255edo: Difference between revisions

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== 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 ===

=== Subsets and supersets ===

Since 255 factors into {{factorization|255}}, 255edo has subset edos {{EDOs| 3, 5, 15, 17, 51, and 85 }}.

== Regular temperament properties ==

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| 2.3

| {{monzo| -404 255 }}

| [{{~~val~~| 255 404 }}]

| {{mapping| 255 404 }}

| +0.246

| 0.246

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| 2.3.5

| {{monzo| 8 14 -13 }}, {{monzo| -36 11 8 }}

| [{{~~val~~| 255 404 592 }}]

| {{mapping| 255 404 592 }}

| +0.226

| 0.203

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| 2.3.5.7

| 16875/16807, 19683/19600, 65625/65536

| [{{~~val~~| 255 404 592 716 }}]

| {{mapping| 255 404 592 716 }}

| +0.117

| 0.257

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| 2.3.5.7.11

| 540/539, 1375/1372, 8019/8000, 65625/65536

| [{{~~val~~| 255 404 592 716 882 }}]

| {{mapping| 255 404 592 716 882 }}

| +0.136

| 0.233

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[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]]

@@ Line 3: / Line 3: @@
 == Theory ==
-et 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 ===
 {{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 ==
@@ Line 21: / Line 24: @@
 | 2.3
 | {{monzo| -404 255 }}
-| [{{val| 255 404 }}]
+| {{mapping| 255 404 }}
 | +0.246
 | 0.246
@@ Line 28: / Line 31: @@
 | 2.3.5
 | {{monzo| 8 14 -13 }}, {{monzo| -36 11 8 }}
-| [{{val| 255 404 592 }}]
+| {{mapping| 255 404 592 }}
 | +0.226
 | 0.203
@@ Line 35: / Line 38: @@
 | 2.3.5.7
 | 16875/16807, 19683/19600, 65625/65536
-| [{{val| 255 404 592 716 }}]
+| {{mapping| 255 404 592 716 }}
 | +0.117
 | 0.257
@@ Line 42: / Line 45: @@
 | 2.3.5.7.11
 | 540/539, 1375/1372, 8019/8000, 65625/65536
-| [{{val| 255 404 592 716 882 }}]
+| {{mapping| 255 404 592 716 882 }}
 | +0.136
 | 0.233
@@ Line 52: / Line 55: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 105: / Line 108: @@
 | [[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]]