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.

It [[support]]s the [[bison]] temperament and the [[~~commatic~~]] 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.

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.

=== Prime harmonics ===

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=== Subsets and supersets ===

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

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

== Regular temperament properties ==

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

|-

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

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

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

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

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

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

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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|-

| 2.3

| {{~~monzo~~| 279 -176 }}

| {{Monzo| 279 -176 }}

| {{~~mapping~~| 176 279 }}

| {{Mapping| 176 279 }}

| -0.100

| −0.100

| 0.100

| 1.47

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| 2.3.5

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

| {{~~mapping~~| 176 279 409 }}

| {{Mapping| 176 279 409 }}

| -0.400

| −0.400

| 0.432

| 6.34

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| 2.3.5.7

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

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

| {{Mapping| 176 279 409 494 }}

| -0.243

| −0.243

| 0.463

| 6.79

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| 2.3.5.7.11

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

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

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

| -0.250

| −0.250

| 0.414

| 6.08

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| 2.3.5.7.11.13

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

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

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

| -0.123

| −0.123

| 0.473

| 6.93

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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 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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| 565.91

| 13/9

| [[~~Tricot]] / [[trident~~]]

| [[Alphatrident]]

|-

| 2

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| 20.45

| 81/80

| [[~~Commatic~~]]

| [[Bicommatic]]

|-

| 2

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| 497.73 (102.27)

| 4/3 (35/33)

| [[~~Undim~~]]

| [[Unlit]]

|-

| 8

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

|}

<nowiki>*~~</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.
+{{ED 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.
-It [[support]]s the [[bison]] temperament and the [[commatic]] 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.
+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.
 === Prime harmonics ===
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
-Since 176 factors into {{factorization|176}}, 176edo has subset edos {{EDOs| 2, 4, 8, 11, 22, 44, and 88 }}.
+Since 176 factors into primes as {{nowrap| 2<sup>4</sup> × 11 }}, 176edo has subset edos {{EDOs| 2, 4, 8, 11, 16, 22, 44, and 88 }}.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 25: / Line 26: @@
 |-
 | 2.3
-| {{monzo| 279 -176 }}
+| {{Monzo| 279 -176 }}
-| {{mapping| 176 279 }}
+| {{Mapping| 176 279 }}
-| -0.100
+| −0.100
 | 0.100
 | 1.47
@@ Line 33: / Line 34: @@
 | 2.3.5
 | 78732/78125, {{monzo| 41 -20 -4 }}
-| {{mapping| 176 279 409 }}
+| {{Mapping| 176 279 409 }}
-| -0.400
+| −0.400
 | 0.432
 | 6.34
@@ Line 40: / Line 41: @@
 | 2.3.5.7
 | 6144/6125, 10976/10935, 50421/50000
-| {{mapping| 176 279 409 494 }}
+| {{Mapping| 176 279 409 494 }}
-| -0.243
+| −0.243
 | 0.463
 | 6.79
@@ Line 47: / Line 48: @@
 | 2.3.5.7.11
 | 441/440, 3388/3375, 6144/6125, 8019/8000
-| {{mapping| 176 279 409 494 609 }}
+| {{Mapping| 176 279 409 494 609 }}
-| -0.250
+| −0.250
 | 0.414
 | 6.08
@@ Line 54: / Line 55: @@
 | 2.3.5.7.11.13
 | 351/350, 364/363, 441/440, 2197/2187, 3146/3125
-| {{mapping| 176 279 409 494 609 651 }}
+| {{Mapping| 176 279 409 494 609 651 }}
-| -0.123
+| −0.123
 | 0.473
 | 6.93
@@ Line 62: / Line 63: @@
 === 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 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 97: / Line 99: @@
 | 565.91
 | 13/9
-| [[Tricot]] / [[trident]]
+| [[Alphatrident]]
 |-
 | 2
@@ Line 103: / Line 105: @@
 | 20.45
 | 81/80
-| [[Commatic]]
+| [[Bicommatic]]
 |-
 | 2
@@ Line 115: / Line 117: @@
 | 497.73<br>(102.27)
 | 4/3<br>(35/33)
-| [[Undim]]
+| [[Unlit]]
 |-
 | 8
@@ Line 141: / Line 143: @@
 | [[Icosidillic]] / [[major arcana]]
 |}
-<nowiki>*</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]]