Mathematical Concepts in Music

Often, you hear the phrase " Music and Mathematics are the same". This book addresses some of the basic underlying mathematical concepts and connections with the music we listen to. The focus here is quite rigorous and pragmatic and very much unlike most typical books in music.The Pythagorean, the just intoned, and the equal tempered scales are the most prominent octave-based musical scales that simply differ in the distribution of successive note ratios in going from the tonic up to...
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Often, you hear the phrase " Music and Mathematics are the same". This book addresses some of the basic underlying mathematical concepts and connections with the music we listen to. The focus here is quite rigorous and pragmatic and very much unlike most typical books in music.The Pythagorean, the just intoned, and the equal tempered scales are the most prominent octave-based musical scales that simply differ in the distribution of successive note ratios in going from the tonic up to the octave. The mathematical basis for the note ratios that define these three scales is developed here.The 12 tone scale is the most prominent of all equal tempered scales. A procedure for generating alternate numbered equal tempered intervals is shown for targeting preferred ratios such as perfect fifths and major thirds within scales.Vibrating strings are ubiquitous in the concert piano, string instruments, and guitars. The Fourier analysis and harmonic synthesis of vibrating waves on strings demonstrates the unique dependence of harmonic structure and sound quality upon the initial setup conditions of the string.Beat phenomena within intervals of simple tones leads to the analysis of intervals of complex harmonic notes and the interactions of the upper partials of these notes. Based on work done by Helmholtz and, more recently, by Plomp and Levelt, we present the development of a mathematical model that provides a quantitative measure of the sensory perception of consonance and dissonance in tonal harmony.The concept of tonality is explored in terms of the mathematical structure of triads and seventh chords. This mathematical approach lends itself very nicely to the numeric representation and quantitative analysis of musical objects.The author is an emeritus professor of mathematics at Rutgers University and is an avid jazz musician.