166edo: Difference between revisions

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The '''166 equal ~~temperament~~''' (~~in short~~ 166-[[~~EDO~~]]) divides the [[octave]] into 166 equal steps of size 7.229 [[cent]]s each. Its principle interest lies in the usefulness of its approximations; it tempers out 1600000/1594323, 225/224, 385/384, 540/539, 4000/3993, 325/324 and 729/728. It is an excellent tuning for the rank three temperament [[Marvel|marvel]], in both the [[11-limit]] and in the 13-limit extension [[Marvel_family#Hecate|hecate]], and the rank two temperament wizard, which also tempers out 4000/3993, 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 [[Marvel temperaments|gizzard]], the 72&94 temperament, for which 166 is an excellent tuning through the [[19-limit]].

The '''166 equal divisions of the octave''' ('''166edo'''), or the '''166(-tone) equal temperament''' ('''166tet''', '''166et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 166 equal steps of size 7.229 [[cent]]s each.

~~Its prime factorization~~ is ~~166 =~~ [[~~2edo|~~2]] * [[~~83edo|83~~]].

== Theory ==

The principle interest of 166edo lies in the usefulness of its approximations; it tempers out [[1600000/1594323]], [[225/224]], [[385/384]], [[540/539]], [[4000/3993]], [[325/324]] and [[729/728]]. It is an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], and the [[rank-2 temperament]] [[wizard]], which also tempers out 4000/3993, 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 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]]. It's 0.15121 cent flat of the just interval 7:4.

Its prime factorization is 166 = [[2edo|2]] × [[83edo|83]].

166edo (as 83edo) contains a very good approximation of the [[7/4|harmonic 7th]]. It is 0.15121 cent flat of the just interval 7:4.

=== Prime harmonics ===

== Scales ==

* [[prisun]]

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-The '''166 equal temperament''' (in short 166-[[EDO]]) divides the [[octave]] into 166 equal steps of size 7.229 [[cent]]s each. Its principle interest lies in the usefulness of its approximations; it tempers out 1600000/1594323, 225/224, 385/384, 540/539, 4000/3993, 325/324 and 729/728. It is an excellent tuning for the rank three temperament [[Marvel|marvel]], in both the [[11-limit]] and in the 13-limit extension [[Marvel_family#Hecate|hecate]], and the rank two temperament wizard, which also tempers out 4000/3993, 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 [[Marvel temperaments|gizzard]], the 72&amp;94 temperament, for which 166 is an excellent tuning through the [[19-limit]].
+The '''166 equal divisions of the octave''' ('''166edo'''), or the '''166(-tone) equal temperament''' ('''166tet''', '''166et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 166 equal steps of size 7.229 [[cent]]s each.
-Its prime factorization is 166 = [[2edo|2]] * [[83edo|83]].
+== Theory ==
+The principle interest of 166edo lies in the usefulness of its approximations; it tempers out [[1600000/1594323]], [[225/224]], [[385/384]], [[540/539]], [[4000/3993]], [[325/324]] and [[729/728]]. It is an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], and the [[rank-2 temperament]] [[wizard]], which also tempers out 4000/3993, 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 72&amp;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]]. It's 0.15121 cent flat of the just interval 7:4.
+Its prime factorization is 166 = [[2edo|2]] × [[83edo|83]].
+edo (as 83edo) contains a very good approximation of the [[7/4|harmonic 7th]]. It is 0.15121 cent flat of the just interval 7:4.
+=== Prime harmonics ===
+{{Primes in edo|166}}
 == Scales ==
 * [[prisun]]