166edo: Difference between revisions

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{{Infobox ET

~~| Prime factorization = 2 × 83~~

~~| Step size = 7.22892¢~~

~~| Fifth = 97\166 (701.20¢)~~

~~| Major 2nd = 28\166 (202¢)~~

~~| Minor 2nd = 13\166 (94¢)~~

~~| Augmented 1sn = 15\166 (108¢)~~

}}

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]], 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~~]].~~

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]].

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

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 ===

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

== Regular temperament properties ==

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

! rowspan="2" | Subgroup

|-

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

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

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

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

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| 2.3

| {{monzo| -263 166 }}

| [{{~~val~~| 166 263 }}]

| {{mapping| 166 263 }}

| +0.237

| 0.237

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| 2.3.5

| 1600000/1594323, {{monzo| -31 2 12 }}

| [{{~~val~~| 166 263 385 }}]

| {{mapping| 166 263 385 }}

| +0.615

| 0.568

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| 2.3.5.7

| 225/224, 118098/117649, 1250000/1240029

| [{{~~val~~| 166 263 385 466 }}]

| {{mapping| 166 263 385 466 }}

| +0.474

| 0.549

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| 2.3.5.7.11

| 225/224, 385/384, 4000/3993, 322102/321489

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

| {{mapping| 166 263 385 466 574 }}

| +0.490

| 0.492

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| 2.3.5.7.11.13

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

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

| {{mapping| 166 263 385 466 574 614 }}

| +0.498

| 0.449

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=== Rank-2 temperaments ===

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

|+Table of rank-2 temperaments by generator

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

! Periods per ~~octave~~

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

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| 339.76

| 243/200

| [[~~Amity]] / [[houborizic~~]]

| [[Houborizic]]

|-

| 1

Line 91:

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| 585.54

| 7/5

| [[Merman]]

| [[Merman]] (7-limit)

|-

| 2

Line 99:

Line 98:

| [[Wizard]] / gizzard

|}

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

== Scales ==

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* [[Fifteentofourteen]]

~~[[Category:Theory]]~~

~~[[Category:Equal divisions of the octave]]~~

~~[[Category:166edo| ]] ~~

[[Category:Wizard]]

~~[[Category:Gizzard]]~~

[[Category:Houborizic]]

[[Category:Marvel]]

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 2 × 83
+{{ED intro}}
-| Step size = 7.22892¢
-| Fifth = 97\166 (701.20¢)
-| Major 2nd = 28\166 (202¢)
-| Minor 2nd = 13\166 (94¢)
-| Augmented 1sn = 15\166  (108¢)
-}}
-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 ==
-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]].
+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]].
-Its prime factorization is 166 = [[2edo|2]] × [[83edo|83]].
 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 ===
-{{Primes in edo|166}}
+{{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 factors into primes as 2 × 83, 166edo contains [[2edo]] and [[83edo]] as subsets.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+|-
+! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 32: / Line 30: @@
 | 2.3
 | {{monzo| -263 166 }}
-| [{{val| 166 263 }}]
+| {{mapping| 166 263 }}
 | +0.237
 | 0.237
@@ Line 39: / Line 37: @@
 | 2.3.5
 | 1600000/1594323, {{monzo| -31 2 12 }}
-| [{{val| 166 263 385 }}]
+| {{mapping| 166 263 385 }}
 | +0.615
 | 0.568
@@ Line 46: / Line 44: @@
 | 2.3.5.7
 | 225/224, 118098/117649, 1250000/1240029
-| [{{val| 166 263 385 466 }}]
+| {{mapping| 166 263 385 466 }}
 | +0.474
 | 0.549
@@ Line 53: / Line 51: @@
 | 2.3.5.7.11
 | 225/224, 385/384, 4000/3993, 322102/321489
-| [{{val| 166 263 385 466 574 }}]
+| {{mapping| 166 263 385 466 574 }}
 | +0.490
 | 0.492
@@ Line 60: / Line 58: @@
 | 2.3.5.7.11.13
 | 225/224, 325/324, 385/384, 1573/1568, 2200/2197
-| [{{val| 166 263 385 466 574 614 }}]
+| {{mapping| 166 263 385 466 574 614 }}
 | +0.498
 | 0.449
@@ Line 68: / Line 66: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per octave
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 85: / Line 84: @@
 | 339.76
 | 243/200
-| [[Amity]] / [[houborizic]]
+| [[Houborizic]]
 |-
 | 1
@@ Line 91: / Line 90: @@
 | 585.54
 | 7/5
-| [[Merman]]
+| [[Merman]] (7-limit)
 |-
 | 2
@@ Line 99: / Line 98: @@
 | [[Wizard]] / gizzard
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==
@@ Line 111: / Line 111: @@
 * [[Fifteentofourteen]]
-[[Category:Theory]]
-[[Category:Equal divisions of the octave]]
-[[Category:166edo| ]] <!-- main article -->
 [[Category:Wizard]]
-[[Category:Gizzard]]
 [[Category:Houborizic]]
 [[Category:Marvel]]