255edo: Difference between revisions
Cleanup and +prime error table |
+infobox, improve intro, +RTT table and rank-2 temperaments |
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{{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]] |