91edo: Difference between revisions

{{EDO intro|91}}

The [[harmonic]]s [[3/1|3]], [[5/1|5]] and [[7/1|7]] for 91edo are on the flat side, making this a mostly flat system. The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit, [[225/224]] and [[4375/4374]] in the 7-limit, [[245/242]], [[385/384]] in the 11-limit, and [[105/104]], [[144/143]], [[196/195]] in the 13-limit. It provides the [[optimal patent val]] for 11- and 13-limit [[septimin]] temperament, and the 13-limit rank-3 [[tripod]] temperament, as well as the 11-limit rank-4 temperament tempering out 245/242 and the 13-limit rank-5 temperament tempering out 105/104, or rank-4 tempering out 105/104 and 144/143, or else 105/104 and 196/195 and hence 225/224 also.  

Using the 91c val, it is audibly indistinguishable from a closed system of [[1/7-comma meantone]], with a 5th only 0.018 cents sharper. The chromatic semitone in this scale corresponds to 135/128, the [[eigenmonzo|eigenmonzo (unchanged-interval)]] of [[1/7-comma meantone]]. Being 7 steps, what is also remarkable is that in this instance the chromatic semitone is equal to one step of [[13edo]]. Since 135/128 is also equal to 1/13 of the octave, the 91c [[val]] tempers out the [[aluminium comma]] in the 5-limit.  

The equal temperament also tempers out the {{monzo| -11 26 -13 }}, the tridecatonic comma, which assigns [[10/9]] to 2/13 of the octave, and it supports [[trideci]] in the 7-limit, tempering out 4375/4374 and 83349/81920. It supports a variant of [[semaphore]] temperament which tempers out the {{monzo| -42 23 2 }} comma in the 2.3.7 [[subgroup]], and is generated by a 19\91 generator. It is the second highest in a series of four consecutive edos that temper out [[quartisma]] ({{monzo| 24 -6 0 1 -5 }}), and as a corollary it is a tuning for the [[quartkeenlig]] temperament, which can also act as a [[23edo and octave stretching|stretched 23edo]]. In the 13-limit, it supports [[vidar]] and gives a reasonable tuning for its size.

| ''[[Semaphore]] variant'' (24 & 91) †

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

^&dagger; derived from scales in the Scales section, official name not decided upon yet.

[[User:Eliora|Eliora]], who believes the diatonic way of naming intervals in 91edo is not useful due to the fact that other temperaments and techniques for 91edo are more prominent, proposes a way of naming that merges the factors 7 and 13&mdash;7 equidistant notes are named do, re, mi, and 13 are named by some other virtue. The proposition is to use Old Slavic letter names, since no one uses them for naming or in mathematics. The {{nowrap|7 + 13}} naming convention can be called a duality notation. Intervals can be named through Latin ones for the 7-note scale, and Greek ones for the 13-note.

* ''Sadness - Nope'' (2022) &ndash; [https://mercuryamalgam.bandcamp.com/track/sadness-nope-the-molecular-agoge-pt-2 Bandcamp] | [https://www.youtube.com/watch?v=_5WS7AGZxm4 YouTube]

* ''DPRK ISON CHASE'' (2014) &ndash; [http://chrisvaisvil.com/dprk-ison-chase-12-of-91-edo-ambient/ blog] | [https://www.youtube.com/watch?v=StCR6hcm5tM YouTube]