176edo: Difference between revisions
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== Theory == | == Theory == | ||
176edo is [[consistent]] to the [[11-odd-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 | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 23: | Line 26: | ||
| 2.3 | | 2.3 | ||
| {{monzo| 279 -176 }} | | {{monzo| 279 -176 }} | ||
| | | {{mapping| 176 279 }} | ||
| -0.100 | | -0.100 | ||
| 0.100 | | 0.100 | ||
| Line 30: | Line 33: | ||
| 2.3.5 | | 2.3.5 | ||
| 78732/78125, {{monzo| 41 -20 -4 }} | | 78732/78125, {{monzo| 41 -20 -4 }} | ||
| | | {{mapping| 176 279 409 }} | ||
| -0.400 | | -0.400 | ||
| 0.432 | | 0.432 | ||
| Line 37: | Line 40: | ||
| 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 }} | ||
| -0.243 | | -0.243 | ||
| 0.463 | | 0.463 | ||
| Line 44: | Line 47: | ||
| 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 }} | ||
| -0.250 | | -0.250 | ||
| 0.414 | | 0.414 | ||
| Line 51: | Line 54: | ||
| 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 }} | ||
| -0.123 | | -0.123 | ||
| 0.473 | | 0.473 | ||
| Line 61: | Line 64: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br> | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 138: | Line 141: | ||
| [[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:Countermiracle]] | [[Category:Countermiracle]] | ||