Paul Cummins

Date of Award

2017

Document Type

Thesis

Degree Name

Bachelors

Department

Natural Sciences

First Advisor

Dancigers, Mark

Area of Concentration

Natural Sciences

Abstract

Music has a long history of being analyzed in terms of the number relationships that create its melodic, harmonic, and rhythmic structure. In this paper, I begin by discussing some of the properties of sound that produce these features, namely the frequency ratios of differing soundwaves. The history and mathematics of how these properties were utilized in creating ideal scales is examined, and culminates in the discovery of the 12-tone equal tempered scale. This scale subdivides the octave into 12 equal parts, and allows for musical objects such as scales, chords, and melodies to be modeled both geometrically and as vectors and matrices. Geometrically, musical objects can be modeled as a set of equidistant points around a circle. In some cases, such as any chord progression containing only the two consonant triads (major and minor), musical actions can be analyzed as symmetry transformations. Also discussed is an algorithm which generates “maximal evenness” amongst pulses arranged in a set of regular time intervals. It is shown that generating rhythms with this property (known as “Euclidean rhythms”) produces many fundamental rhythms from musical cultures around the globe. Musical objects can also be represented as vectors. One application of this is that for certain musical objects, we can construct a basis for R n corresponding to its vector representation, and decompose any element ∈ R n into linear combinations of this object. Musically, this allows us to represent objects as combinations of simpler elements, such as the major 7th chord being expressible as two major thirds, separated by a perfect fifth. Finally, the paper concludes with several examples of code written in the Python language that demonstrate the usefulness of computer sciences to modeling musical objects. Two programs visualize the geometry of pitch class movement, and another the geometry of cyclic rhythms. The programs attempt to sync the playing of a corresponding audio file to the geometric transformations that take place. Also included is a program which generates Euclidean rhythms, which can be used alone or as input for the other functions.

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