Superparticular ratio

It is also possible to convert the Leibniz formula for π into an Euler product of superparticular ratios in which each term has a prime number as its numerator and the nearest multiple of four as its denominator:[5] In graph theory, superparticular numbers (or rather, their reciprocals, 1/2, 2/3, 3/4, etc.)

arise via the Erdős–Stone theorem as the possible values of the upper density of an infinite graph.

Indeed, whether a ratio was superparticular was the most important criterion in Ptolemy's formulation of musical harmony.

[9] Every pair of adjacent positive integers represent a superparticular ratio, and similarly every pair of adjacent harmonics in the harmonic series (music) represent a superparticular ratio.

Many individual superparticular ratios have their own names, either in historical mathematics or in music theory.