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176edo: Difference between revisions

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{{Infobox ET}}
{{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 ==
== 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 [[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 ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== 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 ==
== Regular temperament properties ==
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 279 -176 }}
| {{Monzo| 279 -176 }}
| {{mapping| 176 279 }}
| {{Mapping| 176 279 }}
| &minus;0.100
| −0.100
| 0.100
| 0.100
| 1.47
| 1.47
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| 2.3.5
| 2.3.5
| 78732/78125, {{monzo| 41 -20 -4 }}
| 78732/78125, {{monzo| 41 -20 -4 }}
| {{mapping| 176 279 409 }}
| {{Mapping| 176 279 409 }}
| &minus;0.400
| −0.400
| 0.432
| 0.432
| 6.34
| 6.34
Line 41: Line 41:
| 2.3.5.7
| 2.3.5.7
| 6144/6125, 10976/10935, 50421/50000
| 6144/6125, 10976/10935, 50421/50000
| {{mapping| 176 279 409 494 }}
| {{Mapping| 176 279 409 494 }}
| &minus;0.243
| −0.243
| 0.463
| 0.463
| 6.79
| 6.79
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| 2.3.5.7.11
| 2.3.5.7.11
| 441/440, 3388/3375, 6144/6125, 8019/8000
| 441/440, 3388/3375, 6144/6125, 8019/8000
| {{mapping| 176 279 409 494 609 }}
| {{Mapping| 176 279 409 494 609 }}
| &minus;0.250
| −0.250
| 0.414
| 0.414
| 6.08
| 6.08
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 351/350, 364/363, 441/440, 2197/2187, 3146/3125
| 351/350, 364/363, 441/440, 2197/2187, 3146/3125
| {{mapping| 176 279 409 494 609 651 }}
| {{Mapping| 176 279 409 494 609 651 }}
| &minus;0.123
| −0.123
| 0.473
| 0.473
| 6.93
| 6.93
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 565.91
| 565.91
| 13/9
| 13/9
| [[Tricot]] / [[trident]]
| [[Alphatrident]]
|-
|-
| 2
| 2
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| 20.45
| 20.45
| 81/80
| 81/80
| [[Commatic]]
| [[Bicommatic]]
|-
|-
| 2
| 2
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|-
|-
| 4
| 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]]
| [[Unlit]]
|-
|-
| 8
| 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]]
| [[Twilight]]
|-
|-
| 8
| 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)
| [[Octowerck]] (176f) / octowerckis (176)
|-
|-
| 11
| 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]]
| [[Hendecatonic]]
|-
|-
| 22
| 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]]
| [[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]]
[[Category:Countermiracle]]
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