255edo: Difference between revisions

Cleanup and +prime error table
+infobox, improve intro, +RTT table and rank-2 temperaments
Line 1: Line 1:
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]]