Xenharmonic Wiki

255edo: Difference between revisions

← Older edit
VisualWikitext
Update
 
(12 intermediate revisions by 7 users not shown)
Line 1: Line 1:
{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3 × 5 × 17
{{ED intro}}
| Step size = 4.70589¢
| Fifth = 149\255 (701.18¢)
| Semitones = 23:20 (108.24¢ : 94.12¢)
| Consistency = 11
}}
{{EDO intro|255}}


== 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 19683/19600 ([[cataharry comma]]), 16875/16807 ([[canopic comma]]), and 65625/65536 ([[horwell comma]]) 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 {{nowrap| 3 × 5 × 17 }}, 255edo has subset edos {{EDOs| 3, 5, 15, 17, 51, and 85 }}.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 26: Line 24:
|-
|-
| 2.3
| 2.3
| {{monzo| -404 255 }}
| {{Monzo| -404 255 }}
| [{{val| 255 404 }}]
| {{Mapping| 255 404 }}
| +0.246
| +0.246
| 0.246
| 0.246
Line 33: Line 31:
|-
|-
| 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
Line 40: Line 38:
|-
|-
| 2.3.5.7
| 2.3.5.7
| 1687/16807, 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
Line 48: Line 46:
| 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
Line 56: Line 54:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br>per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
| 1
| 39\255
| 183.53
| 10/9
| [[Mirkat]] (255f)
|-
|-
| 1
| 1
Line 79: Line 72:
| 315.29
| 315.29
| 6/5
| 6/5
| [[Parakleismic]] (5-limit)
| [[Counterpara]]
|-
|-
| 1
| 1
Line 88: Line 81:
|-
|-
| 3
| 3
| 82\255<br>(3\255)
| 39\255
| 385.88<br>(14.12)
| 183.53
| 5/4<br>(126/125)
| 10/9
| [[Mirkat]] (255f)
|-
| 3
| 3\255
| 14.12
| 126/125
| [[Mutt]] (7-limit)
| [[Mutt]] (7-limit)
|-
|-
| 5
| 5
| 53\255<br>(2\255)
| 2\255
| 249.41<br>(9.41)
| 9.41
| 81/70<br>(176/175)
| 176/175
| [[Hemipental]] / hemipent (255) / hemipentalis (255f)
| [[Hemiquintile]] / hemiquint (255) / hemiquintilis (255f)
|-
|-
| 5
| 5
| 106\255<br>(4\255)
| 4\255
| 498.82<br>(18.82)
| 18.82
| 4/3<br>(81/80)
| 81/80
| [[Pental]] (5-limit)
| [[Quintile]]
|-
|-
| 17
| 17
| 53\255<br>(7\255)
| 7\255
| 249.41<br>(32.94)
| 32.94
| {{monzo| -25 -9 17 }}<br>(1990656/1953125)
| 1990656/1953125
| [[Chlorine]] (5-limit)
| [[Chlorine]] (5-limit)
|}
|}
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


[[Category:Equal divisions of the octave]]
[[Category:Mirkat]]
[[Category:Mirkat]]
Retrieved from "https://en.xen.wiki/w/255edo"