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255edo: Difference between revisions

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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
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== Theory ==
== Theory ==
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]].
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 ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 255 factors into {{factorization|255}}, 255edo has subset edos {{EDOs| 3, 5, 15, 17, 51, and 85 }}.  
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 ==
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! Associated<br>ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
| 1
| 39\255
| 183.53
| 10/9
| [[Mirkat]] (255f)
|-
|-
| 1
| 1
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| 315.29
| 315.29
| 6/5
| 6/5
| [[Parakleismic]] (5-limit)
| [[Counterpara]]
|-
|-
| 1
| 1
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|-
|-
| 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
| [[Hemiquintile]] / hemiquint (255) / hemiquintilis (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
| [[Quintile]]
| [[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/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


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