401edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|401}} ==Theory== 401et tempers out 283115520/282475249 and 703125/702464 in the 7-limit; 35156250/35153041, 2097152/2096325, 117649/117612, 2264..." |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
==Theory== | |||
== Theory == | |||
It provides the optimal patent val for diatessic temperament. | 401edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. Otherwise, it has a reasonable approximation to harmonics [[5/1|5]], [[7/1|7]], [[9/1|9]], [[11/1|11]], and [[13/1|13]], making it suitable for a 2.9.5.7.11.13 [[subgroup]] interpretation. | ||
Using the [[patent val]] nonetheless, the equal temperament [[tempering out|tempers out]] 283115520/282475249 and [[703125/702464]] in the 7-limit; 35156250/35153041, 2097152/2096325, 117649/117612, 226492416/226474325, 9765625/9732096, 42875/42768, 1375/1372, [[5632/5625]], 15488/15435, 202397184/201768035, 102487/102400 and 805255/802816 in the 11-limit. It provides the [[optimal patent val]] for [[diatessic]], the {{nowrap|140 & 261}} temperament. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|401}} | {{Harmonics in equal|401}} | ||
==Scales== | |||
*[[Starlingtet11]] | === Subsets and supersets === | ||
*[[Diatessic27]] | 401edo is the 79th [[prime edo]] | ||
== 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| 636 -401 }} | |||
| {{mapping| 401 636 }} | |||
| −0.4060 | |||
| 0.4058 | |||
| 13.56 | |||
|- | |||
| 2.3.5 | |||
| 15625/15552, {{monzo| 107 -66 -1 }} | |||
| {{mapping| 401 636 931 }} | |||
| −0.2307 | |||
| 0.4139 | |||
| 13.83 | |||
|- | |||
| 2.3.5.7 | |||
| 10976/10935, 15625/15552, 67108864/66706983 | |||
| {{mapping| 401 636 931 1126 }} | |||
| −0.2400 | |||
| 0.3588 | |||
| 11.99 | |||
|- | |||
| 2.3.5.7.11 | |||
| 2200/2187, 1375/1372, 5632/5625, 102487/102400 | |||
| {{mapping| 401 636 931 1126 1387 }} | |||
| −0.1518 | |||
| 0.3661 | |||
| 12.23 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 325/324, 352/351, 625/624, 1375/1372, 3276800/3270267 | |||
| {{mapping| 401 636 931 1126 1387 1484 }} | |||
| −0.1431 | |||
| 0.3348 | |||
| 11.19 | |||
|} | |||
=== 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 | |||
| 169\401 | |||
| 505.74 | |||
| 75/56 | |||
| [[Diatessic]] | |||
|} | |||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
== Scales == | |||
* [[Starlingtet11]] | |||
* [[Diatessic27]] | |||
== Music == | |||
; [[Francium]] | |||
* [https://www.youtube.com/watch?v=hh-cgGe_zas ''Diana Tessa''] (2023) – diatessic in 403edo tuning | |||
[[Category:Listen]] | |||
[[Category:Diatessic]] | |||