61edo

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61edo

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Prime factorization	61 (prime)
Step size	19.6721¢
Fifth	36\61 (708.197¢)
Semitones (A1:m2)	8:3 (157.4¢ : 59.02¢)
Consistency limit	5
Distinct consistency limit	5

61 equal divisions of the octave (abbreviated 61edo or 61ed2), also called 61-tone equal temperament (61tet) or 61 equal temperament (61et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 61 equal parts of about 19.7 ¢ each. Each step represents a frequency ratio of 2^1/61, or the 61st root of 2.

Theory

As an equal temperament, 61et is characterized by tempering out 20000/19683 (tetracot comma) and 262144/253125 (passion comma) in the 5-limit. In the 7-limit, the patent val ⟨61 97 142 171] supports valentine (15 & 46), and is the optimal patent val for freivald (24 & 37) in the 7-, 11- and 13-limit. The 61d val ⟨61 97 142 172] is a great tuning for modus and quasisuper, and is a simple but out-of-tune edo tuning for parakleismic. Peter Kosmorsky has an interesting poem about its tuning profile, as follows.

Introductory poem

These 61 equal divisions of the octave,

though rare are assuredly a ROCK-tave (har har),

while the 3rd and 5th harmonics are about six cents sharp,

(and the flattish 15th poised differently on the harp),

the 7th and 11th err by less, around three,

and thus mayhap, a good orgone tuning found to be;

slightly sharp as well, is the 13th harmonic's place,

but the 9th and 17th lack near so much grace,

interestingly the 19th is good but a couple cents flat,

and the 21st and 23rd are but a cent or two sharp!

Odd harmonics

Approximation of odd harmonics in 61edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+6.24	+7.13	-4.89	-7.19	-0.50	+5.37	-6.30	-6.59	-2.43	+1.35	+1.23
	Relative (%)	+31.7	+36.2	-24.9	-36.5	-2.5	+27.3	-32.0	-33.5	-12.4	+6.9	+6.3
Steps (reduced)		97 (36)	142 (20)	171 (49)	193 (10)	211 (28)	226 (43)	238 (55)	249 (5)	259 (15)	268 (24)	276 (32)

Subsets and supersets

61edo is the 18th prime edo, after 59edo and before 67edo. 183edo, which triples it, corrects its approximation to many of the lower harmonics.

Intervals

Steps	Cents	Approximate Ratios	Ups and Downs Notation
0	0	1/1	D
1	19.672		^D, vvE♭
2	39.344		^^D, vE♭
3	59.016	29/28, 32/31	^3D, E♭
4	78.689	22/21, 23/22	^4D, v7E
5	98.361	35/33	^5D, v6E
6	118.033	31/29	^6D, v5E
7	137.705	13/12	^7D, v4E
8	157.377	23/21, 34/31, 35/32	D♯, v3E
9	177.049	21/19, 31/28	^D♯, vvE
10	196.721	19/17	^^D♯, vE
11	216.393	26/23	E
12	236.066		^E, vvF
13	255.738	22/19	^^E, vF
14	275.41	34/29	F
15	295.082	19/16	^F, vvG♭
16	314.754	6/5	^^F, vG♭
17	334.426	17/14, 23/19	^3F, G♭
18	354.098		^4F, v7G
19	373.77	26/21	^5F, v6G
20	393.443		^6F, v5G
21	413.115	14/11, 33/26	^7F, v4G
22	432.787		F♯, v3G
23	452.459	13/10	^F♯, vvG
24	472.131	21/16	^^F♯, vG
25	491.803		G
26	511.475	35/26	^G, vvA♭
27	531.148	19/14	^^G, vA♭
28	550.82	11/8	^3G, A♭
29	570.492	25/18, 32/23	^4G, v7A
30	590.164	31/22	^5G, v6A
31	609.836		^6G, v5A
32	629.508	23/16	^7G, v4A
33	649.18	16/11, 35/24	G♯, v3A
34	668.852	28/19	^G♯, vvA
35	688.525		^^G♯, vA
36	708.197		A
37	727.869	29/19, 32/21, 35/23	^A, vvB♭
38	747.541	20/13	^^A, vB♭
39	767.213		^3A, B♭
40	786.885	11/7	^4A, v7B
41	806.557	35/22	^5A, v6B
42	826.23	21/13	^6A, v5B
43	845.902	31/19	^7A, v4B
44	865.574	28/17, 33/20	A♯, v3B
45	885.246	5/3	^A♯, vvB
46	904.918	32/19	^^A♯, vB
47	924.59	29/17	B
48	944.262	19/11	^B, vvC
49	963.934		^^B, vC
50	983.607	23/13	C
51	1003.279	34/19	^C, vvD♭
52	1022.951		^^C, vD♭
53	1042.623	31/17	^3C, D♭
54	1062.295	24/13	^4C, v7D
55	1081.967		^5C, v6D
56	1101.639		^6C, v5D
57	1121.311	21/11	^7C, v4D
58	1140.984	31/16	C♯, v3D
59	1160.656		^C♯, vvD
60	1180.328		^^C♯, vD
61	1200	2/1	D