419edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|419}}. ==Theory== 419et tempers out 32805/32768, 29360128/29296875, 1959552/1953125, 420175/419904 and 2100875/2097152 in the 7-limit and 3..." |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
==Theory== | |||
== Theory == | |||
419edo is a decent 7-limit system, and is [[consistent]] to the [[9-odd-limit]]. The equal temperament [[tempering out|tempers out]] 32805/32768 ([[schisma]]) in the 5-limit; 235298/234375 (triwellisma), 420175/419904 (wizma) in the 7-limit. It [[support]]s and provides the [[optimal patent val]] for [[sextilifourths]], the {{nowrap|130 & 289}} temperament, in the 7-limit. | |||
Extending it to the 11-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the 419e [[val]], it tempers out [[3025/3024]], [[5632/5625]], and [[8019/8000]]. Using the [[patent val]], it tempers out [[441/440]], [[4000/3993]], and 14700/14641 in the 11-limit. The patent val supports 11-limit sextilifourths, though [[289edo]] is better suited for that purpose. | |||
The same can be said of the mapping for 13, with the 419e val tempering out [[676/675]], [[1716/1715]], [[4225/4224]], and 4459/4455, and the 419f val tempering out [[729/728]], [[2200/2197]], 2205/2197, 3584/3575, and 4459/4455. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|419}} | |||
=== Subsets and supersets === | |||
419edo is the 81th [[prime edo]]. | 419edo is the 81th [[prime edo]]. | ||
{{ | |||
==Scales== | == 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| -664 419 }} | |||
| {{mapping| 419 664 }} | |||
| +0.0897 | |||
| 0.0897 | |||
| 3.13 | |||
|- | |||
| 2.3.5 | |||
| 32805/32768, {{monzo| 41 43 -47 }} | |||
| {{mapping| 419 664 973 }} | |||
| +0.0137 | |||
| 0.1301 | |||
| 4.54 | |||
|- | |||
| 2.3.5.7 | |||
| 32805/32768, 235298/234375, 420175/419904 | |||
| {{mapping| 419 664 973 1176 }} | |||
| +0.0821 | |||
| 0.1635 | |||
| 5.71 | |||
|- | |||
| 2.3.5.7.11 | |||
| 441/440, 4000/3993, 32805/32768, 420175/419904 | |||
| {{mapping| 419 664 973 1176 1450 }} (419) | |||
| −0.0168 | |||
| 0.2460 | |||
| 8.59 | |||
|} | |||
=== 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 | |||
| 29\419 | |||
| 83.05 | |||
| 21/20 | |||
| [[Sextilifourths]] (419f) | |||
|- | |||
| 1 | |||
| 49\419 | |||
| 140.33 | |||
| 243/224 | |||
| [[Tsaharuk]] (7-limit) | |||
|- | |||
| 1 | |||
| 174\419 | |||
| 498.33 | |||
| 162/125 | |||
| [[Helmholtz (temperament)|Helmholtz]] | |||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Scales == | |||
* [[Sextilifourths13]] | |||
== Music == | |||
; [[Francium]] | |||
* [https://www.youtube.com/watch?v=cXvlQxvwUIM ''Cultural Appropriation?''] (2023) | |||
[[Category:Listen]] | |||
[[Category:Sextilifourths]] | |||