164edo: Difference between revisions
m Categories |
Cleanup; +prime error table; +RTT table and rank-2 temperaments |
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The ''164 equal division'' divides the octave into 164 equal parts of 7.317 cents each. In the 5-limit it tempers out the würschmidt comma, 393216/390625, and supplies the [[ | The ''164 equal division'' divides the octave into 164 equal parts of 7.317 cents each. In the 5-limit it tempers out the [[würschmidt comma]], 393216/390625, and supplies the [[optimal patent val]] for the [[würschmidt]] temperament. In higher limits, also supplies the optimal patent val for the 7-limit, 1/41 octave period 41&123 temperament, and the 13-limit the [[Gamelismic family #Portent|momentous]] temperament, the rank-3 temperament tempering out 196/195, 352/351, 385/384 and 441/440. | ||
164 = 4 | 164 = 4 × 41, with divisors 2, 4, 41, 82 | ||
=== Prime harmonics === | |||
{{Harmonics in equal|164}} | |||
== 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]] (%) | |||
|- | |||
| 2.3.5 | |||
| 393216/390625, {{monzo| 24 -21 4 }} | |||
| [{{val| 164 260 381 }}] | |||
| -0.316 | |||
| 0.262 | |||
| 3.58 | |||
|- | |||
| 2.3.5.13 | |||
| 676/675, 256000/255879, 393216/390625 | |||
| [{{val| 164 260 381 607 }}] | |||
| -0.300 | |||
| 0.229 | |||
| 3.13 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per Otave | |||
! Generator<br>(Reduced) | |||
! Cents<br>(Reduced) | |||
! Associated<br>Ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 47\164 | |||
| 343.90 | |||
| 8000/6561 | |||
| [[Geb]] | |||
|- | |||
| 1 | |||
| 49\164 | |||
| 358.54 | |||
| 16/13 | |||
| [[Restles]] (164) | |||
|- | |||
| 1 | |||
| 53\164 | |||
| 387.80 | |||
| 5/4 | |||
| [[Würschmidt]] | |||
|- | |||
| 1 | |||
| 53\164 | |||
| 475.61 | |||
| 320/243 | |||
| [[Vulture]] | |||
|- | |||
| 1 | |||
| 69\164 | |||
| 504.88 | |||
| 104976/78125 | |||
| [[Countermeantone]] | |||
|- | |||
| 2 | |||
| 17\164 | |||
| 124.39 | |||
| 275/256 | |||
| [[Semivulture]] (164) | |||
|- | |||
| 4 | |||
| 68\164<br>(14\164) | |||
| 497.56<br>(102.44) | |||
| 4/3<br>(35/33) | |||
| [[Undim]] (164deff) / [[unlit]] (164f) | |||
|- | |||
| 41 | |||
| 53\164<br>(1\164) | |||
| 387.80<br>(7.32) | |||
| 5/4<br>(32805/32768) | |||
| [[Counterpyth]] | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Würschmidt]] | |||