Two theorems concerning rational approximations

The two theorems of the title constitute the mathematical results underlying well-formed scale theory. This paper includes the purely mathematical portion of a manuscript from 1988, which the authors cited the following year in N. Carey and D. Clampitt [Aspects of well-formed scales, Music Theory Spectrum 11 (1989), pp. 187–206]. Both theorems concern finite sets of the fractional parts of multiples of a positive real number θ. Such sets may be taken to represent musical scales generated by a single interval, and they have nice properties when their cardinalities are the denominators of convergents or semi-convergents (intermediate convergents) in the continued fraction representation of θ. Here, the results and their proofs are brought together in a notationally consistent framework.