ON THE EUCLIDEAN ALGORITHM: RHYTHM WITHOUT RECURSION

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...

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Bibliographic Details
Published in:Bulletin of the Australian Mathematical Society Vol. 107; no. 3; pp. 361 - 367
Main Author: MORRILL, THOMAS
Format: Journal Article
Language:English
Published: Cambridge, UK Cambridge University Press 01.06.2023
Subjects:
Algorithms
Arrays
Combinatorial analysis
Composers
Modular construction
Rhythm
Software
Euclidean algorithm
computation by hand
11A05
97F60
binary rhythms
ISSN:0004-9727, 1755-1633
Online Access:Get full text
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Summary: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.
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ISSN:0004-9727
1755-1633
DOI:10.1017/S0004972722000909