62edo: Difference between revisions

62 = 2 × 31 and the patent val of 62edo is a contorted 31edo through the 11-limit, but it makes for a good tuning in the higher limits. In the 13-limit it tempers out 169/168, 1188/1183, 847/845 and 676/675; in the 17-limit 221/220, 273/272, and 289/288; in the 19-limit 153/152, 171/170, 209/208, 286/285, and 361/360. Unlike 31edo, which has a sharp profile for primes 13, 17, 19 and 23, 62edo has a flat profile for these, as it removes the distinction of otonal and utonal superparticular pairs of the primes (e.g. 13/12 vs 14/13 for prime 13) by tempering out the corresponding square-particulars. This flat tendency extends to higher primes too, as the first prime harmonic that is tuned sharper than its 5/4 is its 59/32. Interestingly, the size differences between consecutive harmonics are monotonically decreasing for all first 24 harmonics, and 62edo is one of the few meantone edos that achieve this, great for those who seek higher-limit meantone harmony.

It provides the optimal patent val for gallium, semivalentine and hemimeantone temperaments.

Using the 35\62 generator, which leads to the ⟨62 97 143 173] val, 62edo is also an excellent tuning for septimal mavila temperament; alternatively ⟨62 97 143 172] supports hornbostel.

Odd harmonics

Approximation of odd harmonics in 62edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-5.18	+0.78	-1.08	+8.99	-9.38	-8.27	-4.40	-8.18	-7.19	-6.26	-8.92
Error	Relative (%)	-26.8	+4.0	-5.6	+46.5	-48.5	-42.7	-22.7	-42.3	-37.2	-32.4	-46.1
Steps (reduced)		98 (36)	144 (20)	174 (50)	197 (11)	214 (28)	229 (43)	242 (56)	253 (5)	263 (15)	272 (24)	280 (32)

Subsets and supersets

Since 62 factors into 2 × 31, 62edo does not contain nontrivial subset edos other than 2edo and 31edo. 186edo and 248edo are notable supersets.

Miscellaneous properties

62 years is the amount of years in a leap week calendar cycle which corresponds to a year of 365 days 5 hours 48 minutes 23 seconds, meaning it is both a simple cycle for a calendar, and 62 being a multiple of 31 makes it a harmonically useful and playable cycle. The corresponding maximal evenness scales are 15 & 62 and 11 & 62.

The 11 & 62 temperament is called mabon, named so because its associated year length corresponds to an autumnal equinoctial year. In the 2.9.7 subgroup tempers out 44957696/43046721, and the three generators of 17\62 correspond to 16/9. It is possible to extend this to the 11-limit with comma basis {896/891, 1331/1296}, where 17\62 is mapped to 11/9 and two of them make 16/11. In addition, three generators make the patent val 9/8, which is also created by combining the flat patent val fifth from 31edo with the sharp 37\62 fifth.

The 15 & 62 temperament, corresponding to the leap day cycle, is demivalentine in the 13-limit.

