166edo: Difference between revisions

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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 about 7.23 [[cent]]s each, a size close to [[243/242]], the rastma.

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

166edo is [[consistent]] through the [[13-odd-limit]]. It has a flat tendency, with [[harmonic]]s 3 to 13 all tuned flat. Its principal 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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=== Prime harmonics ===

=== Octave stretch ===

166edo's approximated harmonics 3, 5, 7, 11, and 13 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[263edt]] or [[429ed6]].

=== Subsets and supersets ===

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

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

== Regular temperament properties ==

{~~{comma basis begin}}~~

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br />8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

Line 50:

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| 0.449

| 6.21

~~{{comma basis end}~~}

|}

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods<br />per 8ve

! Generator*

! Cents*

! Associated<br />ratio*

! Temperaments

|-

| 1

Line 78:

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| 17/15

| [[Wizard]] / gizzard

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-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 about 7.23 [[cent]]s each, a size close to [[243/242]], the rastma.
+{{ED intro}}
 == 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]], the equal temperament [[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 72 &amp; 94 temperament, for which 166 is an excellent tuning through the [[19-limit]].
+edo is [[consistent]] through the [[13-odd-limit]]. It has a flat tendency, with [[harmonic]]s 3 to 13 all tuned flat. Its principal 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 9: / Line 9: @@
 === Prime harmonics ===
 {{Harmonics in equal|166|intervals=prime}}
+=== Octave stretch ===
+edo's approximated harmonics 3, 5, 7, 11, and 13 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[263edt]] or [[429ed6]].
 === Subsets and supersets ===
-Since 166 = 2 × 83, 166edo contains [[2edo]] and [[83edo]] as subsets.
+Since 166 factors into primes as 2 × 83, 166edo contains [[2edo]] and [[83edo]] as subsets.
 == Regular temperament properties ==
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
@@ Line 50: / Line 62: @@
 | 0.449
 | 6.21
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
 |-
 | 1
@@ Line 78: / Line 97: @@
 | 17/15
 | [[Wizard]] / gizzard
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==