255edo: Difference between revisions

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+infobox, improve intro, +RTT table and rank-2 temperaments
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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]], {{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]].
{{Infobox ET
| Prime factorization = 3 × 5 × 17
| Step size = 4.70589¢
| Fifth = 149\255 (701.18¢)
| Semitones = 23:20 (108.24¢ : 94.12¢)
| Consistency = 11
}}
{{EDO intro|255}}
 
== Theory ==
255et 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 ===
=== Prime harmonics ===
{{Harmonics in equal|255}}
{{Harmonics in equal|255}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -404 255 }}
| [{{val| 255 404 }}]
| +0.246
| 0.246
| 5.22
|-
| 2.3.5
| {{monzo| 8 14 -13 }}, {{monzo| -36 11 8 }}
| [{{val| 255 404 592 }}]
| +0.226
| 0.203
| 4.30
|-
| 2.3.5.7
| 1687/16807, 19683/19600, 65625/65536
| [{{val| 255 404 592 716 }}]
| +0.117
| 0.257
| 5.46
|-
| 2.3.5.7.11
| 540/539, 1375/1372, 8019/8000, 65625/65536
| [{{val| 255 404 592 716 882 }}]
| +0.136
| 0.233
| 4.95
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! 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
| 106\255<br>(4\255)
| 498.82<br>(18.82)
| 4/3<br>(81/80)
| [[Pental]] (5-limit)
|-
| 17
| 53\255<br>(7\255)
| 249.41<br>(32.94)
| {{monzo| -25 -9 17 }}<br>(1990656/1953125)
| [[Chlorine]] (5-limit)
|}


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:Mirkat]]
[[Category:Mirkat]]

Revision as of 20:36, 2 April 2022

← 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

Template:EDO intro

Theory

255et 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)

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 1687/16807, 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 octave 		Generator
(reduced) 		Cents
(reduced) 		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)
5 106\255
(4\255) 		498.82
(18.82) 		4/3
(81/80) 		Pental (5-limit)
17 53\255
(7\255) 		249.41
(32.94) 		[-25 -9 17⟩
(1990656/1953125) 		Chlorine (5-limit)
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