401edo: Difference between revisions
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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 == | ||
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. | 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 140 & 261 temperament. | 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 === | === Odd harmonics === | ||
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== 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]] | ||
! 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]] (¢) | ||
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| {{monzo| 636 -401 }} | | {{monzo| 636 -401 }} | ||
| {{mapping| 401 636 }} | | {{mapping| 401 636 }} | ||
| | | −0.4060 | ||
| 0.4058 | | 0.4058 | ||
| 13.56 | | 13.56 | ||
| Line 34: | Line 35: | ||
| 15625/15552, {{monzo| 107 -66 -1 }} | | 15625/15552, {{monzo| 107 -66 -1 }} | ||
| {{mapping| 401 636 931 }} | | {{mapping| 401 636 931 }} | ||
| | | −0.2307 | ||
| 0.4139 | | 0.4139 | ||
| 13.83 | | 13.83 | ||
| Line 41: | Line 42: | ||
| 10976/10935, 15625/15552, 67108864/66706983 | | 10976/10935, 15625/15552, 67108864/66706983 | ||
| {{mapping| 401 636 931 1126 }} | | {{mapping| 401 636 931 1126 }} | ||
| | | −0.2400 | ||
| 0.3588 | | 0.3588 | ||
| 11.99 | | 11.99 | ||
| Line 48: | Line 49: | ||
| 2200/2187, 1375/1372, 5632/5625, 102487/102400 | | 2200/2187, 1375/1372, 5632/5625, 102487/102400 | ||
| {{mapping| 401 636 931 1126 1387 }} | | {{mapping| 401 636 931 1126 1387 }} | ||
| | | −0.1518 | ||
| 0.3661 | | 0.3661 | ||
| 12.23 | | 12.23 | ||
| Line 55: | Line 56: | ||
| 325/324, 352/351, 625/624, 1375/1372, 3276800/3270267 | | 325/324, 352/351, 625/624, 1375/1372, 3276800/3270267 | ||
| {{mapping| 401 636 931 1126 1387 1484 }} | | {{mapping| 401 636 931 1126 1387 1484 }} | ||
| | | −0.1431 | ||
| 0.3348 | | 0.3348 | ||
| 11.19 | | 11.19 | ||
| Line 62: | Line 63: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | 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 75: | Line 77: | ||
| [[Diatessic]] | | [[Diatessic]] | ||
|} | |} | ||
<nowiki>* | <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 == | == Scales == | ||
| Line 82: | Line 84: | ||
== Music == | == Music == | ||
; [[ | ; [[Francium]] | ||
* [https://www.youtube.com/watch?v=hh-cgGe_zas ''Diana Tessa''] (2023) – diatessic in 403edo tuning | * [https://www.youtube.com/watch?v=hh-cgGe_zas ''Diana Tessa''] (2023) – diatessic in 403edo tuning | ||
[[Category:Listen]] | |||
[[Category:Diatessic]] | |||