62edo: Difference between revisions
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== Theory == | == Theory == | ||
62 = 2 × 31 and the [[patent val]] is a contorted [[31edo]] through the 11-limit | 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 [[13/1|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. 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 [[31 comma temperaments #Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[Meantone family #Hemimeantone|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]]. | 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]]. | ||
Revision as of 11:53, 16 August 2023
| ← 61edo | 62edo | 63edo → |
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
| 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
| 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 |
Regular temperament properties
62edo is contorted 31edo through the 11-limit.
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 13\62 | 251.61 | 15/13 | Hemimeantone |
| 1 | 17\62 | 329.03 | 16/11 | Mabon |
| 2 | 4\62 | 77.42 | 21/20 | Semivalentine |
| 31 | 1\62 | 19.35 | 196/195 | Kumhar |
| 31 | 1\62 | 19.35 | 16807/16640 | Gallium |