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285edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|285}} ==Theory== {{Primes in edo|285}}"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|285}}
{{ED intro}}
==Theory==
 
{{Primes in edo|285}}
== Theory ==
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 ===
{{Harmonics in equal|285|intervals=prime}}
 
=== Subsets and supersets ===
Since 285 factors into {{factorisation|285}}, 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]]
! 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 }}
| {{mapping| 285 452 }}
| −0.3795
| 0.3794
| 9.01
|-
| 2.3.5
| 67108864/66430125, {{monzo| -14 -19 19 }}
| {{mapping| 285 452 662 }}
| −0.4043
| 0.3117
| 7.41
|-
| 2.3.5.7
| 3136/3125, 5120/5103, 40353607/39858075
| {{mapping| 285 452 662 800 }}
| −0.2673
| 0.3596
| 8.54
|-
| 2.3.5.7.11
| 3025/3024, 3136/3125, 5120/5103, 12005/11979
| {{mapping| 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
| {{mapping| 285 452 662 800 986 1055 }}
| −0.2618
| 0.3107
| 7.38
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! 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)
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
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