Mathematics and Music (Mathematical World, Vol. 28)
David Curtis Wright · 2010
It is a commonplace that there are links between the world of mathematics and the world of music. But in the literature on these connections, the two areas play asymmetric roles. The reader is usually assumed to have some mathematical background: Mathematical terms and theories are used with little explanation. These investigations are hardly accessible to nonspecialists. Mathematics and Music is written in a different spirit. It reviews some basic concepts in both mathematics and music from the very beginning, presuming no background in either of these fields. It’s addressed to students of all fields who are interested in both subjects. The 12 chapters cover a wide variety of mathematical and musical themes. Chapter 1 is devoted to ‘‘basic concepts.’’ Here, the various sets of numbers are introduced (N, Q, etc.), and one also learns, for example, that the integers are well ordered, how to visualize functions by their graphs, and how an equivalence relation is defined. Basics for the musical counterpart include the translation of pitches to notes by the treble and bass clefs, musical intervals (for example, the fifth or the octave), and the use of accidentals. At the end of this chapter, cyclic permutations are introduced to explain how the different modes (Ionian, Dorian, etc.) can be derived from a single scale. Chapter 2 is concerned with ‘‘horizontal structures.’’ How are whole notes, half notes, and so on written, which symbols are used for rests, and how do dots change the length of a note? I never realized before that the length d of a note increases to d(2 - 1/2 m ) if the note is m-dotted, a fact proved here by geometric series. It is also explained that translation (resp. transposition, resp. retrogression) of patterns corresponds to replacing f(x )b yf(x - c) (resp. f(x) + c, resp. -f(x)) for functions f. Let’s turn to Chapter 3: Harmony and Related Numerology. The mathematics starts with the algebraic structure of Z12. In this setting, a major chord is just the sequence (4, 3, 5) of modular intervals. Similarly, diminished chords and many others are described and correctly translated to musical notation. (That is, one must write E # and not F in the major chord of C # .) Chapter 4 introduces ratios as equivalence classes which one can hear as pitches: The octave, for example, corre