58edo: Difference between revisions

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58 equal divisions of the octave (58edo), or 58-tone equal temperament (58tet), 58 equal temperament (58et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 58 equal parts of about 20.7 ¢ each, an interval close to the syntonic comma. Each step represents the frequency ratio of 21/58, or the 58th root of 2.

== Theory ==

58edo 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 distinctly consistent in the [[11-odd-limit]] (the first equal temperament to map the entire 11-odd-limit [[tonality diamond]] to distinct scale steps), and hence the first which can define a tempered version of the famous 43-note [[Harry Partch related scales|Genesis scale]] of [[Harry Partch]].

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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.

Of all edos which map the syntonic comma ([[81/80]]) to 1 step by patent val, 58edo is the one with the step size closest to 81/80, with one step of 58edo being less than 1{{cent}} narrower than the just interval.

=== Prime harmonics ===

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 {{Infobox ET}}
 {{Wikipedia|58 equal temperament}}
-equal divisions of the octave (58edo), or 58-tone equal temperament (58tet), 58 equal temperament (58et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 58 equal parts of about 20.7 ¢ each, an interval close to the syntonic comma. Each step represents the frequency ratio of 21/58, or the 58th root of 2.
+{{EDO intro|58}}
 == Theory ==
 edo 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 distinctly consistent in the [[11-odd-limit]] (the first equal temperament to map the entire 11-odd-limit [[tonality diamond]] to distinct scale steps), and hence the first which can define a tempered version of the famous 43-note [[Harry Partch related scales|Genesis scale]] of [[Harry Partch]].
@@ Line 8: / Line 9: @@
 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.
+Of all edos which map the syntonic comma ([[81/80]]) to 1 step by patent val, 58edo is the one with the step size closest to 81/80, with one step of 58edo being less than 1{{cent}} narrower than the just interval.
 === Prime harmonics ===