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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]]

Latest revision as of 14:02, 16 March 2025

← 175edo 176edo 177edo →
Prime factorization 24 × 11
Step size 6.81818 ¢ 
Fifth 103\176 (702.273 ¢)
Semitones (A1:m2) 17:13 (115.9 ¢ : 88.64 ¢)
Consistency limit 11
Distinct consistency limit 11

176 equal divisions of the octave (abbreviated 176edo or 176ed2), also called 176-tone equal temperament (176tet) or 176 equal temperament (176et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 176 equal parts of about 6.82 ¢ each. Each step represents a frequency ratio of 21/176, or the 176th root of 2.

Theory

176edo is consistent to the 11-odd-limit. The equal temperament tempers out 78732/78125 (sensipent comma) and [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 Archytas' comma to 1/44th of the octave (4 steps) and as a corollary supports the ruthenium temperament. It supports 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

Approximation of prime harmonics in 176edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.32 +2.32 -0.64 +0.95 -1.89 -2.68 +2.49 -1.00 -0.03 +0.42
Relative (%) +0.0 +4.7 +34.1 -9.4 +14.0 -27.7 -39.3 +36.5 -14.7 -0.5 +6.1
Steps
(reduced) 		176
(0) 		279
(103) 		409
(57) 		494
(142) 		609
(81) 		651
(123) 		719
(15) 		748
(44) 		796
(92) 		855
(151) 		872
(168)

Subsets and supersets

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

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [279 -176⟩ [⟨176 279]] −0.100 0.100 1.47
2.3.5 78732/78125, [41 -20 -4⟩ [⟨176 279 409]] −0.400 0.432 6.34
2.3.5.7 6144/6125, 10976/10935, 50421/50000 [⟨176 279 409 494]] −0.243 0.463 6.79
2.3.5.7.11 441/440, 3388/3375, 6144/6125, 8019/8000 [⟨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 [⟨176 279 409 494 609 651]] −0.123 0.473 6.93

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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
(15\176) 		497.73
(102.27) 		4/3
(35/33) 		Unlit
8 73\176
(7\176) 		497.73
(47.73) 		4/3
(36/35) 		Twilight
8 83\176
(5\176) 		565.91
(34.09) 		168/121
(55/54) 		Octowerck (176f) / octowerckis (176)
11 73\176
(7\176) 		497.73
(47.73) 		4/3
(36/35) 		Hendecatonic
22 73\176
(1\176) 		497.73
(6.82) 		4/3
(385/384) 		Icosidillic / major arcana

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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