166edo: Difference between revisions

Line 2:

== 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 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 72&94 temperament, for which 166 is an excellent tuning through the [[19-limit]].

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~~.

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

=== Prime harmonics ===

Line 51:

|-

| 2.3.5.7.11.13

| 225/224, 325/324, 385/384, ~~1575~~/~~1573~~, 2200/2197

| 225/224, 325/324, 385/384, 1573/1568, 2200/2197

| [{{val| 166 263 385 466 574 614 }}]

| +0.498

Line 98:

[[Category:Equal divisions of the octave]]

[[Category:166edo| ]]

[[Category:Wizard]]

[[Category:Gizzard]]

[[Category:Houborizic]]

[[Category:Marvel]]

~~[[Category:Wizard]]~~

@@ Line 2: / Line 2: @@
 == 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 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 72&amp;94 temperament, for which 166 is an excellent tuning through the [[19-limit]].
 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.
+edo (as 83edo) contains a very good approximation of the [[7/4|harmonic 7th]], of which it is only flat by 0.15121 cent.
 === Prime harmonics ===
@@ Line 51: / Line 51: @@
 |-
 | 2.3.5.7.11.13
-| 225/224, 325/324, 385/384, 1575/1573, 2200/2197
+| 225/224, 325/324, 385/384, 1573/1568, 2200/2197
 | [{{val| 166 263 385 466 574 614 }}]
 | +0.498
@@ Line 98: / Line 98: @@
 [[Category:Equal divisions of the octave]]
 [[Category:166edo| ]] <!-- main article -->
+[[Category:Wizard]]
 [[Category:Gizzard]]
+[[Category:Houborizic]]
 [[Category:Marvel]]
-[[Category:Wizard]]