81/80: Difference between revisions

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.