The Similarities Between Music Theory, And Pythagoras

Abstract—Music is the art of combining sounds to produce a composition having rhythm, melody, or harmony. Mathematics is the scientific study of numerical patterns, quantity, and space. By the definitions of music and mathematics one tends to separate the two fields into separate categories. However, music and mathematics have more in common than one may think. From reading musical notes to the wavelengths of sounds, mathematics is always a part of music. This paper will focus on the connections between mathematics and music.

Index Terms—Fibonacci, Fourier series, Golden Ratio, Harmonic Series, Music Theory, Pythagoras.

Introduction

Most people assume that the arts and sciences are opposites. For the most part, those who are talented in

Fascinated by the numbers and music, he turned to the vibrating string, explored the ideas of the length of strings and pitches, and found simple ratios relating harmonizing tones [1].

Harmonics are tones that have frequencies that are integer multiples of the original tone, the fundamental tone [1]. Harmonics and the original tone are naturally compatible to sound good together. Western music is based on harmonics because each tone has a harmonic series. When a musician plays a note on an instrument one is able to hear the fundamental tone along with all other harmonics due to the physics of sound waves [1].

After Pythagoras’ discovery of harmonics, a mathematician by the name of Marin Mersenne helped define harmonics further. He mathematically defined the first six harmonics as ratios of the fundamental frequency, 1/1, 2/1, 3/1, 4/1, 5/1 and 6/1 and he also worked on tuning systems based on harmonics [1]. Jean-Philippe Rameau, a French music theorist also attributed with defining harmonics. He published a Treatise on Harmony based on him being able to hear more than one harmonic sound when a note was played. This treatise revolutionized music theory.

Sound

Fascinatingly, arithmetic and geometry play an important part in music composition.

In 1201, an Italian mathematician by the name of Leonardo Fibonacci introduced a mathematical theory that constructs and infinite series of integers. The Fibonacci sequence begins with the number 1 followed by another 1 and each successive term is constructed by adding the two previous terms [4]. For example the first ten numbers of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55. Fibonacci ratios are a series of numbers produced when one divides a number in the Fibonacci sequence by the previous number. Table 1 shows the first seven terms of the Fibonacci ratios.

TABLE I: Fibonacci Ratios r(1) = 1/1 =