91edo: Difference between revisions
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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. It is the second highest it a series of four consecutive EDOs that temper out [[quartisma]] (117440512/117406179). 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 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. It is the second highest it a series of four consecutive EDOs that temper out [[quartisma]] (117440512/117406179). 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. | ||
=== Naive modes === | |||
91edo possesses naive versions of heptatonic and tridecatonic scales. | |||
For example, it can recreate diatonic major by first making an equiheptatonic scale with step size 13, and then raising III, VI and VII by a desired amount. Similar approximation can be done in [[28edo]] or any edo divisible by 7. | |||
Likewise, it can also recreate Orwells from tridecatonic scale. However, there is not a "trivial" way to do it due to larger amount of notes. The "correct" way to recreate a mode in this fashion would be using the differences of [[Irvic scale|Irvian mode]] and principal mode - that is by applying the generator or construction at the original tonic. For example, [[22edo]]'s Irvian mode for Orwell[13] is 2212212221221, while for 7/6 generator from the tonic is 1221221221222, and applying the differences results in 5795795777779 scale. This scale is quite bulky for musical performance since it contains an even row of 5x7 steps. A more vibrant possible variant is 5795797597579, which is derived from differences of 2212212212221 and 1221222122122 22edo Orwells. | |||
Since 7 and 13 are the only factors of 91, these numbers are the only ones which can produce naive scales. From an aesthetic standpoint, this represents a kind of "yin-yang" for 91edo since 7 symbolizes luck and 13 misfortune. | |||
== Scales == | == Scales == | ||
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* NaiveMajor[7]: 13 16 10 13 16 13 10 | * NaiveMajor[7]: 13 16 10 13 16 13 10 | ||
* NaiveMajor[7]: 13 16 10 15 14 13 10 - fifth adjusted to match with | * NaiveMajor[7]: 13 16 10 15 14 13 10 - fifth adjusted to match with NaiveOrwell[13] | ||
* NaiveMinor[7]: 13 10 16 13 10 13 16 | |||
* NaiveOrwell[13]: 5795797597579 | * NaiveOrwell[13]: 5795797597579 | ||
* ArabicNaiveOrwell[13]: 1 11 9 5 1 15 7 5 9 7 1 11 9 - above scale with 2, 6, and 12 degrees lowered 4 steps | * ArabicNaiveOrwell[13]: 1 11 9 5 1 15 7 5 9 7 1 11 9 - above scale with 2, 6, and 12 degrees lowered 4 steps | ||
Revision as of 08:22, 21 October 2021
91edo, the 91 equal division divides the octave into 91 parts of 13.187 cents each.
91 is the smallest composite number whose composite character is not immediately evident in the decimal system; it is, in fact, the product of 7 and 13.
Theory
Script error: No such module "primes_in_edo".
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. It is the second highest it a series of four consecutive EDOs that temper out quartisma (117440512/117406179). 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.
Naive modes
91edo possesses naive versions of heptatonic and tridecatonic scales.
For example, it can recreate diatonic major by first making an equiheptatonic scale with step size 13, and then raising III, VI and VII by a desired amount. Similar approximation can be done in 28edo or any edo divisible by 7.
Likewise, it can also recreate Orwells from tridecatonic scale. However, there is not a "trivial" way to do it due to larger amount of notes. The "correct" way to recreate a mode in this fashion would be using the differences of Irvian mode and principal mode - that is by applying the generator or construction at the original tonic. For example, 22edo's Irvian mode for Orwell[13] is 2212212221221, while for 7/6 generator from the tonic is 1221221221222, and applying the differences results in 5795795777779 scale. This scale is quite bulky for musical performance since it contains an even row of 5x7 steps. A more vibrant possible variant is 5795797597579, which is derived from differences of 2212212212221 and 1221222122122 22edo Orwells.
Since 7 and 13 are the only factors of 91, these numbers are the only ones which can produce naive scales. From an aesthetic standpoint, this represents a kind of "yin-yang" for 91edo since 7 symbolizes luck and 13 misfortune.
Scales
Non-JI themed scales that are derived from 7 and 13 note scales, the 2 divisors of 91:
- NaiveMajor[7]: 13 16 10 13 16 13 10
- NaiveMajor[7]: 13 16 10 15 14 13 10 - fifth adjusted to match with NaiveOrwell[13]
- NaiveMinor[7]: 13 10 16 13 10 13 16
- NaiveOrwell[13]: 5795797597579
- ArabicNaiveOrwell[13]: 1 11 9 5 1 15 7 5 9 7 1 11 9 - above scale with 2, 6, and 12 degrees lowered 4 steps