Generalized superparticulars

From Xenharmonic Wiki
Jump to navigation Jump to search

According to Thomas Taylor's Theoretic Arithmetic, in Three Books, superparticular ratios are defined as those for which the denominator divides into the numerator once, leaving a remainder of one.

In almost every case, this checks out with the popular usage of superparticular to mean ratios of the form (n+1)/n. In only one case does it deviate: that of 2/1. According to Taylor, 2/1 is not superparticular, because 1 divides into 2 twice, leaving a remainder of 0.

Taylor's book further describes generalizations of the superparticulars:

  • superbiparticulars are those where the denominator divides into the numerator once, but leaves a remainder of two (such as 5/3)
  • double superparticulars are those where the denominator divides into the numerator twice, leaving a remainder of one (such as 5/2)
  • one can go on and on, with e.g. triple supertriparticulars where both the divisions and the remainder are 3 (such as 15/4).

More details can be found on this forum thread here: Generalisation of the terms "epimoric" and "superparticular" as applied to ratios