285edo: Difference between revisions
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== Theory == | == Theory == | ||
285edo has a sharp tendency. | 285edo has a sharp tendency. The equal temperament [[tempering out|tempers out]] the [[misty comma]] and the [[enneadeca]] in the 5-limit; [[3136/3125]] and [[5120/5103]] in the 7-limit; [[3025/3024]] and [[3388/3375]] in the 11-limit; [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], and [[2080/2079]] in the 13-limit. It supports the 13-limit [[hemimist]] temperament. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|285}} | {{Harmonics in equal|285|intervals=prime}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
| Line 24: | Line 24: | ||
| 2.3 | | 2.3 | ||
| {{monzo| 452 -285 }} | | {{monzo| 452 -285 }} | ||
| | | {{mapping| 285 452 }} | ||
| -0.3795 | | -0.3795 | ||
| 0.3794 | | 0.3794 | ||
| Line 31: | Line 31: | ||
| 2.3.5 | | 2.3.5 | ||
| 67108864/66430125, {{monzo| -14 -19 19 }} | | 67108864/66430125, {{monzo| -14 -19 19 }} | ||
| | | {{mapping| 285 452 662 }} | ||
| -0.4043 | | -0.4043 | ||
| 0.3117 | | 0.3117 | ||
| Line 38: | Line 38: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 3136/3125, 5120/5103, 40353607/39858075 | | 3136/3125, 5120/5103, 40353607/39858075 | ||
| | | {{mapping| 285 452 662 800 }} | ||
| -0.2673 | | -0.2673 | ||
| 0.3596 | | 0.3596 | ||
| Line 45: | Line 45: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 3025/3024, 3136/3125, 5120/5103, 12005/11979 | | 3025/3024, 3136/3125, 5120/5103, 12005/11979 | ||
| | | {{mapping| 285 452 662 800 986 }} | ||
| -0.2289 | | -0.2289 | ||
| 0.3307 | | 0.3307 | ||
| Line 52: | Line 52: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 352/351, 676/675, 847/845, 3025/3024, 12005/11979 | | 352/351, 676/675, 847/845, 3025/3024, 12005/11979 | ||
| | | {{mapping| 285 452 662 800 986 1055 }} | ||
| -0.2618 | | -0.2618 | ||
| 0.3107 | | 0.3107 | ||
| Line 62: | Line 62: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 91: | Line 91: | ||
| [[Enneadecal]] (5-limit) | | [[Enneadecal]] (5-limit) | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
Revision as of 05:20, 8 March 2024
| ← 284edo | 285edo | 286edo → |
Theory
285edo has a sharp tendency. The equal temperament tempers out the misty comma and the enneadeca in the 5-limit; 3136/3125 and 5120/5103 in the 7-limit; 3025/3024 and 3388/3375 in the 11-limit; 352/351, 676/675, 847/845, 1001/1000, and 2080/2079 in the 13-limit. It supports the 13-limit hemimist temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +1.20 | +1.05 | -0.40 | +0.26 | +1.58 | +0.31 | +1.43 | -0.91 | +2.00 | +0.23 |
| Relative (%) | +0.0 | +28.6 | +25.0 | -9.6 | +6.2 | +37.5 | +7.3 | +34.1 | -21.5 | +47.5 | +5.4 | |
| Steps (reduced) |
285 (0) |
452 (167) |
662 (92) |
800 (230) |
986 (131) |
1055 (200) |
1165 (25) |
1211 (71) |
1289 (149) |
1385 (245) |
1412 (272) | |
Subsets and supersets
Since 285 factors into 3 × 5 × 19, 285edo has subset edos 3, 5, 15, 19, 57, and 95.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [452 -285⟩ | [⟨285 452]] | -0.3795 | 0.3794 | 9.01 |
| 2.3.5 | 67108864/66430125, [-14 -19 19⟩ | [⟨285 452 662]] | -0.4043 | 0.3117 | 7.41 |
| 2.3.5.7 | 3136/3125, 5120/5103, 40353607/39858075 | [⟨285 452 662 800]] | -0.2673 | 0.3596 | 8.54 |
| 2.3.5.7.11 | 3025/3024, 3136/3125, 5120/5103, 12005/11979 | [⟨285 452 662 800 986]] | -0.2289 | 0.3307 | 7.85 |
| 2.3.5.7.11.13 | 352/351, 676/675, 847/845, 3025/3024, 12005/11979 | [⟨285 452 662 800 986 1055]] | -0.2618 | 0.3107 | 7.38 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 109\285 | 458.95 | 125/96 | Majvam |
| 3 | 59\285 (36\285) |
248.42 (151.58) |
15/13 (12/11) |
Hemimist |
| 3 | 59\285 (23\285) |
496.84 (96.84) |
4/3 (256/243) |
Misty |
| 19 | 118\285 (2\285) |
496.84 (8.42) |
4/3 (15625/15552) |
Enneadecal (5-limit) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct