Constructing an optimal circulating temperament based on a set of musical requirements

Abstract

A novel parametric circulating temperament is presented using a constructive approach. The temperament is optimal with respect to a heuristically chosen set of musical requirements. It is parametric in the sense that the tempering of the narrowest (i.e. closest to pure) major third can be freely chosen. Equal temperament arises as a limiting case. The temperament is optimal in the sense that the tempering of the widest major third and the narrowest fifth is minimized given the size of the least tempered major third. Also, under this constraint, the tempering of the major thirds closest to the least tempered third along the circle of fifths is minimized. The remaining degree of freedom is used to minimize the number of unique intervals. The resulting temperament exhibits various symmetries, and has, in general, two differently sized fifths and five differently sized major thirds. The temperament has no historic relevance as such, but can find good uses within all keyboard music from early baroque till today due to the selection of optimization criteria that closely follow historical requirements for good temperaments. With some exercise it can be tuned by ear.

Keywords:

• tuning
• temperament
• optimal
• fifths
• thirds

Acknowledgment

The author would like to thank the editors and the reviewers for fruitful comments and suggestions for the manuscript.