62edo
| ← 61edo | 62edo | 63edo → |
Theory
62edo's 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 relative size differences between consecutive odd harmonics are well preserved for the first eleven[note 1] odd-particulars (with the exception of 9/7), and 62edo is one of the few meantone edos to achieve this—great for those who seek higher-limit (though no-twos) 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
| 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) | |
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 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 | 0.00 | 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 |  |
 |
 |
 |
 |
 |
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 |
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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 | |
|---|---|---|---|---|---|
| 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* |
Temperament |
|---|---|---|---|---|
| 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 distinct