Mathematics and Music

Composition, Perception, and Performance

James S. Walker, Gary W. Don

• Publisher: Chapman and Hall/CRC
• Year: 2019
• ISBN:
  9780429013553
• Edition: 2
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Mathematics and Music: Composition, Perception, and Performance, Second Edition includes many new sections and more consistent expectations of a student's experience. The new edition of this popular text is more accessible for students with limited musical backgrounds and only high school mathematics is required.

The new edition includes more illustrations than the previous one and the added sections deal with the XronoMorph rhythm generator, musical composition, and analyzing personal performance.

The text teaches the basics of reading music, explaining how various patterns in music can be described with mathematics, providing mathematical explanations for musical scales, harmony, and rhythm. The book gives students a deeper appreciation showing how music is informed by both its mathematical and aesthetic structures.

Highlights of the Second Edition:

• Now updated for more consistent expectations of students' backgrounds

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• More accessible for students with limited musical backgrounds

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• Full-color presentation

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• Includes more thorough coverage of spectrograms for analyzing recorded music

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• Provides a basic introduction to reading music

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• Features new coverage of building and evaluating rhythms

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Chapter 1

Pitch, Frequency, and Musical Scales

…in nature itself, a single note sets up a harmony of its own; and this harmonic series has been the (unconscious) basis of Western European harmony, and the tonal system.

—Deryk Cooke (1959)

The scientific study of music was put on a firm footing with the seminal work of Hermann von Helmholtz in the middle of the 19th century. His masterpiece, Helmholtz (1954), is still worth studying today. In this chapter we describe Helmholtz’s ideas and show how they provide a rationale for musical scales.

1.1Pitch and Frequency

There is a close connection between pitches in musical tones and the mathematical concept of frequency. In the 19th century, Helmholtz did an experiment with tuning forks. He attached a pen to one of the tines of a tuning fork and drew the fork across a piece of paper while it was sounding a specific pitch. The vibration of the pen traced out a simple waveform. We will refer to it as a pure tone waveform. See Figure 1.1.

Figure 1.1. Illustration of the famous experiment of Helmholtz. A pen is attached to a tine of tuning fork. As the tuning fork is struck and drawn across a piece of paper at a uniform speed, the pen traces out a pure tone waveform. Distance marked between two peaks of a waveform is called a cycle.

The most fundamental aspect of a pure tone waveform is that it repeats itself periodically. In physics, the distance from one peak of the wave to the next is called its wavelength. Another term for wavelength is cycle. We have marked one cycle for the pure tone waveform in Figure 1.1. The number of cycles in the pure tone waveform that occur in 1 second is called its frequency. We have this formula for frequency:

frequency=number of cyclessecond.

The unit of cycles/sec for measuring frequency is also called Hz, which is short for Hertz (another German physicist who did fundamental work in the study of frequency). For example, if t...