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== Theory ==
== Theory ==
176edo is [[consistent]] to the [[11-odd-limit]], tempering 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.  
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.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|176}}  
{{Harmonics in equal|176}}
 
=== Subsets and supersets ===
Since 176 factors into {{factorization|176}}, 176edo has subset edos {{EDOs| 2, 4, 8, 11, 22, 44, and 88 }}.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 2.3
| 2.3
| {{monzo| 279 -176 }}
| {{monzo| 279 -176 }}
| [{{val| 176 279 }}]
| {{mapping| 176 279 }}
| -0.100
| -0.100
| 0.100
| 0.100
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| 2.3.5
| 2.3.5
| 78732/78125, {{monzo| 41 -20 -4 }}
| 78732/78125, {{monzo| 41 -20 -4 }}
| [{{val| 176 279 409 }}]
| {{mapping| 176 279 409 }}
| -0.400
| -0.400
| 0.432
| 0.432
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| 2.3.5.7
| 2.3.5.7
| 6144/6125, 10976/10935, 50421/50000
| 6144/6125, 10976/10935, 50421/50000
| [{{val| 176 279 409 494 }}]
| {{mapping| 176 279 409 494 }}
| -0.243
| -0.243
| 0.463
| 0.463
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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
| [{{val| 176 279 409 494 609 }}]
| {{mapping| 176 279 409 494 609 }}
| -0.250
| -0.250
| 0.414
| 0.414
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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
| [{{val| 176 279 409 494 609 651 }}]
| {{mapping| 176 279 409 494 609 651 }}
| -0.123
| -0.123
| 0.473
| 0.473
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Icosidillic]] / [[major arcana]]
| [[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


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Countermiracle]]
[[Category:Countermiracle]]

Revision as of 09:51, 25 April 2024

← 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

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 equal division of the octave into 176 parts of about 6.82 cents each, a size close to 243/242, the rastma.

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.

It supports 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.

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 24 × 11, 176edo has subset edos 2, 4, 8, 11, 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 Tricot / trident
2 23\176 20.45 81/80 Commatic
2 23\176 156.82 35/32 Bison
4 73\176
(15\176) 		497.73
(102.27) 		4/3
(35/33) 		Undim
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 it is distinct

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