In Search of Universal Properties of Musical Scales

Abstract

Musical scales have both general and culture-specific properties. While most common scales use octave equivalence and discrete pitch relationships, there seem to be no other universal properties. This paper presents an additional property across the world's musical scales that may qualify for universality. When the intervals of 998 (just intonation) scales from the Scala Archive are represented on an Euler lattice, 96.7% of them form star-convex structures. For the subset of traditional scales this percentage is even 100%. We present an attempted explanation for the star-convexity feature, suggesting that the mathematical search for universal musical properties has not yet reached its limits.

Acknowledgements

The authors wish to thank two anonymous reviewers. This research was supported by grant 277–70–006 of the Netherlands Foundation for Scientific Research (NWO).

Notes

¹This lattice representation and minor variants of it appear in numerous discussions on tuning systems, for example Von Helmholtz (1863/1954), Riemann (1914), Fokker (1949), and Longuet-Higgins (1962a, 1962b). Fokker (1949) attributes this lattice representation originally to Leonhard Euler, whence Euler lattice.

²The dataset that we used is available from http://staff.science.uva.nl/∼ahoningh/data.html

³The duplicate pairs are: (1) sal-farabi_diat2.scl and ptolemy_diat.scl, (2) hexany6.scl and smithgw_pel2.scl, (3) hirajoshi2.scl and pelog_jc.scl, (4) ptolemy.scl and zarlino.scl.

⁴Remember, however, that an equal tempered scale can only be visualized as a Fokker block if it forms an approximation of a just intonation scale. Thus, not all equal tempered scales can be evaluated in terms of convexity.

⁵There is of course not one way to do this, since consonance is not unambiguously defined.

⁶Any positive integer a can be written as a unique product of positive integer powers e_i of primes p ₁ <p ₂ < … <p_n . Euler's Gradus function is defined as:

and for the ratio x/y (which should be given in lowest terms) the value is Γ(x · y).