58edo: Difference between revisions

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The '''58 equal ~~temperament~~''', ~~often abbreviated~~ 58-~~tET~~, ~~58-EDO, or 58-ET~~, is the ~~scale~~ derived by dividing the [[octave]] into 58 ~~equally~~-sized steps~~. Each step represents a frequency ratio~~ of 20.69 ~~cents~~.

The '''58 equal divisions of the octave''' ('''58edo'''), or '''58(-tone) equal temperament''' ('''58tet''', '''58et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 58 [[equal]]ly-sized steps of 20.69 [[cent]]s each.

== Theory ==

58et tempers out 2048/2025, 126/125, 1728/1715, 144/143, 176/175, 896/891, 243/242, 5120/5103, 351/350, 364/363, 441/440, and 540/539, and is a strong system in the [[11-limit|11]]-, [[13-limit|13]]- and [[17-limit]]. It is the smallest [[edo]] which is [[consistent]] through the [[17-odd-limit]], and is also the smallest uniquely consistent in the [[11-odd-limit]] (the first equal temperament to map the entire 11-limit [[tonality diamond]] to distinct scale steps), and hence the first et which can define a version of the famous 43-note [[Harry Partch related scales|Genesis scale]] of [[Harry Partch]]. It supports [[hemififths]], [[myna]], [[diaschismic]], [[harry]], [[mystery]], [[buzzard]] and [[thuja]] [[Regular temperament|temperament]]s, and supplies the [[optimal patent val]] for 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank-3 temperaments [[thrush]], [[bluebird]], [[aplonis]] and [[jofur]].

~~58edo~~ tempers out 2048/2025, 126/125, 1728/1715, 144/143, 176/175, 896/891, 243/242, 5120/5103, 351/350, 364/363, 441/440, and 540/539, and is a strong system in the [[11-limit|11]]-, [[13-limit|13]]- and [[17-limit]]. It is the smallest [[edo]] which is [[consistent]] through the [[17-odd-limit]], and is also the smallest uniquely consistent in the [[11-odd-limit]] (the first equal temperament to map the entire 11-limit [[tonality diamond]] to distinct scale steps), and hence the first et which can define a version of the famous 43-note [[Harry Partch related scales|Genesis scale]] of [[Harry Partch]]. It supports [[hemififths]], [[myna]], [[diaschismic]], [[harry]], [[mystery]], [[buzzard]] and [[thuja]] [[Regular temperament|temperament]]s, and supplies the [[optimal patent val]] for 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank-3 temperaments [[thrush]], [[bluebird]], [[aplonis]] and [[jofur]].

While the 17th harmonic is a cent and a half flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system. 58 = 2 × 29, and 58 shares the same excellent fifth with [[29edo]].

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=== Rank-2 temperaments ===

{| class="wikitable center-1 center-2"

|+Table of rank-2 temperaments by generator

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== Scales ==

* [[Hemif7]]

* [[~~hemif7~~]]

* [[Hemif10]]

* [[~~hemif10~~]]

* [[Hemif17]]

* [[~~hemif17~~]]

[[Category:58edo| ]]

~~[[Category:buzzard]]~~

~~[[Category:diaschismic]]~~

[[Category:Equal divisions of the octave]]

[[Category:~~genesis~~]]

[[Category:Buzzard]]

[[Category:~~harry~~]]

[[Category:Diaschismic]]

[[Category:~~hemififths~~]]

[[Category:Harry]]

[[Category:~~myna~~]]

[[Category:Hemififths]]

[[Category:~~mystery~~]]

[[Category:Myna]]

[[Category:~~partch~~]]

[[Category:Mystery]]

[[Category:Genesis]]

[[Category:Partch]]

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-The '''58 equal temperament''', often abbreviated 58-tET, 58-EDO, or 58-ET, is the scale derived by dividing the [[octave]] into 58 equally-sized steps. Each step represents a frequency ratio of 20.69 cents.
+The '''58 equal divisions of the octave''' ('''58edo'''), or '''58(-tone) equal temperament''' ('''58tet''', '''58et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 58 [[equal]]ly-sized steps of 20.69 [[cent]]s each.
 == Theory ==
+et tempers out 2048/2025, 126/125, 1728/1715, 144/143, 176/175, 896/891, 243/242, 5120/5103, 351/350, 364/363, 441/440, and 540/539, and is a strong system in the [[11-limit|11]]-, [[13-limit|13]]- and [[17-limit]]. It is the smallest [[edo]] which is [[consistent]] through the [[17-odd-limit]], and is also the smallest uniquely consistent in the [[11-odd-limit]] (the first equal temperament to map the entire 11-limit [[tonality diamond]] to distinct scale steps), and hence the first et which can define a version of the famous 43-note [[Harry Partch related scales|Genesis scale]] of [[Harry Partch]]. It supports [[hemififths]], [[myna]], [[diaschismic]], [[harry]], [[mystery]], [[buzzard]] and [[thuja]] [[Regular temperament|temperament]]s, and supplies the [[optimal patent val]] for 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank-3 temperaments [[thrush]], [[bluebird]], [[aplonis]] and [[jofur]].
-edo tempers out 2048/2025, 126/125, 1728/1715, 144/143, 176/175, 896/891, 243/242, 5120/5103, 351/350, 364/363, 441/440, and 540/539, and is a strong system in the [[11-limit|11]]-, [[13-limit|13]]- and [[17-limit]]. It is the smallest [[edo]] which is [[consistent]] through the [[17-odd-limit]], and is also the smallest uniquely consistent in the [[11-odd-limit]] (the first equal temperament to map the entire 11-limit [[tonality diamond]] to distinct scale steps), and hence the first et which can define a version of the famous 43-note [[Harry Partch related scales|Genesis scale]] of [[Harry Partch]]. It supports [[hemififths]], [[myna]], [[diaschismic]], [[harry]], [[mystery]], [[buzzard]] and [[thuja]] [[Regular temperament|temperament]]s, and supplies the [[optimal patent val]] for 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank-3 temperaments [[thrush]], [[bluebird]], [[aplonis]] and [[jofur]].
 While the 17th harmonic is a cent and a half flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system. 58 = 2 × 29, and 58 shares the same excellent fifth with [[29edo]].
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 === Rank-2 temperaments ===
 {| class="wikitable center-1 center-2"
 |+Table of rank-2 temperaments by generator
@@ Line 429: / Line 427: @@
 == Scales ==
+* [[Hemif7]]
-* [[hemif7]]
+* [[Hemif10]]
-* [[hemif10]]
+* [[Hemif17]]
-* [[hemif17]]
 [[Category:58edo| ]] <!-- main article -->
-[[Category:buzzard]]
-[[Category:diaschismic]]
 [[Category:Equal divisions of the octave]]
-[[Category:genesis]]
+[[Category:Buzzard]]
-[[Category:harry]]
+[[Category:Diaschismic]]
-[[Category:hemififths]]
+[[Category:Harry]]
-[[Category:myna]]
+[[Category:Hemififths]]
-[[Category:mystery]]
+[[Category:Myna]]
-[[Category:partch]]
+[[Category:Mystery]]
+[[Category:Genesis]]
+[[Category:Partch]]