285edo: Difference between revisions
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CompactStar (talk | contribs) Created page with "{{Infobox ET}} {{EDO intro|285}} ==Theory== {{Primes in edo|285}}" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|285}} | {{EDO intro|285}} | ||
==Theory== | |||
{{ | == Theory == | ||
285edo has a sharp tendency. It 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 === | |||
{{Harmonics in equal|285}} | |||
=== Subsets and supersets === | |||
Since 285 factors into 3 × 5 × 19, 285edo has subset edos {{EDOs| 3, 5, 15, 19, 57, and 95 }}. | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list|Comma List]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve Stretch (¢) | |||
! colspan="2" | Tuning Error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| 452 -285 }} | |||
| [{{val| 285 452 }}] | |||
| -0.3795 | |||
| 0.3794 | |||
| 9.01 | |||
|- | |||
| 2.3.5 | |||
| 67108864/66430125, {{monzo| -14 -19 19 }} | |||
| [{{val| 285 452 662 }}] | |||
| -0.4043 | |||
| 0.3117 | |||
| 7.41 | |||
|- | |||
| 2.3.5.7 | |||
| 3136/3125, 5120/5103, 40353607/39858075 | |||
| [{{val| 285 452 662 800 }}] | |||
| -0.2673 | |||
| 0.3596 | |||
| 8.54 | |||
|- | |||
| 2.3.5.7.11 | |||
| 3025/3024, 3136/3125, 5120/5103, 12005/11979 | |||
| [{{val| 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 | |||
| [{{val| 285 452 662 800 986 1055 }}] | |||
| -0.2618 | |||
| 0.3107 | |||
| 7.38 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per 8ve | |||
! Generator<br>(Reduced) | |||
! Cents<br>(Reduced) | |||
! Associated<br>Ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 109\285 | |||
| 458.95 | |||
| 125/96 | |||
| [[Majvam]] | |||
|- | |||
| 3 | |||
| 59\285<br>(36\285) | |||
| 248.42<br>(151.58) | |||
| 15/13<br>(12/11) | |||
| [[Hemimist]] | |||
|- | |||
| 3 | |||
| 59\285<br>(23\285) | |||
| 496.84<br>(96.84) | |||
| 4/3<br>(256/243) | |||
| [[Misty]] | |||
|- | |||
| 19 | |||
| 118\285<br>(2\285) | |||
| 496.84<br>(8.42) | |||
| 4/3<br>(15625/15552) | |||
| [[Enneadecal]] (5-limit) | |||
|} | |||
Revision as of 07:31, 11 June 2023
| ← 284edo | 285edo | 286edo → |
Theory
285edo has a sharp tendency. It 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 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.20 | +1.05 | -0.40 | -1.80 | +0.26 | +1.58 | -1.95 | +0.31 | +1.43 | +0.80 | -0.91 |
| Relative (%) | +28.6 | +25.0 | -9.6 | -42.9 | +6.2 | +37.5 | -46.4 | +7.3 | +34.1 | +19.0 | -21.5 | |
| Steps (reduced) |
452 (167) |
662 (92) |
800 (230) |
903 (48) |
986 (131) |
1055 (200) |
1113 (258) |
1165 (25) |
1211 (71) |
1252 (112) |
1289 (149) | |
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 (Reduced) |
Cents (Reduced) |
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) |