62edo: Difference between revisions

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Cleanup on armodue notation
→Notation: since this section is about armodue, the ratios should also be derived from the armodue mapping
Line 50: Line 50:
| 19.4
| 19.4
| 1Ɨ
| 1Ɨ
| 90/89
|  
|-
|-
| 2
| 2
Line 56: Line 56:
| 38.7
| 38.7
| 1‡ (9#)
| 1‡ (9#)
| 45/44
|  
|-
|-
| 3
| 3
Line 62: Line 62:
| 58.1
| 58.1
| 2b
| 2b
| 30/29
|  
|-
|-
| 4
| 4
Line 68: Line 68:
| 77.4
| 77.4
| 1◊2
| 1◊2
| 23/22
|  
|-
|-
| 5
| 5
Line 74: Line 74:
| 96.8
| 96.8
| 1#
| 1#
| 37/35, 18/17, 19/18
|  
|-
|-
| 6
| 6
Line 80: Line 80:
| 116.1
| 116.1
| 2v
| 2v
| 31/29, 15/14, 16/15
|  
|-
|-
| 7
| 7
Line 86: Line 86:
| 135.5
| 135.5
| 2⌐
| 2⌐
| 27/25, 13/12, 14/13
|  
|-
|-
| 8
| 8
Line 92: Line 92:
| 154.8
| 154.8
| 2
| 2
| 12/11
| 11/10~12/11
|-
|-
| 9
| 9
Line 98: Line 98:
| 174.2
| 174.2
| 2Ɨ
| 2Ɨ
| 11/10
|
|-
|-
| 10
| 10
Line 104: Line 104:
| 193.5
| 193.5
| 2‡
| 2‡
| 19/17, 9/8, 10/9
|  
|-
|-
| 11
| 11
Line 110: Line 110:
| 212.9
| 212.9
| 3b
| 3b
| 17/15, 9/8
| 8/7
|-
|-
| 12
| 12
Line 116: Line 116:
| 232.3
| 232.3
| 2◊3
| 2◊3
| 8/7
|  
|-
|-
| 13
| 13
Line 122: Line 122:
| 251.6
| 251.6
| 2#
| 2#
| 15/13
|  
|-
|-
| 14
| 14
Line 128: Line 128:
| 271.0
| 271.0
| 3v
| 3v
| 7/6
|  
|-
|-
| 15
| 15
Line 140: Line 140:
| 309.7
| 309.7
| 3
| 3
| 6/5
| 6/5~7/6
|-
|-
| 17
| 17
Line 152: Line 152:
| 348.4
| 348.4
| 3‡
| 3‡
| 11/9
|  
|-
|-
| 19
| 19
Line 158: Line 158:
| 367.7
| 367.7
| 4b
| 4b
|  
| 5/4
|-
|-
| 20
| 20
Line 164: Line 164:
| 387.1
| 387.1
| 3◊4
| 3◊4
| 5/4
|  
|-
|-
| 21
| 21
Line 200: Line 200:
| 503.2
| 503.2
| 5⌐ (4‡)
| 5⌐ (4‡)
| 4/3
|  
|-
|-
| 27
| 27
Line 206: Line 206:
| 522.6
| 522.6
| 5
| 5
| 4/3~11/8
|-
|-
| 28
| 28
Line 223: Line 224:
| 580.6
| 580.6
| 6b
| 6b
| 7/5
| 10/7
|-
|-
| 31
| 31
Line 235: Line 236:
| 619.4
| 619.4
| 5#
| 5#
| 10/7
| 7/5
|-
|-
| 33
| 33
Line 253: Line 254:
| 677.4
| 677.4
| 6
| 6
|  
| 3/2~16/11
|-
|-
| 36
| 36
Line 259: Line 260:
| 696.8
| 696.8
| 6Ɨ
| 6Ɨ
| 3/2
|  
|-
|-
| 37
| 37
Line 295: Line 296:
| 812.9
| 812.9
| 7⌐
| 7⌐
| 8/5
|  
|-
|-
| 43
| 43
Line 301: Line 302:
| 832.3
| 832.3
| 7
| 7
|  
| 8/5
|-
|-
| 44
| 44
Line 307: Line 308:
| 851.6
| 851.6
| 7Ɨ
| 7Ɨ
| 18/11
|  
|-
|-
| 45
| 45
Line 319: Line 320:
| 890.3
| 890.3
| 8b
| 8b
| 5/3
| 5/3~12/7
|-
|-
| 47
| 47
Line 331: Line 332:
| 929.0
| 929.0
| 7#
| 7#
| 12/7
|  
|-
|-
| 49
| 49
Line 337: Line 338:
| 948.4
| 948.4
| 8v
| 8v
| 26/15
|  
|-
|-
| 50
| 50
Line 343: Line 344:
| 967.7
| 967.7
| 8⌐
| 8⌐
| 7/4
|  
|-
|-
| 51
| 51
Line 349: Line 350:
| 987.1
| 987.1
| 8
| 8
| 16/9
| 7/4
|-
|-
| 52
| 52
Line 367: Line 368:
| 1045.2
| 1045.2
| 9b
| 9b
|  
| 11/6~20/11
|-
|-
| 55
| 55