Intervals

Steps	Cents	Approximate ratios*	Ups and downs notation
0	0.00	1/1	D
1	19.35	65/64, 66/65, 78/77, 91/90, 105/104	^D, vE♭♭
2	38.71	33/32, 36/35, 45/44, 49/48, 50/49, 55/54, 56/55, 64/63	^^D, E♭♭
3	58.06	26/25, 27/26	vD♯, ^E♭♭
4	77.42	21/20, 22/21, 23/22, 24/23, 25/24, 28/27	D♯, vvE♭
5	96.77	17/16, 18/17, 19/18, 20/19	^D♯, vE♭
6	116.13	15/14, 16/15	^^D♯, E♭
7	135.48	13/12, 14/13	vD𝄪, ^E♭
8	154.84	11/10, 12/11, 23/21	D𝄪, vvE
9	174.19	21/19	^D𝄪, vE
10	193.55	9/8, 10/9, 19/17, 28/25	E
11	212.90	17/15	^E, vF♭
12	232.26	8/7	^^E, F♭
13	251.61	15/13, 22/19	vE♯, ^F♭
14	270.97	7/6	E♯, vvF
15	290.32	13/11, 19/16, 20/17	^E♯, vF
16	309.68	6/5	F
17	329.03	17/14, 23/19	^^F, G♭♭
18	348.39	11/9, 27/22, 28/23	^^F, G♭♭
19	367.74	16/13, 21/17, 26/21	vF♯, ^G♭♭
20	387.10	5/4	F♯, vvG♭
21	406.45	19/15, 24/19	^F♯, vG♭
22	425.81	9/7, 14/11, 23/18, 32/25	^^F♯, G♭
23	445.16	13/10, 22/17	vF𝄪, ^G♭
24	464.52	17/13, 21/16, 30/23	F𝄪, vvG
25	483.87	25/19	^F𝄪, vG
26	503.23	4/3	G
27	522.58	19/14, 23/17	^G, vA♭♭
28	541.94	11/8, 15/11, 26/19	^^G, A♭♭
29	561.29	18/13	vG♯, ^A♭♭
30	580.65	7/5, 25/18, 32/23	G♯, vvA♭
31	600.00	17/12, 24/17	E
32	619.35	10/7, 23/16, 36/25	^^G♯, A♭
33	638.71	13/9	vG𝄪, ^A♭
34	658.06	16/11, 19/13, 22/15	G𝄪, vvA
35	677.42	28/19, 34/23	^G𝄪, vA
36	696.77	3/2	A
37	716.13	38/25	^A, vB♭♭
38	735.48	23/15, 26/17, 32/21	^^A, B♭♭
39	754.84	17/11, 20/13	vA♯, ^B♭♭
40	774.19	11/7, 14/9, 25/16, 36/23	A♯, vvB♭
41	793.55	19/12, 30/19	^A♯, vB♭
42	812.90	8/5	^^A♯, B♭
43	832.26	13/8, 21/13, 34/21	vA𝄪, ^B♭
44	851.61	18/11, 23/14, 44/27	A𝄪, vvB
45	870.97	28/17, 38/23	^A𝄪, vB
46	890.32	5/3	B
47	909.68	17/10, 22/13, 32/19	^B, vC♭
48	929.03	12/7	^^B, C♭
49	948.39	19/11, 26/15	vB♯, ^C♭
50	967.74	7/4	B♯, vvC
51	987.10	30/17	^B♯, vC
52	1006.45	9/5, 16/9, 25/14, 34/19	C
53	1025.81	38/21	^C, vD♭♭
54	1045.16	11/6, 20/11, 42/23	^^C, D♭♭
55	1064.52	13/7, 24/13	vC♯, ^D♭♭
56	1083.87	15/8, 28/15	C♯, vvD♭
57	1103.23	17/9, 19/10, 32/17, 36/19	^C♯, vD♭
58	1122.58	21/11, 23/12, 27/14, 40/21, 44/23, 48/25	^^C♯, D♭
59	1141.94	25/13, 52/27	vC𝄪, ^D♭
60	1161.29	35/18, 49/25, 55/28, 63/32, 64/33, 88/45, 96/49, 108/55	C𝄪, vvD
61	1180.65	65/33, 77/39, 128/65, 180/91, 208/105	^C𝄪, vD
62	1200.00	2/1	D

* 23-limit patent val, inconsistent intervals in italic

Notation

Ups and downs notation

62edo can be notated with quarter-tone accidentals and ups and downs. This can be done by combining sharps and flats with arrows borrowed from extended Helmholtz–Ellis notation:

Step offset	0	1	2	3	4	5	6	7	8	9
Sharp symbol
Flat symbol

Sagittal notation

This notation uses the same sagittal sequence as EDOs 69 and 76, and is a superset of the notation for 31-EDO.

