Local minima of dissonance functions

Abstract

When the same sound is produced simultaneously with two different fundamental frequencies, auditory roughness is observed. If the first sound is fixed and the fundamental frequency of the second is varied continuously, auditory roughness also varies continuously. A vowel sound is distinguished by its spectral envelope – which is independent of the fundamental frequency. This is a motivation to define the metric space of timbres. Each timbre is associated with a dissonance function which has local minima at certain intervals of local consonance related to the timbre. This is related to the music-theoretical notion of consonant intervals and scales. For the subspace consisting of all timbres with an interval of local consonance at a chosen point β, the main theorem describes certain points on the boundary by the vanishing of one-sided derivatives of dissonance functions at β.