A different kind of meantone temperament and a novel mapping from the harmonic series to Pythagorean pitch classes

Abstract

The logarithm mapping of natural numbers is a sum of products of coefficients. If these coefficients are arbitrary parameters, a new mapping of natural numbers to some subset of real numbers appears. This mapping preserves some crucial logarithm properties and constructs a new musical sound with a spectrum of inharmonic overtones. The simplest one-parameter mapping to a subset of polynomials with integer coefficients is constructed. The parameter defines a new “perfect fifth” similarly to the meantone temperament. It is an interesting case when the mapping parameter is defined from the linear relation between the new “major third” and the new “perfect fifth.” The most interesting case of a linear relation is the condition of zeroing the syntonic comma. Here, the new meantone temperament, based on the new “perfect fifth,” simultaneously coincides with Pythagorean- and five-limit-like tunings.