is extremely close to Pi with only +0.023 cents of error.

Interval table

Steps	Cents	Ups and downs notation (dual flat fifth 38\66)	Ups and downs notation (dual sharp fifth 39\66)	Approximate ratios
0	0	D	D	1/1
1	18.1818	↑D, ↓E♭♭♭♭	↑D, ↓↓E♭
2	36.3636	D♯, E♭♭♭♭	↑↑D, ↓E♭	50/49, 56/55
3	54.5455	↑D♯, ↓E♭♭♭	↑3D, E♭	33/32
4	72.7273	D𝄪, E♭♭♭	↑4D, ↓8E	26/25, 80/77
5	90.9091	↑D𝄪, ↓E♭♭	↑5D, ↓7E	21/20, 55/52
6	109.091	D♯𝄪, E♭♭	↑6D, ↓6E	16/15, 52/49
7	127.273	↑D♯𝄪, ↓E♭	↑7D, ↓5E	14/13
8	145.455	D𝄪𝄪, E♭	↑8D, ↓4E
9	163.636	↑D𝄪𝄪, ↓E	D♯, ↓3E	11/10
10	181.818	E	↑D♯, ↓↓E	49/44
11	200	↑E, ↓F♭♭♭	↑↑D♯, ↓E	28/25, 55/49
12	218.182	E♯, F♭♭♭	E	25/22
13	236.364	↑E♯, ↓F♭♭	↑E, ↓↓F	8/7
14	254.545	E𝄪, F♭♭	↑↑E, ↓F	15/13, 65/56
15	272.727	↑E𝄪, ↓F♭	F	75/64
16	290.909	E♯𝄪, F♭	↑F, ↓↓G♭	13/11, 33/28, 77/65
17	309.091	↑E♯𝄪, ↓F	↑↑F, ↓G♭
18	327.273	F	↑3F, G♭	40/33
19	345.455	↑F, ↓G♭♭♭♭	↑4F, ↓8G	39/32, 49/40
20	363.636	F♯, G♭♭♭♭	↑5F, ↓7G	16/13, 26/21
21	381.818	↑F♯, ↓G♭♭♭	↑6F, ↓6G	5/4
22	400	F𝄪, G♭♭♭	↑7F, ↓5G	44/35
23	418.182	↑F𝄪, ↓G♭♭	↑8F, ↓4G	14/11, 33/26
24	436.364	F♯𝄪, G♭♭	F♯, ↓3G
25	454.545	↑F♯𝄪, ↓G♭	↑F♯, ↓↓G	13/10
26	472.727	F𝄪𝄪, G♭	↑↑F♯, ↓G	21/16
27	490.909	↑F𝄪𝄪, ↓G	G	4/3, 65/49
28	509.091	G	↑G, ↓↓A♭	35/26, 75/56
29	527.273	↑G, ↓A♭♭♭♭	↑↑G, ↓A♭
30	545.455	G♯, A♭♭♭♭	↑3G, A♭	11/8
31	563.636	↑G♯, ↓A♭♭♭	↑4G, ↓8A
32	581.818	G𝄪, A♭♭♭	↑5G, ↓7A	7/5
33	600	↑G𝄪, ↓A♭♭	↑6G, ↓6A
34	618.182	G♯𝄪, A♭♭	↑7G, ↓5A	10/7
35	636.364	↑G♯𝄪, ↓A♭	↑8G, ↓4A	75/52
36	654.545	G𝄪𝄪, A♭	G♯, ↓3A	16/11
37	672.727	↑G𝄪𝄪, ↓A	↑G♯, ↓↓A	65/44, 77/52
38	690.909	A	↑↑G♯, ↓A	52/35
39	709.091	↑A, ↓B♭♭♭♭	A	3/2
40	727.273	A♯, B♭♭♭♭	↑A, ↓↓B♭	32/21
41	745.455	↑A♯, ↓B♭♭♭	↑↑A, ↓B♭	20/13, 77/50
42	763.636	A𝄪, B♭♭♭	↑3A, B♭
43	781.818	↑A𝄪, ↓B♭♭	↑4A, ↓8B	11/7, 52/33
44	800	A♯𝄪, B♭♭	↑5A, ↓7B	35/22
45	818.182	↑A♯𝄪, ↓B♭	↑6A, ↓6B	8/5
46	836.364	A𝄪𝄪, B♭	↑7A, ↓5B	13/8, 21/13
47	854.545	↑A𝄪𝄪, ↓B	↑8A, ↓4B	64/39, 80/49
48	872.727	B	A♯, ↓3B	33/20
49	890.909	↑B, ↓C♭♭♭	↑A♯, ↓↓B
50	909.091	B♯, C♭♭♭	↑↑A♯, ↓B	22/13, 56/33
51	927.273	↑B♯, ↓C♭♭	B	75/44
52	945.455	B𝄪, C♭♭	↑B, ↓↓C	26/15
53	963.636	↑B𝄪, ↓C♭	↑↑B, ↓C	7/4
54	981.818	B♯𝄪, C♭	C	44/25
55	1000	↑B♯𝄪, ↓C	↑C, ↓↓D♭	25/14
56	1018.18	C	↑↑C, ↓D♭
57	1036.36	↑C, ↓D♭♭♭♭	↑3C, D♭	20/11
58	1054.55	C♯, D♭♭♭♭	↑4C, ↓8D
59	1072.73	↑C♯, ↓D♭♭♭	↑5C, ↓7D	13/7
60	1090.91	C𝄪, D♭♭♭	↑6C, ↓6D	15/8, 49/26
61	1109.09	↑C𝄪, ↓D♭♭	↑7C, ↓5D	40/21
62	1127.27	C♯𝄪, D♭♭	↑8C, ↓4D	25/13, 77/40
63	1145.45	↑C♯𝄪, ↓D♭	C♯, ↓3D	64/33
64	1163.64	C𝄪𝄪, D♭	↑C♯, ↓↓D	49/25, 55/28
65	1181.82	↑C𝄪𝄪, ↓D	↑↑C♯, ↓D
66	1200	D	D	2/1