285edo: Difference between revisions

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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

Latest revision as of 06:28, 21 February 2025

← 284edo 285edo 286edo →
Prime factorization 3 × 5 × 19
Step size 4.21053 ¢ 
Fifth 167\285 (703.158 ¢)
Semitones (A1:m2) 29:20 (122.1 ¢ : 84.21 ¢)
Consistency limit 7
Distinct consistency limit 7

285 equal divisions of the octave (abbreviated 285edo or 285ed2), also called 285-tone equal temperament (285tet) or 285 equal temperament (285et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 285 equal parts of about 4.21 ¢ each. Each step represents a frequency ratio of 21/285, or the 285th root of 2.

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

Approximation of prime harmonics in 285edo
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

Table of rank-2 temperaments by generator
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 distinct

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