Rational numbers with small numerators and denominators play a special role in music. For example, the frequency ratio corresponds to the interval of an octave, whilst the frequency ratios  correspond to the intervals of a major triad. These have led to various tunings of the set of notes on a stave involving rational ratios, known as just intonation. The differences between these tunings are quite subtle. Here, by the process of generalisation, we explore some other ways of using rational numbers with small numerators and denominators in music. We include a collection of tunings of the notes on the stave obtained in this way, where the differences between the tunings are not so subtle.

rational numbers, tunings of the notes.

Received: April 5, 2022; Accepted: May 19, 2022; Published: June 28, 2022

How to cite this article: Will Turner, On rational numbers with small numerators and denominators in music, Far East Journal of Mathematical Education 22 (2022), 33-50. http://dx.doi.org/10.17654/0973563122007

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] J. S. Bach, Two Part Invention No.1, BWV 772.
[2] J. S. Bach, Two Part Invention No.9, BWV 780.
[3] D. J. Benson, Music: A Mathematical Offering, Cambridge University Press, November 2006.
[4] C. Huygens, Brief betreffende de harmonische cyclus, Histoire des Ouvrages des Sçavans, Rotterdam, October 1691, pp. 78-88.
[5] K. Stange, C. Wick and H. Hinrichsen, Playing music in just intonation: a dynamically adaptive tuning scheme, Computer Music Journal 42(3) (2018), 47-62.
[6] W. Turner, On representing consonance structures.
http://homepages.abdn.ac.uk/w.turner/pages/.
[7] W. Turner, On reading timbre and tempo from the score.
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[8] W. Turner, Examples 1-23. http://homepages.abdn.ac.uk/w.turner/pages/.