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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 6.8182 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
176edo is [[consistent]] to the [[11-odd-limit]], tempering out 78732/78125 ([[sensipent comma]]) and 2199023255552/2179240250625 ([[undim comma]]) in the 5-limit; [[6144/6125]], [[10976/10935]], and 50421/50000 in the 7-limit; [[441/440]], 3388/3375, 6912/6875, and 8019/8000 in the 11-limit, supporting the [[bison]] temperament and the [[commatic]] temperament.
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.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|176}}
{{Harmonics in equal|176}}
 
=== Subsets and supersets ===
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]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 279 -176 }}
| {{Mapping| 176 279 }}
| −0.100
| 0.100
| 1.47
|-
| 2.3.5
| 78732/78125, {{monzo| 41 -20 -4 }}
| {{Mapping| 176 279 409 }}
| −0.400
| 0.432
| 6.34
|-
| 2.3.5.7
| 6144/6125, 10976/10935, 50421/50000
| {{Mapping| 176 279 409 494 }}
| −0.243
| 0.463
| 6.79
|-
| 2.3.5.7.11
| 441/440, 3388/3375, 6144/6125, 8019/8000
| {{Mapping| 176 279 409 494 609 }}
| −0.250
| 0.414
| 6.08
|-
| 2.3.5.7.11.13
| 351/350, 364/363, 441/440, 2197/2187, 3146/3125
| {{Mapping| 176 279 409 494 609 651 }}
| −0.123
| 0.473
| 6.93
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
| 1
| 17\176
| 115.91
| 77/72
| [[Mercy]] / [[countermiracle]] / counterbenediction / countermanna
|-
| 1
| 35\176
| 238.64
| 147/128
| [[Tokko]]
|-
| 1
| 65\176
| 443.18
| 162/125
| [[Sensipent]]
|-
| 1
| 73\176
| 497.73
| 4/3
| [[Gary]] / [[cotoneum]]
|-
| 1
| 83\176
| 565.91
| 13/9
| [[Alphatrident]]
|-
| 2
| 23\176
| 20.45
| 81/80
| [[Bicommatic]]
|-
| 2
| 23\176
| 156.82
| 35/32
| [[Bison]]
|-
| 4
| 73\176<br>(15\176)
| 497.73<br>(102.27)
| 4/3<br>(35/33)
| [[Unlit]]
|-
| 8
| 73\176<br>(7\176)
| 497.73<br>(47.73)
| 4/3<br>(36/35)
| [[Twilight]]
|-
| 8
| 83\176<br>(5\176)
| 565.91<br>(34.09)
| 168/121<br>(55/54)
| [[Octowerck]] (176f) / octowerckis (176)
|-
| 11
| 73\176<br>(7\176)
| 497.73<br>(47.73)
| 4/3<br>(36/35)
| [[Hendecatonic]]
|-
| 22
| 73\176<br>(1\176)
| 497.73<br>(6.82)
| 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 distinct


[[Category:Equal divisions of the octave]]
[[Category:Countermiracle]]
Retrieved from "https://en.xen.wiki/w/176edo"