255edo: Difference between revisions

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== Theory ==
== 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]].
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 ===
=== Prime harmonics ===
{{Harmonics in equal|255}}
{{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 ==
== Regular temperament properties ==
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| 2.3
| 2.3
| {{monzo| -404 255 }}
| {{monzo| -404 255 }}
| [{{val| 255 404 }}]
| {{mapping| 255 404 }}
| +0.246
| +0.246
| 0.246
| 0.246
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| 2.3.5
| 2.3.5
| {{monzo| 8 14 -13 }}, {{monzo| -36 11 8 }}
| {{monzo| 8 14 -13 }}, {{monzo| -36 11 8 }}
| [{{val| 255 404 592 }}]
| {{mapping| 255 404 592 }}
| +0.226
| +0.226
| 0.203
| 0.203
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| 2.3.5.7
| 2.3.5.7
| 16875/16807, 19683/19600, 65625/65536
| 16875/16807, 19683/19600, 65625/65536
| [{{val| 255 404 592 716 }}]
| {{mapping| 255 404 592 716 }}
| +0.117
| +0.117
| 0.257
| 0.257
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| 2.3.5.7.11
| 2.3.5.7.11
| 540/539, 1375/1372, 8019/8000, 65625/65536
| 540/539, 1375/1372, 8019/8000, 65625/65536
| [{{val| 255 404 592 716 882 }}]
| {{mapping| 255 404 592 716 882 }}
| +0.136
| +0.136
| 0.233
| 0.233
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Chlorine]] (5-limit)
| [[Chlorine]] (5-limit)
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


[[Category:Mirkat]]
[[Category:Mirkat]]

Revision as of 08:55, 14 March 2024

← 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

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) 		Hemipental / hemipent (255) / hemipentalis (255f)
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)

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

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