Revision as of 05:12, 8 August 2024

← 61edo 62edo 63edo →
Prime factorization 2 × 31
Step size 19.3548 ¢ 
Fifth 36\62 (696.774 ¢) (→ 18\31)
Semitones (A1:m2) 4:6 (77.42 ¢ : 116.1 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

62 = 2 × 31 and the patent val 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. Interestingly, the relative size differences of 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.

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
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)

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 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.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 19.4 ^D, vE♭♭
2 38.7 ^^D, E♭♭
3 58.1 29/28, 30/29, 31/30, 32/31 vD♯, ^E♭♭
4 77.4 22/21, 23/22, 24/23 D♯, vvE♭
5 96.8 18/17, 19/18 ^D♯, vE♭
6 116.1 15/14, 31/29 ^^D♯, E♭
7 135.5 13/12 vD𝄪, ^E♭
8 154.8 12/11, 23/21, 35/32 D𝄪, vvE
9 174.2 21/19, 31/28, 32/29 ^D𝄪, vE
10 193.5 19/17, 28/25 E
11 212.9 17/15, 26/23, 35/31 ^E, vF♭
12 232.3 8/7 ^^E, F♭
13 251.6 15/13, 22/19 vE♯, ^F♭
14 271 7/6 E♯, vvF
15 290.3 13/11 ^E♯, vF
16 309.7 F
17 329 23/19, 29/24, 35/29 ^F, vG♭♭
18 348.4 11/9 ^^F, G♭♭
19 367.7 21/17, 26/21 vF♯, ^G♭♭
20 387.1 5/4 F♯, vvG♭
21 406.5 19/15, 24/19 ^F♯, vG♭
22 425.8 23/18, 32/25 ^^F♯, G♭
23 445.2 22/17, 31/24 vF𝄪, ^G♭
24 464.5 17/13 F𝄪, vvG
25 483.9 ^F𝄪, vG
26 503.2 G
27 522.6 23/17 ^G, vA♭♭
28 541.9 26/19 ^^G, A♭♭
29 561.3 18/13, 29/21 vG♯, ^A♭♭
30 580.6 7/5 G♯, vvA♭
31 600 17/12, 24/17 ^G♯, vA♭
32 619.4 10/7 ^^G♯, A♭
33 638.7 13/9 vG𝄪, ^A♭
34 658.1 19/13 G𝄪, vvA
35 677.4 31/21, 34/23 ^G𝄪, vA
36 696.8 A
37 716.1 ^A, vB♭♭
38 735.5 26/17, 29/19 ^^A, B♭♭
39 754.8 17/11, 31/20 vA♯, ^B♭♭
40 774.2 25/16 A♯, vvB♭
41 793.5 19/12, 30/19 ^A♯, vB♭
42 812.9 8/5 ^^A♯, B♭
43 832.3 21/13, 34/21 vA𝄪, ^B♭
44 851.6 18/11, 31/19 A𝄪, vvB
45 871 ^A𝄪, vB
46 890.3 B
47 909.7 22/13 ^B, vC♭
48 929 12/7 ^^B, C♭
49 948.4 19/11, 26/15 vB♯, ^C♭
50 967.7 7/4 B♯, vvC
51 987.1 23/13, 30/17 ^B♯, vC
52 1006.5 25/14, 34/19 C
53 1025.8 29/16 ^C, vD♭♭
54 1045.2 11/6 ^^C, D♭♭
55 1064.5 24/13 vC♯, ^D♭♭
56 1083.9 28/15 C♯, vvD♭
57 1103.2 17/9 ^C♯, vD♭
58 1122.6 21/11, 23/12 ^^C♯, D♭
59 1141.9 29/15, 31/16 vC𝄪, ^D♭
60 1161.3 C𝄪, vvD
61 1180.6 ^C𝄪, vD
62 1200 2/1 D

Notation

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

Regular temperament properties

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

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
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

Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 3\62 58.06 27/26 Hemisecordite
1 7\62 135.48 13/12 Doublethink
1 13\62 251.61 15/13 Hemimeantone
1 17\62 329.03 16/11 Mabon
2 3\62 58.06 27/26 Semihemisecordite
2 4\62 77.42 21/20 Semivalentine
2 6\62 116.13 15/14 Semimiracle
2 26\62 503.22 4/3 Semimeantone
31 29\62
(1\62) 		561.29
(19.35) 		11/8
(196/195) 		Kumhar (62e)
31 19\62
(1\62) 		367.74
(19.35) 		16/13
(77/76) 		Gallium

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

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