Evo flavor

Revo flavor

Evo-SZ flavor

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

Armodue notation

Armodue nomenclature 8;3 relation

Ɨ = Thick (1/8-tone up)
‡ = Semisharp (1/4-tone up)
b = Flat (5/8-tone down)
◊ = Node (sharp/flat blindspot 1/2-tone)
# = Sharp (5/8-tone up)
v = Semiflat (1/4-tone down)
⌐ = Thin (1/8-tone down)

#		Cents	Armodue notation	Associated ratio
0		0.0	1
1		19.4	1Ɨ
2		38.7	1‡ (9#)
3		58.1	2b
4		77.4	1◊2
5		96.8	1#
6		116.1	2v
7		135.5	2⌐
8		154.8	2	11/10~12/11
9		174.2	2Ɨ
10		193.5	2‡
11		212.9	3b	8/7
12		232.3	2◊3
13		251.6	2#
14		271.0	3v
15		290.3	3⌐
16		309.7	3	6/5~7/6
17		329.0	3Ɨ
18		348.4	3‡
19	·	367.7	4b	5/4
20		387.1	3◊4
21		406.5	3#
22		425.8	4v (5b)
23		445.2	4⌐
24		464.5	4
25		483.9	4Ɨ (5v)
26		503.2	5⌐ (4‡)
27	·	522.6	5	4/3~11/8
28		541.9	5Ɨ
29		561.3	5‡ (4#)
30		580.6	6b	10/7
31		600.0	5◊6
32		619.4	5#	7/5
33		638.7	6v
34		658.1	6⌐
35	·	677.4	6	3/2~16/11
36		696.8	6Ɨ
37		716.1	6‡
38		735.5	7b
39		754.8	6◊7
40		774.2	6#
41		793.5	7v
42		812.9	7⌐
43	·	832.3	7	8/5
44		851.6	7Ɨ
45		871.0	7‡
46		890.3	8b	5/3~12/7
47		909.7	7◊8
48		929.0	7#
49		948.4	8v
50		967.7	8⌐
51		987.1	8	7/4
52		1006.5	8Ɨ
53		1025.8	8‡
54		1045.2	9b	11/6~20/11
55		1064.5	8◊9
56		1083.9	8#
57		1103.2	9v (1b)
58		1122.6	9⌐
59		1141.9	9
60		1161.3	9Ɨ (1v)
61		1180.6	1⌐ (9‡)
62		1200.0	1

Approximation to JI

Zeta peak index

Tuning			Strength			Closest edo		Integer limit
ZPI	Steps per octave	Step size (cents)	Height	Integral	Gap	Edo	Octave (cents)	Consistent	Distinct
314zpi	61.9380472360525	19.3741981471691	6.262952	0.952068	15.026453	62edo	1201.20028512448	8	8

Regular temperament properties

62edo is contorted 31edo through the 11-limit.

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3.5.7.11.13	81/80, 99/98, 121/120, 126/125, 169/168	[⟨62 98 144 174 214 229]]	+1.38	1.41	7.28
2.3.5.7.11.13.17	81/80, 99/98, 121/120, 126/125, 169/168, 221/220	[⟨62 98 144 174 214 229 253]]	+1.47	1.32	6.83
2.3.5.7.11.13.17.19	81/80, 99/98, 121/120, 126/125, 153/152, 169/168, 209/208	[⟨62 98 144 174 214 229 253 263]]	+1.50	1.24	6.40
2.3.5.7.11.13.17.19.23	81/80, 99/98, 121/120, 126/125, 153/152, 161/160, 169/168, 209/208	[⟨62 98 144 174 214 229 253 263 280]]	+1.55	1.18	6.09

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	3\62	58.1	27/26	Hemisecordite
1	7\62	135.5	13/12	Doublethink
1	13\62	251.6	15/13	Hemimeantone
1	17\62	329.0	16/11	Mabon
1	29\62	561.3	18/13	Demivalentine
2	3\62	58.1	27/26	Semihemisecordite
2	4\62	77.4	21/20	Semivalentine
2	6\62	116.1	15/14	Semimiracle
2	26\62	503.2	4/3	Semimeantone
31	29\62 (1\62)	561.3 (19.4)	11/8 (196/195)	Kumhar (62e)
31	19\62 (1\62)	367.7 (19.4)	16/13 (77/76)	Gallium

