197edo: Difference between revisions
m Infobox ET added |
+regular temperament properties |
||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|197}} | |||
==Theory== | |||
197edo gives excellent results for tuning both marvel, the planar temperament tempering out 225/224, and [[Kleismic_family|catakleismic]], the temperament [[tempering_out|tempering out]] both 225/224 and 4375/4374, which has wedgie <<6 5 22 -6 18 37||. Among [[Patent_val|patent vals]], in fact, it gives the best results for both. In fact, the [[11-limit]] patent val <197 312 457 553 682| has a [[comma basis]] [225/224, 4375/4374, 441/440, 65536/65219], so taking 225/224 and 441/440 together ([[Marvel family|prodigy temperament]]) also works well with 197edo, and taking 225/224, 441/440, and 4375/4374 (an alternative 11-limit catakleismic) is once again excellently tuned by 197edo. | |||
If we use 197e, the <197 312 457 553 681| val, we can also use 197edo as an excellent tuning for the 11-limit version of marvel temperament, tempering out 385/384 as well as 225/224. If we add 4375/4374 to the comma list for 11-limit [[Marvel family|marvel]], we get 11-limit catakleismic, and 197edo with the above val is also an excellent tuning for that. | If we use 197e, the <197 312 457 553 681| val, we can also use 197edo as an excellent tuning for the 11-limit version of marvel temperament, tempering out 385/384 as well as 225/224. If we add 4375/4374 to the comma list for 11-limit [[Marvel family|marvel]], we get 11-limit catakleismic, and 197edo with the above val is also an excellent tuning for that. | ||
===Odd harmonics=== | |||
{{Harmonics in equal|197}} | |||
===Subsets and supersets=== | |||
197edo is the 45th [[prime_numbers|prime]] edo. | |||
==Regular temperament properties== | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" |[[Subgroup]] | |||
! rowspan="2" |[[Comma list|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|-312 197}} | |||
|{{val|197 312}} | |||
| 0.4566 | |||
| 0.4568 | |||
| 7.50 | |||
|- | |||
|2.3.5 | |||
|15625/15552, {{monzo|-53 32 1}} | |||
|{{val|197 312 457}} | |||
| 0.6717 | |||
| 0.4813 | |||
| 7.90 | |||
|- | |||
|2.3.5.7 | |||
|225/224, 4375/4374, 15625/15552 | |||
|{{val|197 312 457 553}} | |||
| 0.5302 | |||
| 0.4834 | |||
| 7.94 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per 8ve | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
|1 | |||
|52\197 | |||
|316.75 | |||
|6/5 | |||
|[[Hanson]] / [[catakleismic]] | |||
|- | |||
|1 | |||
|53\197 | |||
|322.84 | |||
|3087/2560 | |||
|[[Seniority]] | |||
|} | |||
==Scales== | ==Scales== | ||