166edo: Difference between revisions

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== Theory ==

166edo is [[consistent]] through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it [[tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the {{nowrap|72 &~~amp;~~ 94}} temperament, for which 166 is an excellent tuning through the [[19-limit]].

166edo is [[consistent]] through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it [[tempering out|tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the {{nowrap|72 & 94}} temperament, for which 166 is an excellent tuning through the [[19-limit]].

166edo (as 83edo) contains a very good approximation of the [[7/4|harmonic 7th]], of which it is only flat by 0.15121 cent.

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=== Subsets and supersets ===

Since 166 factors into ~~{{factorisation|166}}~~, 166edo contains [[2edo]] and [[83edo]] as subsets.

Since 166 factors into 2 × 83, 166edo contains [[2edo]] and [[83edo]] as subsets.

== Regular temperament properties ==

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 == Theory ==
-edo is [[consistent]] through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it [[tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the {{nowrap|72 &amp; 94}} temperament, for which 166 is an excellent tuning through the [[19-limit]].
+edo is [[consistent]] through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it [[tempering out|tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the {{nowrap|72 & 94}} temperament, for which 166 is an excellent tuning through the [[19-limit]].
 edo (as 83edo) contains a very good approximation of the [[7/4|harmonic 7th]], of which it is only flat by 0.15121 cent.
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
-Since 166 factors into {{factorisation|166}}, 166edo contains [[2edo]] and [[83edo]] as subsets.
+Since 166 factors into 2 × 83, 166edo contains [[2edo]] and [[83edo]] as subsets.
 == Regular temperament properties ==