91edo: Difference between revisions
these aren't objective facts so moving them to miscellaneous properties |
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== Theory == | == 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. | 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. | ||
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]] of 1/7 comma meantone, both being 7 steps. | 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]] 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. | ||
91edo tempers out the [[Decaononic|devil's tridecalimma]], which | 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. | ||
It is the second highest it a series of four consecutive edos that temper out [[quartisma]] ({{monzo| 24 -6 0 1 -5 }}) | 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 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. | 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. | ||
91edo | In the 13-limit, 91edo provides the [[optimal patent val]] for [[vidar]] temperament, although it is no longer consistent in the 11-odd-limit. | ||
=== Odd harmonics === | === Odd harmonics === | ||