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

@@ Line 3: / Line 3: @@
 == Theory ==
-{{Nowrap| 62 {{=}} 2 × 31 }} and the [[patent val]] of 62edo is a [[contorsion|contorted]] [[31edo]] through the [[11-limit]], but it makes for a good tuning in the higher limits. In the 13-limit it [[tempering out|tempers out]] [[169/168]], [[1188/1183]], [[847/845]] and [[676/675]]; in the [[17-limit]] [[221/220]], [[273/272]], and [[289/288]]; in the [[19-limit]] [[153/152]], [[171/170]], [[209/208]], [[286/285]], and [[361/360]]. Unlike 31edo, which has a sharp profile for primes [[13/1|13]], [[17/1|17]], [[19/1|19]] and [[23/1|23]], 62edo has a flat profile for these, as it removes the distinction of otonal and utonal [[superparticular]] pairs of the primes (e.g. 13/12 vs 14/13 for prime 13) by tempering out the corresponding [[square-particular]]s. This flat tendency extends to higher primes too, as the first prime harmonic that is tuned sharper than its [[5/4]] is its [[59/32]]. Interestingly, the relative size differences between consecutive harmonics are well preserved for all first 24 harmonics, and 62edo is one of the few [[meantone]] edos that achieve this, great for those who seek higher-limit meantone harmony.
+{{Nowrap| 62 {{=}} 2 × 31 }} and the [[patent val]] of 62edo is a [[contorsion|contorted]] [[31edo]] through the [[11-limit]], but it makes for a good tuning in the higher limits. In the 13-limit it [[tempering out|tempers out]] [[169/168]], [[1188/1183]], [[847/845]] and [[676/675]]; in the [[17-limit]] [[221/220]], [[273/272]], and [[289/288]]; in the [[19-limit]] [[153/152]], [[171/170]], [[209/208]], [[286/285]], and [[361/360]]. Unlike 31edo, which has a sharp profile for primes [[13/1|13]], [[17/1|17]], [[19/1|19]] and [[23/1|23]], 62edo has a flat profile for these, as it removes the distinction of otonal and utonal [[superparticular]] pairs of the primes (e.g. 13/12 vs 14/13 for prime 13) by tempering out the corresponding [[square-particular]]s. This flat tendency extends to higher primes too, as the first prime harmonic that is tuned sharper than its [[5/4]] is its [[59/32]]. Interestingly, the size differences between consecutive harmonics are monotonically decreasing for all first 24 harmonics, and 62edo is one of the few [[meantone]] edos that achieve this, great for those who seek higher-limit meantone harmony.
-It provides the [[optimal patent val]] for [[31st-octave temperaments #Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[Meantone family #Hemimeantone|hemimeantone]] temperaments.
+It provides the [[optimal patent val]] for [[gallium]], [[semivalentine]] and [[hemimeantone]] temperaments.
 Using the 35\62 generator, which leads to the {{val| 62 97 143 173 }} val, 62edo is also an excellent tuning for septimal [[mavila]] temperament; alternatively {{val| 62 97 143 172 }} [[support]]s [[hornbostel]].
@@ Line 13: / Line 13: @@
 === Subsets and supersets ===
-Since 62 factors into {{factorization|62}}, 62edo does not contain nontrivial subset edos other than [[2edo]] and 31edo. [[186edo]] and [[248edo]] are notable supersets.
+Since 62 factors into 2 × 31, 62edo does not contain nontrivial subset edos other than [[2edo]] and 31edo. [[186edo]] and [[248edo]] are notable supersets.
 === Miscellaneous properties ===
@@ Line 20: / Line 20: @@
 The 11 & 62 temperament is called mabon, named so because its associated year length corresponds to an autumnal equinoctial year. In the 2.9.7 subgroup tempers out 44957696/43046721, and the three generators of 17\62 correspond to [[16/9]]. It is possible to extend this to the 11-limit with comma basis {896/891, 1331/1296}, where 17\62 is mapped to [[11/9]] and two of them make [[16/11]]. In addition, three generators make the patent val 9/8, which is also created by combining the flat patent val fifth from 31edo with the sharp 37\62 fifth.
-The 15 & 62 temperament, corresponding to the leap day cycle, is an unnamed extension to [[valentine]] in the 13-limit.
+The 15 & 62 temperament, corresponding to the leap day cycle, is [[demivalentine]] in the 13-limit.
 == Intervals ==

62edo: Difference between revisions

Revision as of 09:15, 26 January 2025

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