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Continued Fraction Approximations Demonstrated Through The Musical Chromatic Scale
Knowles, Nicole Knowles, Nicole
Knowles, Nicole
Knowles, Nicole
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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.
Title
Continued Fraction Approximations Demonstrated Through The Musical Chromatic Scale
Date
2024-05-01
Subject
Continued fraction
Euclodean algorithm
Continued fraction algorithm
Chromatic scale
Approximations
Irrational numbers
Euclodean algorithm
Continued fraction algorithm
Chromatic scale
Approximations
Irrational numbers
Material type
Collections
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knowlesnicole.pdf
Adobe PDF, 429.06 KB
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.
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Department
Mathematics
Degree
Bachelor of Science (BS)