255edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
Line 49: Line 41:
| 0.233
| 0.233
| 4.95
| 4.95
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 85: Line 71:
|-
|-
| 3
| 3
| 82\255<br>(3\255)
| 82\255<br />(3\255)
| 385.88<br>(14.12)
| 385.88<br />(14.12)
| 5/4<br>(126/125)
| 5/4<br />(126/125)
| [[Mutt]] (7-limit)
| [[Mutt]] (7-limit)
|-
|-
| 5
| 5
| 53\255<br>(2\255)
| 53\255<br />(2\255)
| 249.41<br>(9.41)
| 249.41<br />(9.41)
| 81/70<br>(176/175)
| 81/70<br />(176/175)
| [[Hemipental]] / hemipent (255) / hemipentalis (255f)
| [[Hemipental]] / hemipent (255) / hemipentalis (255f)
|-
|-
| 5
| 5
| 106\255<br>(4\255)
| 106\255<br />(4\255)
| 498.82<br>(18.82)
| 498.82<br />(18.82)
| 4/3<br>(81/80)
| 4/3<br />(81/80)
| [[Pental (temperament)|Pental]] (5-limit)
| [[Pental (temperament)|Pental]] (5-limit)
|-
|-
| 17
| 17
| 53\255<br>(7\255)
| 53\255<br />(7\255)
| 249.41<br>(32.94)
| 249.41<br />(32.94)
| {{monzo| -25 -9 17 }}<br>(1990656/1953125)
| {{monzo| -25 -9 17 }}<br />(1990656/1953125)
| [[Chlorine]] (5-limit)
| [[Chlorine]] (5-limit)
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


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

Revision as of 04:01, 16 November 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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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) Template:Rank-2 end Template:Orf

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