255edo: Difference between revisions

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The ''255 equal division'' divides the octave into 255 equal parts of 4.706 cents each. It tempers out the parakleisma, |8 14 -13>, and the septendecima, |-52 -17 34>, in the 5-limit. In the 7-limit it tempers out cataharry, 19683/19600, mirkwai, 16875/16807 and horwell, 65625/65536, so that it supports [[Mirkwai_clan#Mirkat|mirkat temperament]], and in fact provides the [[Optimal_patent_val|optimal patent val]]. It also gives the optimal patent val for mirkat in the 11-limit, tempering out 540/539, 1375/1372, 3025/3024 and 8019/8000. In the 13-limit it tempers out 847/845, 625/624, 1575/1573 and 1716/1715.
{{Infobox ET}}
{{ED intro}}
 
== Theory ==
The equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }}, and the [[septendecima]], {{monzo| -52 -17 34 }}, in the 5-limit. In the 7-limit it tempers out [[cataharry]], 19683/19600, [[mirkwai]], 16875/16807 and [[horwell]], 65625/65536, so that it [[support]]s the [[mirkat]] temperament, and in fact provides the [[optimal patent val]]. It also gives the optimal patent val for mirkat in the 11-limit, tempering out [[540/539]], 1375/1372, [[3025/3024]] and [[8019/8000]]. In the 13-limit it tempers out [[847/845]], [[625/624]], [[1575/1573]] and [[1716/1715]].
 
=== Prime harmonics ===
{{Harmonics in equal|255}}
 
=== Subsets and supersets ===
Since 255 factors into {{factorization|255}}, 255edo has subset edos {{EDOs| 3, 5, 15, 17, 51, and 85 }}.
 
== 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| -404 255 }}
| {{Mapping| 255 404 }}
| +0.246
| 0.246
| 5.22
|-
| 2.3.5
| {{Monzo| 8 14 -13 }}, {{monzo| -36 11 8 }}
| {{Mapping| 255 404 592 }}
| +0.226
| 0.203
| 4.30
|-
| 2.3.5.7
| 16875/16807, 19683/19600, 65625/65536
| {{Mapping| 255 404 592 716 }}
| +0.117
| 0.257
| 5.46
|-
| 2.3.5.7.11
| 540/539, 1375/1372, 8019/8000, 65625/65536
| {{Mapping| 255 404 592 716 882 }}
| +0.136
| 0.233
| 4.95
|}
 
=== 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
| 39\255
| 183.53
| 10/9
| [[Mirkat]] (255f)
|-
| 1
| 52\255
| 244.71
| 15/13
| [[Subsemifourth]] (255)
|-
| 1
| 67\255
| 315.29
| 6/5
| [[Parakleismic]] (5-limit)
|-
| 1
| 74\255
| 348.24
| 11/9
| [[Eris]] (255)
|-
| 3
| 82\255<br>(3\255)
| 385.88<br>(14.12)
| 5/4<br>(126/125)
| [[Mutt]] (7-limit)
|-
| 5
| 53\255<br>(2\255)
| 249.41<br>(9.41)
| 81/70<br>(176/175)
| [[Hemiquintile]] / hemiquint (255) / hemiquintilis (255f)
|-
| 5
| 106\255<br>(4\255)
| 498.82<br>(18.82)
| 4/3<br>(81/80)
| [[Quintile]]
|-
| 17
| 53\255<br>(7\255)
| 249.41<br>(32.94)
| {{Monzo| -25 -9 17 }}<br>(1990656/1953125)
| [[Chlorine]] (5-limit)
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
[[Category:Mirkat]]

Latest revision as of 17:53, 19 February 2025

← 254edo 255edo 256edo →
Prime factorization 3 × 5 × 17
Step size 4.70588 ¢ 
Fifth 149\255 (701.176 ¢)
Semitones (A1:m2) 23:20 (108.2 ¢ : 94.12 ¢)
Consistency limit 11
Distinct consistency limit 11

255 equal divisions of the octave (abbreviated 255edo or 255ed2), also called 255-tone equal temperament (255tet) or 255 equal temperament (255et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 255 equal parts of about 4.71 ¢ each. Each step represents a frequency ratio of 21/255, or the 255th root of 2.

Theory

The equal temperament tempers out the parakleisma, [8 14 -13⟩, and the septendecima, [-52 -17 34⟩, in the 5-limit. In the 7-limit it tempers out cataharry, 19683/19600, mirkwai, 16875/16807 and horwell, 65625/65536, so that it supports the mirkat temperament, and in fact provides the optimal patent val. It also gives the optimal patent val for mirkat in the 11-limit, tempering out 540/539, 1375/1372, 3025/3024 and 8019/8000. In the 13-limit it tempers out 847/845, 625/624, 1575/1573 and 1716/1715.

Prime harmonics

Approximation of prime harmonics in 255edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.78 -0.43 +0.59 -0.73 +1.83 -1.43 -1.04 +2.31 +1.01 -1.51
Relative (%) +0.0 -16.5 -9.2 +12.4 -15.5 +38.8 -30.3 -22.2 +49.2 +21.5 -32.0
Steps
(reduced) 		255
(0) 		404
(149) 		592
(82) 		716
(206) 		882
(117) 		944
(179) 		1042
(22) 		1083
(63) 		1154
(134) 		1239
(219) 		1263
(243)

Subsets and supersets

Since 255 factors into 3 × 5 × 17, 255edo has subset edos 3, 5, 15, 17, 51, and 85.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-404 255⟩ [⟨255 404]] +0.246 0.246 5.22
2.3.5 [8 14 -13⟩, [-36 11 8⟩ [⟨255 404 592]] +0.226 0.203 4.30
2.3.5.7 16875/16807, 19683/19600, 65625/65536 [⟨255 404 592 716]] +0.117 0.257 5.46
2.3.5.7.11 540/539, 1375/1372, 8019/8000, 65625/65536 [⟨255 404 592 716 882]] +0.136 0.233 4.95

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 39\255 183.53 10/9 Mirkat (255f)
1 52\255 244.71 15/13 Subsemifourth (255)
1 67\255 315.29 6/5 Parakleismic (5-limit)
1 74\255 348.24 11/9 Eris (255)
3 82\255
(3\255) 		385.88
(14.12) 		5/4
(126/125) 		Mutt (7-limit)
5 53\255
(2\255) 		249.41
(9.41) 		81/70
(176/175) 		Hemiquintile / hemiquint (255) / hemiquintilis (255f)
5 106\255
(4\255) 		498.82
(18.82) 		4/3
(81/80) 		Quintile
17 53\255
(7\255) 		249.41
(32.94) 		[-25 -9 17⟩
(1990656/1953125) 		Chlorine (5-limit)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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