Consistent Pitch Height Forms: A commentary on Daniel Muzzulini's contribution Isaac Newton's Microtonal Approach to Just Intonation

Abstract

This text revisits selected aspects of Muzzulini's article and reformulates them on the basis of a three-dimensional interval space E and its dual E^*. The pitch height of just intonation is conceived as an element h of the dual space. From octave-fifth-third coordinates it becomes transformed into chromatic coordinates. The dual chromatic basis is spanned by the duals a^*of a minor second a and the duals b^* and c^*  of two kinds of augmented primes b and c. Then for every natural number n a modified pitch height form h_n is derived from h by augmenting its coordinates with the factor n, followed by rounding to nearest integers. Of particular interest are the octave-consitent forms h_n  mapping the octave to the value n. The three forms h_n for n = 612, 118, 53 (yielding smallest deviations from the respective values of n h) form the Muzzulini basis of E^*. The respective transformation matrix T^* between the coordinate representations of linear forms in the Muzzulini basis and the dual chromatic basis is unimodular and a Pisot matrix with the dominant eigen-co-vector very close to h. Certain selections of the linear forms h_n are displayed in Muzzuli coordinates as ball-like point clouds within a suitable cuboid containing the origin. As an open problem remains the estimation of the musical relevance of  Newton's chromatic mode, and chromatic modes in general. As a possible direction of further investigation it is proposed to study the exo-mode of Newton's chromatic mode