176edo: Difference between revisions

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~~The '''176 equal divisions of the octave''' ('''176edo'''), or the '''176(-tone) equal temperament''' ('''176tet''', '''176et''') when viewed from a [[regular temperament]] perspective, is the [[~~EDO~~|equal division of the octave]] into 176 parts of about 6.82 [[cent]]s each, a size close to [[243/242]], the rastma.~~

== Theory ==

176edo is [[consistent]] to the [[11-odd-limit]]. The equal temperament [[tempering out|tempers out]] 78732/78125 ([[sensipent comma]]) and {{monzo| 41 -20 -4 }} ([[undim comma]]) in the 5-limit; [[6144/6125]], [[10976/10935]], and [[50421/50000]] in the 7-limit; [[441/440]], [[3388/3375]], 6912/6875, [[8019/8000]], [[9801/9800]] and [[16384/16335]] in the 11-limit. Using the [[patent val]], [[351/350]], [[364/363]], [[2080/2079]], [[2197/2187]], and [[4096/4095]] in the 13-limit.

176edo is [[consistent]] to the [[11-odd-limit]]. The equal temperament [[tempering out|tempers out]] 78732/78125 ([[sensipent comma]]) and {{monzo| 41 -20 -4 }} ([[undim comma]]) in the 5-limit; [[6144/6125]], [[10976/10935]], and [[50421/50000]] in the 7-limit; [[441/440]], [[3388/3375]], 6912/6875, [[8019/8000]], [[9801/9800]], and [[16384/16335]] in the 11-limit. Using the [[patent val]], [[351/350]], [[364/363]], [[2080/2079]], [[2197/2187]], and [[4096/4095]] in the 13-limit.

176edo tempers the [[64/63|Archytas' comma]] to 1/44th of the octave (4 steps) and as a corollary supports the [[ruthenium]] temperament. It [[support]]s the [[bison]] temperament and the [[bicommatic]] temperament, and provides the [[optimal patent val]] for [[countermiracle]] in the 7- and 11-limit, and countermanna, one of the extensions, in the 13-limit.

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=== Subsets and supersets ===

Since 176 factors into ~~24 × 11~~, 176edo has subset edos {{EDOs| 2, 4, 8, 11, 16, 22, 44, and 88 }}.

Since 176 factors into {{Factorisation|176}}, 176edo has subset edos {{EDOs| 2, 4, 8, 11, 16, 22, 44, and 88 }}.

== Regular temperament properties ==

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! 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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| {{monzo| 279 -176 }}

| {{mapping| 176 279 }}

| ~~−0~~.100

| −0.100

| 0.100

| 1.47

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| 78732/78125, {{monzo| 41 -20 -4 }}

| {{mapping| 176 279 409 }}

| ~~−0~~.400

| −0.400

| 0.432

| 6.34

Line 42:

| 6144/6125, 10976/10935, 50421/50000

| {{mapping| 176 279 409 494 }}

| ~~−0~~.243

| −0.243

| 0.463

| 6.79

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| 441/440, 3388/3375, 6144/6125, 8019/8000

| {{mapping| 176 279 409 494 609 }}

| ~~−0~~.250

| −0.250

| 0.414

| 6.08

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| 351/350, 364/363, 441/440, 2197/2187, 3146/3125

| {{mapping| 176 279 409 494 609 651 }}

| ~~−0~~.123

| −0.123

| 0.473

| 6.93

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

|-

! Periods per 8ve

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Associated ratio*

! Temperaments

|-

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|-

| 4

| 73\176 (15\176)

| 73\176 (15\176)

| 497.73 (102.27)

| 497.73 (102.27)

| 4/3 (35/33)

| 4/3 (35/33)

| [[Undim]]

|-

| 8

| 73\176 (7\176)

| 73\176 (7\176)

| 497.73 (47.73)

| 497.73 (47.73)

| 4/3 (36/35)

| 4/3 (36/35)

| [[Twilight]]

|-

| 8

| 83\176 (5\176)

| 83\176 (5\176)

| 565.91 (34.09)

| 565.91 (34.09)

| 168/121 (55/54)

| 168/121 (55/54)

| [[Octowerck]] (176f) / octowerckis (176)

|-

| 11

| 73\176 (7\176)

| 73\176 (7\176)

| 497.73 (47.73)

| 497.73 (47.73)

| 4/3 (36/35)

| 4/3 (36/35)

| [[Hendecatonic]]

|-

| 22

| 73\176 (1\176)

| 73\176 (1\176)

| 497.73 (6.82)

| 497.73 (6.82)

| 4/3 (385/384)

| 4/3 (385/384)

| [[Icosidillic]] / [[major arcana]]

