Hostname: page-component-6bf8c574d5-rwnhh Total loading time: 0 Render date: 2025-02-16T19:43:24.904Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE EUCLIDEAN ALGORITHM: RHYTHM WITHOUT RECURSION

Part of: Arithmetic, number theory Elementary 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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bjorklund, E., ‘A metric for measuring the evenness of timing system rep-rate patterns’, SNS ASD Tech Note, SNS-NOTECNTRL-100, Los Alamos National Laboratory, Los Alamos, 2003.Google Scholar
Bjorklund, E., ‘The theory of rep-rate pattern generation in the SNS timing system’, SNS ASD Tech Note, SNS-NOTE-CNTRL-99, Los Alamos National Laboratory, Los Alamos, 2003.Google Scholar
Demaine, E. D., Gomez-Martin, F., Meijer, H., Rappaport, D., Taslakian, P., Toussaint, G. T., Winograd, T. and Wood, D. R., ‘The distance geometry of music’, Comput. Geom. 42(5) (2009), 429–454.10.1016/j.comgeo.2008.04.005CrossRefGoogle Scholar
Gómez-Martín, F., Taslakian, P. and Toussaint, G., ‘Structural properties of Euclidean rhythms’, J. Math. Music 3(1) (2009), 1–14.10.1080/17459730902819566CrossRefGoogle Scholar
Gómez-Martín, F., Taslakian, P. and Toussaint, G., ‘Interlocking and Euclidean rhythms’, J. Math. Music 3(1) (2009), 15–30.10.1080/17459730902916545CrossRefGoogle Scholar
Graham, R. L., Knuth, D. E., Patashnik, O. and Liu, S., ‘Concrete mathematics: a foundation for computer science’, Comput. Phys. 3 (1989), 106.10.1063/1.4822863CrossRefGoogle Scholar
Harris, M. A. and Reingold, E. M., ‘Line drawing, leap years, and Euclid’, ACM Comput. Surv. (CSUR) 36(1) (2004), 68–80.10.1145/1013208.1013211CrossRefGoogle Scholar
Milne, A. J., Herff, S. A., Bulger, D. W., Sethares, W. A. and Dean, R. T., ‘Xronomorph: algorithmic generation of perfectly balanced and well-formed rhythms’, NIME (2016), 388–393.Google Scholar
Toussaint, G., ‘The Euclidean algorithm generates traditional musical rhythms’, in: Renaissance Banff: Mathematics, Music, Art, Culture (eds. Sarhangi, R. and Moody, R. V.) (The Bridges Organization, Winfield, Kansas, 2005), 47–56.Google Scholar