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Over the course of history, western music has created a unique mathematical problem for itself. From acoustics, we know that two notes sound good together when they are related by simple ratios consisting of low primes. The problem arises when we try to build a finite set of pitches, like the 12 notes on a piano, that are all related by such ratios. We approach the problem by laying out definitions and axioms that seek to identify and generalize desirable properties. We can then apply these ideas to a broadened algebraic framework. Rings in which low prime integers can be factored are of particular interest. Unique viable solutions can be found in various Euclidean domains, including the Gaussian integers and the Eisenstein integers.
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Spitzer, Shay Joel Francis, "Mathematical Structure of Musical Tuning Systems" (2023). Senior Projects Spring 2023. 229.
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