Continued Fraction Approximations Demonstrated Through The Musical Chromatic Scale

Abstract

In this paper, we approximate an irrational number using continued fractions through an example of a musical problem. We first define the chromatic scale. To delve into why the chromatic scale only has twelve notes, we discuss the topic of Pythagorean Tuning and how it utilizes mathematics to create scales. Since using Pythagorean Tuning to approximate the length of a scale results in an irrational number, we introduce the notion of continued fractions. These can be calculated by either using the Euclidean Algorithm or the Continued Fraction Algorithm. We define the term best approximation and finally, we use these components to solve our musical question.