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]], the equal temperament 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]].

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 & 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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| 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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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| 339.76

| 243/200

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

| [[Houborizic]]

|-

| 1

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| 585.54

| 7/5

| [[Merman]]

| [[Merman]] (7-limit)

|-

| 2

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| [[Wizard]] / gizzard

|}

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

== Scales ==

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[[Category:Wizard]]

~~[[Category:Gizzard]]~~

[[Category:Houborizic]]

[[Category:Marvel]]

@@ Line 3: / Line 3: @@
 == 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 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]], 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 (as 83edo) contains a very good approximation of the [[7/4|harmonic 7th]], of which it is only flat by 0.15121 cent.
@@ Line 26: / Line 26: @@
 | 2.3
 | {{monzo| -263 166 }}
-| [{{val| 166 263 }}]
+| {{mapping| 166 263 }}
 | +0.237
 | 0.237
@@ Line 33: / Line 33: @@
 | 2.3.5
 | 1600000/1594323, {{monzo| -31 2 12 }}
-| [{{val| 166 263 385 }}]
+| {{mapping| 166 263 385 }}
 | +0.615
 | 0.568
@@ Line 40: / Line 40: @@
 | 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 47: / Line 47: @@
 | 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 54: / Line 54: @@
 | 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 64: / Line 64: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 79: / Line 79: @@
 | 339.76
 | 243/200
-| [[Amity]] / [[houborizic]]
+| [[Houborizic]]
 |-
 | 1
@@ Line 85: / Line 85: @@
 | 585.54
 | 7/5
-| [[Merman]]
+| [[Merman]] (7-limit)
 |-
 | 2
@@ Line 93: / Line 93: @@
 | [[Wizard]] / gizzard
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 == Scales ==
@@ Line 106: / Line 107: @@
 [[Category:Wizard]]
-[[Category:Gizzard]]
 [[Category:Houborizic]]
 [[Category:Marvel]]