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ON THE EUCLIDEAN ALGORITHM: RHYTHM WITHOUT RECURSION

Part of: Elementary number theory Arithmetic, number theory

Published online by Cambridge University Press:  22 September 2022

THOMAS MORRILL Open the ORCID record for THOMAS MORRILL [Opens in a new window]
THOMAS MORRILL*
Affiliation:
School of Arts and Sciences, Trine University, Angola, Indiana, USA
*
e-mail: morrillt@trine.edu
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Abstract

A modified form of Euclid’s algorithm has gained popularity among musical composers following Toussaint’s 2005 survey of so-called Euclidean rhythms in world music. We offer a method to easily calculate Euclid’s algorithm by hand as a modification of Bresenham’s line-drawing algorithm. Notably, this modified algorithm is a nonrecursive matrix construction, using only modular arithmetic and combinatorics. This construction does not outperform the traditional divide-with-remainder method; it is presented for combinatorial interest and ease of hand computation.

Keywords

Euclidean algorithmbinary rhythmscomputation by hand

MSC classification

Primary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
Secondary: 97F60: Number theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 3 , June 2023 , pp. 361 - 367
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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