81/80: Difference between revisions

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21.506290 cents

The '''syntonic''' or '''Didymus comma''' (frequency ratio '''81/80''') is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit|5-limit]]. Like [[16/15|16/15]], [[625/624|625/624]], [[2401/2400|2401/2400]] and [[4096/4095|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|10/9]] and [[9/8|9/8]], the product of which is the just major third, [[5/4|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. [[105edo|105edo]] tempers it out, while [[15edo|3edo]] does not.

The '''syntonic''' or '''Didymus comma''' (frequency ratio '''81/80''') is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit|5-limit]]. Like [[16/15|16/15]], [[625/624|625/624]], [[2401/2400|2401/2400]] and [[4096/4095|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|10/9]] and [[9/8|9/8]], the product of which is the just major third, [[5/4|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|105edo]] tempers it out, while [[15edo|3edo]] does not.

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]].

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 .506290 cents
-The '''syntonic''' or '''Didymus comma''' (frequency ratio '''81/80''') is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit|5-limit]]. Like [[16/15|16/15]], [[625/624|625/624]], [[2401/2400|2401/2400]] and [[4096/4095|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|10/9]] and [[9/8|9/8]], the product of which is the just major third, [[5/4|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. [[105edo|105edo]] tempers it out, while [[15edo|3edo]] does not.
+The '''syntonic''' or '''Didymus comma''' (frequency ratio '''81/80''') is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit|5-limit]]. Like [[16/15|16/15]], [[625/624|625/624]], [[2401/2400|2401/2400]] and [[4096/4095|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|10/9]] and [[9/8|9/8]], the product of which is the just major third, [[5/4|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|105edo]] tempers it out, while [[15edo|3edo]] does not.
 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]].