81/80: Difference between revisions

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The '''syntonic''' or '''Didymus comma''' (frequency ratio '''81/80''') is ~~the smallest~~ [[~~superparticular|superparticular interval~~]] ~~which belongs to the [[~~5-limit~~]]. Like~~ [[~~16/15~~]]~~, [[625~~/~~624]],~~ [[~~2401/2400~~]] ~~and~~ [[~~4096/4095~~]] ~~it has~~ a ~~fourth power as~~ a ~~numerator~~. ~~Fourth powers are squares, and any comma with~~ a ~~square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between [[10~~/~~9]] and [[9~~/~~8]], the product~~ of ~~which is~~ the ~~just major third~~, ~~[[5/4]]. That the numerator is~~ a ~~fourth power entails that the larger~~ of ~~these two intervals itself has~~ a ~~square numerator; 9~~/8 is ~~the interval between the successive superparticulars 4/3 and 3/2~~. ~~Tempering out~~ a ~~comma does not just depend on an edo's size;~~ [[~~105edo~~]] ~~tempers it out~~, ~~while~~ [[~~15edo~~|~~3edo~~]] ~~does not~~.

The '''syntonic''' or '''Didymus''' or '''meantone comma''' (frequency ratio '''81/80''') is helpful for comparing [[3-limit]] and 5-limit [[just intonation]]. Adding or subtracting this comma to/from any 3-limit [[ratio]] with an [[odd limit]] of 27 or higher creates a 5-limit ratio with a much lower odd-limit. Thus dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. However adding/subtracting this comma to/from any 3-limit ratio of odd limit 3 or less (the 4th, 5th or 8ve), creates a wolf interval of odd limit 27 or higher. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves, thus tempering out 81/80 is desirable. This gives a tuning for the [[Tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone family|meantone temperament]], hence the name meantone comma.

~~Tempering out~~ 81/80 ~~gives a tuning for~~ the [[~~Tone~~|~~whole tone~~]] which is ~~intermediate~~ between 10/9 and 9/8, and ~~leads to~~ [[~~Meantone family~~|~~meantone temperament~~]].

81/80 is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit]]. Like [[16/15]], [[625/624]], [[2401/2400]] and [[4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any superparticular comma with a square numerator is the ratio between two wider successive superparticular intervals, because n<sup>2</sup>/(n<sup>2</sup>-1) = n/(n-1) ÷ (n+1)/n. 81/80 is in fact the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]]. That the numerator is a fourth power entails that the wider of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. Tempering out a comma does not just depend on an edo's size; [[105edo]] tempers it out, while [[15edo|3edo]] does not.

Youtube video of "[http://www.youtube.com/watch?v=IpWiEWFRGAY Five senses of 81/80]", demonstratory video by Jacob Barton.

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-The '''syntonic''' or '''Didymus comma''' (frequency ratio '''81/80''') is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit]]. Like [[16/15]], [[625/624]], [[2401/2400]] and [[4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]]. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. Tempering out a comma does not just depend on an edo's size; [[105edo]] tempers it out, while [[15edo|3edo]] does not.
+The '''syntonic''' or '''Didymus''' or '''meantone comma''' (frequency ratio '''81/80''') is helpful for comparing [[3-limit]] and 5-limit [[just intonation]]. Adding or subtracting this comma to/from any 3-limit [[ratio]] with an [[odd limit]] of 27 or higher creates a 5-limit ratio with a much lower odd-limit. Thus dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. However adding/subtracting this comma to/from any 3-limit ratio of odd limit 3 or less (the 4th, 5th or 8ve), creates a wolf interval of odd limit 27 or higher. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves, thus tempering out 81/80 is desirable. This gives a tuning for the [[Tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone family|meantone temperament]], hence the name meantone comma.
-Tempering out 81/80 gives a tuning for the [[Tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone family|meantone temperament]].
+/80 is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit]]. Like [[16/15]], [[625/624]], [[2401/2400]] and [[4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any superparticular comma with a square numerator is the ratio between two wider successive superparticular intervals, because n<sup>2</sup>/(n<sup>2</sup>-1) = n/(n-1) ÷ (n+1)/n. 81/80 is in fact the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]]. That the numerator is a fourth power entails that the wider of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. Tempering out a comma does not just depend on an edo's size; [[105edo]] tempers it out, while [[15edo|3edo]] does not.
 Youtube video of "[http://www.youtube.com/watch?v=IpWiEWFRGAY Five senses of 81/80]", demonstratory video by Jacob Barton.