91edo: Difference between revisions

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~~The '''~~91 equal divisions of the octave''' ('''91edo'''), or '''91-tone equal temperament''' ('''91tet''', '''91et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 91 parts of 13.187 [[cent]]s each.

== Theory ==

The 3, 5 and 7 for 91 are on the flat side, making this a mostly flat system. It provides the [[optimal patent val]] for 11- and 13-limit [[septimin]] temperament, and the 13-limit rank ~~three~~ [[tripod]] temperament, as well as the 11-limit rank ~~four~~ temperament tempering out [[245/242]] and the 13-limit rank ~~five~~ temperament tempering out [[105/104]], or rank ~~four~~ tempering out 105/104 and [[144/143]], or else 105/104 and [[196/195]] and hence [[225/224]] also. It ~~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. 91edo~~ supports a variant of [[semaphore]] temperament which tempers out the ~~4613015762523/4398046511104~~ comma in the 2.3.7 subgroup, and is ~~produced~~ by a 19\91 generator.

The 3, 5 and 7 for 91 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. 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.

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 also corresponds to 135/128, the [[eigenmonzo]] (unchanged-interval) of [[1/7-comma meantone]]~~, both being~~ 7 steps~~. What's~~ also remarkable is that in this instance the chromatic semitone is equal to one step of [[13edo]]. Since 135/128 is also equal to 1/13th of the octave, 91c val tempers out the [[aluminium comma]] in the 5-limit.

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 also 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 1/13th of the octave, 91c val tempers out the [[aluminium comma]] in the 5-limit.

91edo also tempers out the ~~[[Decaononic~~|~~devil's tridecalimma]]~~, which assigns 10/9 to 2/~~13ths~~ of the octave. It also supports the [[trideci]] temperament, which in the 7-limit tempers out 4375/4374 and 83349/81920.

91edo also tempers out the {{monzo| -11 26 -13 }}, the tridecatonic comma, which assigns [[10/9]] to 2/13 of the octave. It also supports the [[trideci]] temperament, which in the 7-limit tempers out 4375/4374 and 83349/81920.

It is the second highest it 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]].

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 {{Infobox ET}}
-The '''91 equal divisions of the octave''' ('''91edo'''), or '''91-tone equal temperament''' ('''91tet''', '''91et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 91 parts of 13.187 [[cent]]s each.
+{{EDO intro|91}}
 == Theory ==
-The 3, 5 and 7 for 91 are on the flat side, making this a mostly flat system. It provides the [[optimal patent val]] for 11- and 13-limit [[septimin]] temperament, and the 13-limit rank three [[tripod]] temperament, as well as the 11-limit rank four temperament tempering out [[245/242]] and the 13-limit rank five temperament tempering out [[105/104]], or rank four tempering out 105/104 and [[144/143]], or else 105/104 and [[196/195]] and hence [[225/224]] also. It 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. 91edo supports a variant of [[semaphore]] temperament which tempers out the 4613015762523/4398046511104 comma in the 2.3.7 subgroup, and is produced by a 19\91 generator.
+The 3, 5 and 7 for 91 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. 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.
-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 also corresponds to 135/128, the [[eigenmonzo]] (unchanged-interval) of [[1/7-comma meantone]], both being 7 steps. What's also remarkable is that in this instance the chromatic semitone is equal to one step of [[13edo]]. Since 135/128 is also equal to 1/13th of the octave, 91c val tempers out the [[aluminium comma]] in the 5-limit.
+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 also 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 1/13th of the octave, 91c val tempers out the [[aluminium comma]] in the 5-limit.
-edo also tempers out the [[Decaononic|devil's tridecalimma]], which assigns 10/9 to 2/13ths of the octave. It also supports the [[trideci]] temperament, which in the 7-limit tempers out 4375/4374 and 83349/81920.
+edo also tempers out the {{monzo| -11 26 -13 }}, the tridecatonic comma, which assigns [[10/9]] to 2/13 of the octave. It also supports the [[trideci]] temperament, which in the 7-limit tempers out 4375/4374 and 83349/81920.
 It is the second highest it 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]].