Generalized superparticulars

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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