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

Theory

The patent val of 66edo is contorted in the 5-limit, tempering out the same commas (250/243, 2048/2025, 3125/3072, etc.) 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 the 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 the acoustic pi with only +0.023 cents of error.

Odd harmonics

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 (%)	+39.2	-24.7	-28.5	-21.5	-32.2	-22.9	+14.5	+22.7	-36.3	+10.7	+44.5
Steps (reduced)		105 (39)	153 (21)	185 (53)	209 (11)	228 (30)	244 (46)	258 (60)	270 (6)	280 (16)	290 (26)	299 (35)

Subsets and supersets

Since 66 factors into 2 × 3 × 11, 66edo has subset edos 2, 3, 6, 11, 22, and 33. 198edo, which triples it, corrects its approximation to many of the lower harmonics.

Interval table

Steps	Cents	Approximate ratios	Ups and downs notation (Dual flat fifth 38\66)	Ups and downs notation (Dual sharp fifth 39\66)
0	0	1/1	D	D
1	18.2		^D, vE♭♭♭♭	^D, vvE♭
2	36.4		D♯, E♭♭♭♭	^^D, vE♭
3	54.5	31/30, 32/31, 33/32, 34/33	^D♯, vE♭♭♭	^3D, E♭
4	72.7	24/23	D𝄪, E♭♭♭	^4D, ^E♭
5	90.9	20/19	^D𝄪, vE♭♭	v4D♯, ^^E♭
6	109.1	16/15, 33/31	D♯𝄪, E♭♭	v3D♯, ^3E♭
7	127.3	14/13	^D♯𝄪, vE♭	vvD♯, ^4E♭
8	145.5		D𝄪𝄪, E♭	vD♯, v4E
9	163.6	11/10, 34/31	^D𝄪𝄪, vE	D♯, v3E
10	181.8		E	^D♯, vvE
11	200	28/25	^E, vF♭♭♭	^^D♯, vE
12	218.2	17/15, 25/22	E♯, F♭♭♭	E
13	236.4		^E♯, vF♭♭	^E, vvF
14	254.5	22/19	E𝄪, F♭♭	^^E, vF
15	272.7	34/29	^E𝄪, vF♭	F
16	290.9	13/11	E♯𝄪, F♭	^F, vvG♭
17	309.1		^E♯𝄪, vF	^^F, vG♭
18	327.3	29/24	F	^3F, G♭
19	345.5		^F, vG♭♭♭♭	^4F, ^G♭
20	363.6	21/17	F♯, G♭♭♭♭	v4F♯, ^^G♭
21	381.8		^F♯, vG♭♭♭	v3F♯, ^3G♭
22	400	29/23	F𝄪, G♭♭♭	vvF♯, ^4G♭
23	418.2	14/11	^F𝄪, vG♭♭	vF♯, v4G
24	436.4		F♯𝄪, G♭♭	F♯, v3G
25	454.5	13/10	^F♯𝄪, vG♭	^F♯, vvG
26	472.7	21/16, 25/19	F𝄪𝄪, G♭	^^F♯, vG
27	490.9		^F𝄪𝄪, vG	G
28	509.1		G	^G, vvA♭
29	527.3	19/14, 23/17	^G, vA♭♭♭♭	^^G, vA♭
30	545.5	26/19	G♯, A♭♭♭♭	^3G, A♭
31	563.6		^G♯, vA♭♭♭	^4G, ^A♭
32	581.8	7/5	G𝄪, A♭♭♭	v4G♯, ^^A♭
33	600	17/12, 24/17	^G𝄪, vA♭♭	v3G♯, ^3A♭
34	618.2	10/7	G♯𝄪, A♭♭	vvG♯, ^4A♭
35	636.4		^G♯𝄪, vA♭	vG♯, v4A
36	654.5	19/13	G𝄪𝄪, A♭	G♯, v3A
37	672.7	28/19, 31/21, 34/23	^G𝄪𝄪, vA	^G♯, vvA
38	690.9		A	^^G♯, vA
39	709.1		^A, vB♭♭♭♭	A
40	727.3	32/21	A♯, B♭♭♭♭	^A, vvB♭
41	745.5	20/13	^A♯, vB♭♭♭	^^A, vB♭
42	763.6		A𝄪, B♭♭♭	^3A, B♭
43	781.8	11/7	^A𝄪, vB♭♭	^4A, ^B♭
44	800	35/22	A♯𝄪, B♭♭	v4A♯, ^^B♭
45	818.2		^A♯𝄪, vB♭	v3A♯, ^3B♭
46	836.4	34/21	A𝄪𝄪, B♭	vvA♯, ^4B♭
47	854.5		^A𝄪𝄪, vB	vA♯, v4B
48	872.7		B	A♯, v3B
49	890.9		^B, vC♭♭♭	^A♯, vvB
50	909.1	22/13	B♯, C♭♭♭	^^A♯, vB
51	927.3	29/17	^B♯, vC♭♭	B
52	945.5	19/11	B𝄪, C♭♭	^B, vvC
53	963.6		^B𝄪, vC♭	^^B, vC
54	981.8	30/17	B♯𝄪, C♭	C
55	1000	25/14	^B♯𝄪, vC	^C, vvD♭
56	1018.2		C	^^C, vD♭
57	1036.4	20/11, 31/17	^C, vD♭♭♭♭	^3C, D♭
58	1054.5	35/19	C♯, D♭♭♭♭	^4C, ^D♭
59	1072.7	13/7	^C♯, vD♭♭♭	v4C♯, ^^D♭
60	1090.9	15/8	C𝄪, D♭♭♭	v3C♯, ^3D♭
61	1109.1	19/10	^C𝄪, vD♭♭	vvC♯, ^4D♭
62	1127.3	23/12	C♯𝄪, D♭♭	vC♯, v4D
63	1145.5	31/16, 33/17	^C♯𝄪, vD♭	C♯, v3D
64	1163.6		C𝄪𝄪, D♭	^C♯, vvD
65	1181.8		^C𝄪𝄪, vD	^^C♯, vD
66	1200	2/1	D	D