91edo: Difference between revisions

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'''91edo''', ~~the~~ '''91 equal ~~division~~''' divides the octave into 91 parts of 13.187 ~~cents~~ each.

{{Infobox ET

| Prime factorization = 7 × 13

| Step size = 13.1868¢

| Fifth = 53\91 (698.9¢)

| Semitones = 7:8 (92.3¢ : 105.5¢)

| Consistency = 9

}}

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.

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. From an aesthetic standpoint, the factoring of 91 represents a kind of "yin-yang" since historically, the number 7 symbolizes luck and 13 misfortune.

== 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. It is the second highest it a series of four consecutive edos that temper out [[quartisma]] ({{monzo| 24 -6 0 1 -5 }}). 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.

=== Odd harmonics ===

== Regular temperament properties ==

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* [http://chrisvaisvil.com/dprk-ison-chase-12-of-91-edo-ambient/ DPRK ISON CHASE] by [[Chris Vaisvil]]

* [https://www.youtube.com/watch?v=StCR6hcm5tM DPRK ISON CHASE - YouTube]

~~== See also ==~~

* [[Wikipedia: 91 (number)]]

[[Category:Equal divisions of the octave]]

@@ Line 1: / Line 1: @@
-'''91edo''', the '''91 equal division''' divides the octave into 91 parts of 13.187 cents each.
+{{Infobox ET
+| Prime factorization = 7 × 13
+| Step size = 13.1868¢
+| Fifth = 53\91 (698.9¢)
+| Semitones = 7:8 (92.3¢ : 105.5¢)
+| Consistency = 9
+}}
+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.
 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. From an aesthetic standpoint, the factoring of 91 represents a kind of "yin-yang" since historically, the number 7 symbolizes luck and 13 misfortune.
 == Theory ==
-{{harmonics in equal|91}}
+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]] ({{monzo| 24 -6 0 1 -5 }}). 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.
+=== Odd harmonics ===
+{{Harmonics in equal|91}}
 == Regular temperament properties ==
@@ Line 208: / Line 216: @@
 * [http://chrisvaisvil.com/dprk-ison-chase-12-of-91-edo-ambient/ DPRK ISON CHASE] by [[Chris Vaisvil]]
 * [https://www.youtube.com/watch?v=StCR6hcm5tM DPRK ISON CHASE - YouTube]
-== See also ==
-* [[Wikipedia: 91 (number)]]
 [[Category:Equal divisions of the octave]]