71edo: Difference between revisions
→Music: Insert Instruments section before this, now that a Lumatone mapping for 71edo is now available |
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== Theory == | == Theory == | ||
71edo is a [[dual-fifth]] system, with the flat fifth (which is near the fifths of [[26edo]] and [[45edo]]) [[support]]ing [[flattone]] temperament, and the sharp fifth (which is near [[22edo]]'s fifth) supporting [[superpyth]]. Unlike small dual-fifth systems such as [[18edo]], both fifths are close approximations of 3/2. | 71edo is a [[dual-fifth]] system, with the flat fifth (which is near the fifths of [[26edo]] and [[45edo]]) [[support]]ing [[flattone]] temperament with the 71c [[val]], and the sharp fifth (which is near [[22edo]]'s fifth) supporting [[superpyth]] with 71d. Unlike small dual-fifth systems such as [[18edo]], both fifths are close approximations of 3/2. | ||
Using the [[patent val]], the equal temperament [[tempering out|tempers out]] 20480/19683 and [[393216/390625]] in the [[5-limit]], [[875/864]], [[1029/1024]] and [[4000/3969]] in the [[7-limit]], [[100/99]] and [[245/242]] in the [[11-limit]], and [[91/90]] in the [[13-limit]]. In the 13-limit it supplies the optimal [[patent val]] for the 29 & 71 and 34 & 37 temperaments. | Using the [[patent val]], the equal temperament [[tempering out|tempers out]] 20480/19683 and [[393216/390625]] in the [[5-limit]], [[875/864]], [[1029/1024]] and [[4000/3969]] in the [[7-limit]], [[100/99]] and [[245/242]] in the [[11-limit]], and [[91/90]] in the [[13-limit]]. In the 13-limit it supplies the optimal [[patent val]] for the 29 & 71 and 34 & 37 temperaments. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
71edo is the 20th [[prime edo]], following [[67edo]] and before [[73edo]]. [[142edo]], which doubles it, provides correction for the harmonic 3. | 71edo is the 20th [[prime edo]], following [[67edo]] and before [[73edo]]. [[142edo]], which doubles it, provides correction for the harmonic 3. | ||
== Intervals == | == Intervals == | ||
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In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO. | In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this 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| 113 -71 }} | |||
| {{mapping| 71 113 }} | |||
| −2.49 | |||
| 2.49 | |||
| 14.72 | |||
|- | |||
| 2.3.5 | |||
| 20480/19683, 393216/390625 | |||
| {{mapping| 71 113 165 }} | |||
| −2.01 | |||
| 2.14 | |||
| 12.69 | |||
|- | |||
| 2.3.5.7 | |||
| 64/63, 245/243, 2200/2187 | |||
| {{mapping| 71 113 165 200 }} (71d) | |||
| −2.53 | |||
| 2.06 | |||
| 12.19 | |||
|} | |||
== Instruments == | == Instruments == | ||
A [[Lumatone mapping for 71edo]] is available. | A [[Lumatone mapping for 71edo]] is available. | ||
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; [[Bryan Deister]] | ; [[Bryan Deister]] | ||
* [https://www.youtube.com/shorts/nMMSQdIV30I ''71edo improv''] (2025) | * [https://www.youtube.com/shorts/nMMSQdIV30I ''71edo improv''] (2025) | ||
* [https://www.youtube.com/shorts/qeYTgpddpw0 ''Glaceir - Deltarune (microtonal cover in 71edo)''] (2026) | |||
* [https://www.youtube.com/shorts/QMvxgs2UBDM ''The Time - Dancing Line (microtonal cover in 71edo)''] (2026) | |||
* ''Waltz in 71edo'' (2026) | |||
** [https://www.youtube.com/shorts/vj2v8NFlrNo <nowiki>[short]</nowiki>] (with Lumatone view) | |||
** [https://www.youtube.com/watch?v=fT-3gSfxLX8 <nowiki>[full version]</nowiki>] | |||
; [[Francium]] | ; [[Francium]] | ||