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

Thomas Noll


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 cof two kinds of augmented primes b and c. Then for every natural number n a modified pitch height form hn 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 hn  mapping the octave to the value n. The three forms hn 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 hn 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


Linear interval space; Pitch height form; Diatonic modes; Chromatic modes; Exo-modes; Scale Degree Qualia

Full Text:




  • There are currently no refbacks.

Copyright (c) 2021 Thomas Noll

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.


Beginning with Volume 7, No 3-4 (2012), Empirical Musicology Review is published under a Creative Commons Attribution-NonCommercial license

Empirical Musicology Review is published by The Ohio State University Libraries.

If you encounter problems with the site or have comments to offer, including any access difficulty due to incompatibility with adaptive technology, please contact

ISSN: 1559-5749