On the construction, comparison, and exchangeability of tuning systems

Abstract

The aim of this article is to describe mathematically different tuning systems, to study their mathematical properties, and to propose a construction allowing their comparison. In order to reach these goals, we introduce a concept of similarity between tuning systems and then we provide two sufficient conditions for the particular case in which a tuning system generated by an interval and a circulating temperament are compared. Finally, we show by means of an example that, for two tuning systems to be exchangeable, some well-known results determining the suitable number of notes per octave are not enough.

Keywords:

• tuning systems
• continued fractions
• similarity
• Pythagorean tuning
• just intonation
• circulating temperaments

2010 Mathematics Subject Classification:

• 00A65
• 11A55
• 60G09

Acknowledgements

The author would like to thank the anonymous referees and the editors for their comments and suggestions. A special acknowledgement must be made to Professor Marek $\check{\mathrm{Z}}$abka for all his constructive suggestions and comments on the article, which have really improved its overall quality.

Funding

This article has been partially supported by the Spanish Ministry of Science and Innovation [TIN2008-06872-C04-02].

Notes

1 As is well known, for Pythagorean tuning, these intervals correspond with $1/1$ (unison), $2/1$ (octave), $3/2$ (fifth), and $4/3$ (fourth) (Barbour 1951).

2 Not all systems are based on the octave. For instance, Benson (2006), Krantz and Douthett (2011), and Douthett and Krantz (2007) discuss tuning systems which are based on the tritave instead of on the octave.

3 To know more about the historical process regarding the consecution of this correction see, for instance, Goldáraz (2004), Chailley and Challan (1951), and Lattard (1988).

4 Other authors, such as Krantz and Douthett (2011) and Callender, Quinn, and Tymoczko (2008), have proposed similar (but not the same) approaches. However, there is one major difference between these approaches and the approach presented in this article. In our work we compare arbitrary tuning systems, whereas the previous authors essentially measured how close a tuning system is to the equal temperament.

5 In this work, when we use the expression “convergent” we do it in the sense of Baker (1984), which coincides with what Krantz and Douthett (2011) and Douthett and Krantz (2007) called “principal convergent.”

6 Notice that, in terms of Douthett and Krantz (2007), this is “a best approximation of the second kind.”