91edo: Difference between revisions

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== Theory ==

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~~.

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 ~~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.

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.

~~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.

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.

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]].

The [[concoctic scale]] for 91edo is 27 steps, where two concoctic neutral thirds make a sharp fifth of 54\91, representing 3/2 in the 91b val. There are more than one way to interpret this in regular temperament theory. First, in the 13-limit, is to assume that 27\91 is directly equivalent to 16/13 and set a temperament in the 2.3.5.7.13 subgroup, which produces a 27 & 91b temperament with the comma basis 91/90, 6272/6075, {{monzo| 84 13 11 -10 }}. Second is to directly take the 27 & 91b val in the 13-limit, which can also be taken using the 27e & 91b.

The [[concoctic scale]] for 91edo is 27 steps, where two concoctic neutral thirds make a sharp ~~91b val~~ fifth of 54\91. ~~From a regular temperament theory perspective, there is~~ more than one way to interpret this~~, as they're all harmonically not very precise~~. First, in the 13-limit, is to assume that 27\91 is directly equivalent to 16/13 and set a temperament in the 2.3.5.7.~~16/~~13 subgroup, which produces a 27 & 91b temperament with the comma basis 91/90, 6272/6075, {{~~Monzo~~|84 13 11 -10}}. Second is to directly take the 27 & 91b val in the 13-limit, which can also be taken using the 27e & 91b.

In the 13-limit, 91edo is an [[Optimal ET sequence|optimal ET]] for [[vidar]] temperament coming after [[87edo|87d]] val and before [[133edo|133d]] val, although it is no longer consistent in the 11-odd-limit.

=== Odd harmonics ===

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 == Theory ==
-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.
+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 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.
+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.
-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.
+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.
-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]].
+The [[concoctic scale]] for 91edo is 27 steps, where two concoctic neutral thirds make a sharp fifth of 54\91, representing 3/2 in the 91b val. There are more than one way to interpret this in regular temperament theory. First, in the 13-limit, is to assume that 27\91 is directly equivalent to 16/13 and set a temperament in the 2.3.5.7.13 subgroup, which produces a 27 & 91b temperament with the comma basis 91/90, 6272/6075, {{monzo| 84 13 11 -10 }}. Second is to directly take the 27 & 91b val in the 13-limit, which can also be taken using the 27e & 91b.
-The [[concoctic scale]] for 91edo is 27 steps, where two concoctic neutral thirds make a sharp 91b val fifth of 54\91. From a regular temperament theory perspective, there is more than one way to interpret this, as they're all harmonically not very precise. First, in the 13-limit, is to assume that 27\91 is directly equivalent to 16/13 and set a temperament in the 2.3.5.7.16/13 subgroup, which produces a 27 & 91b temperament with the comma basis 91/90, 6272/6075, {{Monzo|84 13 11 -10}}. Second is to directly take the 27 & 91b val in the 13-limit, which can also be taken using the 27e & 91b.
-In the 13-limit, 91edo is an [[Optimal ET sequence|optimal ET]] for [[vidar]] temperament coming after [[87edo|87d]] val and before [[133edo|133d]] val, although it is no longer consistent in the 11-odd-limit.
 === Odd harmonics ===