62edo
Theory
62 = 2 × 31 and the patent val is a contorted 31edo through the 11-limit; in the 13-limit it tempers out 169/168, 1188/1183, 847/845 and 676/675. 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.
Mabon temperament (relation to a calendar reform)
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.
11 & 62 is best interpreted in the 2.9.7 subgroup, where it tempers out 44957696/43046721, and the three generators of 17\62 correspond to 16/9. It's 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 a 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 just contorted valentine, order 2.
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) | |
Intervals
| Armodue Nomenclature 8;3 Relation |
|---|
|
| # | Cents | Armodue notation | Approximate intervals |
|---|---|---|---|
| 0 | 0.000 | 1 | |
| 1 | 19.355 | 1Ɨ | 90/89 |
| 2 | 38.710 | 1‡ (9#) | 45/44 |
| 3 | 58.065 | 2b | 30/29 |
| 4 | 77.419 | 1◊2 | 23/22 |
| 5 | 96.774 | 1# | 37/35, 18/17, 19/18 |
| 6 | 116.129 | 2v | 31/29, 15/14, 16/15 |
| 7 | 135.484 | 2⌐ | 27/25, 13/12, 14/13 |
| 8 | 154.839 | 2 | 12/11 |
| 9 | 174.194 | 2Ɨ | 11/10 |
| 10 | 193.548 | 2‡ | 19/17, 9/8, 10/9 |
| 11 | 212.903 | 3b | 17/15, 9/8 |
| 12 | 232.258 | 2◊3 | 8/7 |
| 13 | 251.613 | 2# | 15/13 |
| 14 | 270.968 | 3v | 7/6 |
| 15 | 290.323 | 3⌐ | |
| 16 | 309.677 | 3 | 6/5 |
| 17 | 329.032 | 3Ɨ | |
| 18 | 348.387 | 3‡ | 11/9 |
| 19 | 367.742 | 4b | · |
| 20 | 387.097 | 3◊4 | 5/4 |
| 21 | 406.452 | 3# | |
| 22 | 425.806 | 4v (5b) | |
| 23 | 445.161 | 4⌐ | |
| 24 | 464.516 | 4 | |
| 25 | 483.871 | 4Ɨ (5v) | |
| 26 | 503.226 | 5⌐ (4‡) | 4/3 |
| 27 | 522.581 | 5 | · |
| 28 | 541.935 | 5Ɨ | |
| 29 | 561.290 | 5‡ (4#) | |
| 30 | 580.645 | 6b | 7/5 |
| 31 | 600.000 | 5◊6 | |
| 32 | 619.355 | 5# | 10/7 |
| 33 | 638.710 | 6v | |
| 34 | 658.065 | 6⌐ | |
| 35 | 677.419 | 6 | · |
| 36 | 696.774 | 6Ɨ | 3/2 |
| 37 | 716.129 | 6‡ | |
| 38 | 735.484 | 7b | |
| 39 | 754.839 | 6◊7 | |
| 40 | 774.194 | 6# | |
| 41 | 793.548 | 7v | |
| 42 | 812.903 | 7⌐ | 8/5 |
| 43 | 832.258 | 7 | · |
| 44 | 851.613 | 7Ɨ | 18/11 |
| 45 | 870.968 | 7‡ | |
| 46 | 890.323 | 8b | 5/3 |
| 47 | 909.677 | 7◊8 | |
| 48 | 929.032 | 7# | 12/7 |
| 49 | 948.387 | 8v | 26/15 |
| 50 | 967.742 | 8⌐ | 7/4 |
| 51 | 987.097 | 8 | 16/9 |
| 52 | 1006.452 | 8Ɨ | |
| 53 | 1025.806 | 8‡ | |
| 54 | 1045.161 | 9b | |
| 55 | 1064.516 | 8◊9 | |
| 56 | 1083.871 | 8# | |
| 57 | 1103.226 | 9v (1b) | |
| 58 | 1122.581 | 9⌐ | |
| 59 | 1141.936 | 9 | |
| 60 | 1161.290 | 9Ɨ (1v) | |
| 61 | 1180.645 | 1⌐ (9‡) | |
| 62 | 1200.000 | 1 |