61edo: Difference between revisions

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.

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!

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
Error	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)

61edo is the 18th prime edo, after 59edo and before 67edo.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	19.7		^D, vvE♭
2	39.3		^^D, vE♭
3	59	29/28, 32/31	^³D, E♭
4	78.7	22/21, 23/22	^⁴D, ^E♭
5	98.4	35/33	v³D♯, ^^E♭
6	118	31/29	vvD♯, ^³E♭
7	137.7	13/12	vD♯, v⁴E
8	157.4	23/21, 34/31, 35/32	D♯, v³E
9	177	21/19, 31/28	^D♯, vvE
10	196.7	19/17	^^D♯, vE
11	216.4	26/23	E
12	236.1		^E, vvF
13	255.7	22/19	^^E, vF
14	275.4	34/29	F
15	295.1	19/16	^F, vvG♭
16	314.8	6/5	^^F, vG♭
17	334.4	17/14, 23/19	^³F, G♭
18	354.1		^⁴F, ^G♭
19	373.8	26/21	v³F♯, ^^G♭
20	393.4		vvF♯, ^³G♭
21	413.1	14/11, 33/26	vF♯, v⁴G
22	432.8		F♯, v³G
23	452.5	13/10	^F♯, vvG
24	472.1	21/16	^^F♯, vG
25	491.8		G
26	511.5	35/26	^G, vvA♭
27	531.1	19/14	^^G, vA♭
28	550.8	11/8	^³G, A♭
29	570.5	25/18, 32/23	^⁴G, ^A♭
30	590.2	31/22	v³G♯, ^^A♭
31	609.8		vvG♯, ^³A♭
32	629.5	23/16	vG♯, v⁴A
33	649.2	16/11, 35/24	G♯, v³A
34	668.9	28/19	^G♯, vvA
35	688.5		^^G♯, vA
36	708.2		A
37	727.9	29/19, 32/21, 35/23	^A, vvB♭
38	747.5	20/13	^^A, vB♭
39	767.2		^³A, B♭
40	786.9	11/7	^⁴A, ^B♭
41	806.6	35/22	v³A♯, ^^B♭
42	826.2	21/13	vvA♯, ^³B♭
43	845.9	31/19	vA♯, v⁴B
44	865.6	28/17, 33/20	A♯, v³B
45	885.2	5/3	^A♯, vvB
46	904.9	32/19	^^A♯, vB
47	924.6	29/17	B
48	944.3	19/11	^B, vvC
49	963.9		^^B, vC
50	983.6	23/13	C
51	1003.3	34/19	^C, vvD♭
52	1023		^^C, vD♭
53	1042.6	31/17	^³C, D♭
54	1062.3	24/13	^⁴C, ^D♭
55	1082		v³C♯, ^^D♭
56	1101.6		vvC♯, ^³D♭
57	1121.3	21/11	vC♯, v⁴D
58	1141	31/16	C♯, v³D
59	1160.7		^C♯, vvD
60	1180.3		^^C♯, vD
61	1200	2/1	D

@@ Line 3: / Line 3: @@
 == Theory ==
-edo provides the [[optimal patent val]] for the [[freivald]] (24 &amp; 37) temperament in the 7-, 11- and 13-limit.
+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]] {{val| 61 97 142 '''171''' }} [[support]]s [[valentine]] (15 & 46), and is the [[optimal patent val]] for [[freivald]] (24 &amp; 37) in the 7-, 11- and 13-limit. The 61d [[val]] {{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 ===
@@ Line 25: / Line 25: @@
 and the 21st and 23rd are but a cent or two sharp!
-—[[Peter Kosmorsky]]
 === Odd harmonics ===
@@ Line 32: / Line 30: @@
 === Subsets and supersets ===
 edo is the 18th [[prime edo]], after [[59edo]] and before [[67edo]].
 == Intervals ==
 {{Interval table}}