|}

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

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

[[Category:Countermiracle]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''176 equal divisions of the octave''' ('''176edo'''), or the '''176(-tone) equal temperament''' ('''176tet''', '''176et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 176 parts of about 6.82 [[cent]]s each, a size close to [[243/242]], the rastma.
+{{EDO intro}}
 == Theory ==
-edo is [[consistent]] to the [[11-odd-limit]]. The equal temperament [[tempering out|tempers out]] 78732/78125 ([[sensipent comma]]) and {{monzo| 41 -20 -4 }} ([[undim comma]]) in the 5-limit; [[6144/6125]], [[10976/10935]], and [[50421/50000]] in the 7-limit; [[441/440]], [[3388/3375]], 6912/6875, [[8019/8000]], [[9801/9800]] and [[16384/16335]] in the 11-limit. Using the [[patent val]], [[351/350]], [[364/363]], [[2080/2079]], [[2197/2187]], and [[4096/4095]] in the 13-limit.
+edo is [[consistent]] to the [[11-odd-limit]]. The equal temperament [[tempering out|tempers out]] 78732/78125 ([[sensipent comma]]) and {{monzo| 41 -20 -4 }} ([[undim comma]]) in the 5-limit; [[6144/6125]], [[10976/10935]], and [[50421/50000]] in the 7-limit; [[441/440]], [[3388/3375]], 6912/6875, [[8019/8000]], [[9801/9800]], and [[16384/16335]] in the 11-limit. Using the [[patent val]], [[351/350]], [[364/363]], [[2080/2079]], [[2197/2187]], and [[4096/4095]] in the 13-limit.
 edo tempers the [[64/63|Archytas' comma]] to 1/44th of the octave (4 steps) and as a corollary supports the [[ruthenium]] temperament. It [[support]]s the [[bison]] temperament and the [[bicommatic]] temperament, and provides the [[optimal patent val]] for [[countermiracle]] in the 7- and 11-limit, and countermanna, one of the extensions, in the 13-limit.
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
-Since 176 factors into 2<sup>4</sup> × 11, 176edo has subset edos {{EDOs| 2, 4, 8, 11, 16, 22, 44, and 88 }}.
+Since 176 factors into {{Factorisation|176}}, 176edo has subset edos {{EDOs| 2, 4, 8, 11, 16, 22, 44, and 88 }}.
 == Regular temperament properties ==
@@ Line 19: / Line 19: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 28: / Line 28: @@
 | {{monzo| 279 -176 }}
 | {{mapping| 176 279 }}
-| &minus;0.100
+| −0.100
 | 0.100
 | 1.47
@@ Line 35: / Line 35: @@
 | 78732/78125, {{monzo| 41 -20 -4 }}
 | {{mapping| 176 279 409 }}
-| &minus;0.400
+| −0.400
 | 0.432
 | 6.34
@@ Line 42: / Line 42: @@
 | 6144/6125, 10976/10935, 50421/50000
 | {{mapping| 176 279 409 494 }}
-| &minus;0.243
+| −0.243
 | 0.463
 | 6.79
@@ Line 49: / Line 49: @@
 | 441/440, 3388/3375, 6144/6125, 8019/8000
 | {{mapping| 176 279 409 494 609 }}
-| &minus;0.250
+| −0.250
 | 0.414
 | 6.08
@@ Line 56: / Line 56: @@
 | 351/350, 364/363, 441/440, 2197/2187, 3146/3125
 | {{mapping| 176 279 409 494 609 651 }}
-| &minus;0.123
+| −0.123
 | 0.473
 | 6.93
@@ Line 65: / Line 65: @@
 |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
-! Periods<br>per 8ve
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 114: / Line 114: @@
 |-
 | 4
-| 73\176<br>(15\176)
+| 73\176<br />(15\176)
-| 497.73<br>(102.27)
+| 497.73<br />(102.27)
-| 4/3<br>(35/33)
+| 4/3<br />(35/33)
 | [[Undim]]
 |-
 | 8
-| 73\176<br>(7\176)
+| 73\176<br />(7\176)
-| 497.73<br>(47.73)
+| 497.73<br />(47.73)
-| 4/3<br>(36/35)
+| 4/3<br />(36/35)
 | [[Twilight]]
 |-
 | 8
-| 83\176<br>(5\176)
+| 83\176<br />(5\176)
-| 565.91<br>(34.09)
+| 565.91<br />(34.09)
-| 168/121<br>(55/54)
+| 168/121<br />(55/54)
 | [[Octowerck]] (176f) / octowerckis (176)
 |-
 | 11
-| 73\176<br>(7\176)
+| 73\176<br />(7\176)
-| 497.73<br>(47.73)
+| 497.73<br />(47.73)
-| 4/3<br>(36/35)
+| 4/3<br />(36/35)
 | [[Hendecatonic]]
 |-
 | 22
-| 73\176<br>(1\176)
+| 73\176<br />(1\176)
-| 497.73<br>(6.82)
+| 497.73<br />(6.82)
-| 4/3<br>(385/384)
+| 4/3<br />(385/384)
 | [[Icosidillic]] / [[major arcana]]
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 [[Category:Countermiracle]]