Mathematical Foundations of Complex Tonality

Author : 6AM
August 04, 2023

Mathematical Foundations of Complex Tonality

For centuries, musicians and mathematicians have struggled with the challenges of temperament and tonality. The problem is that there is a paradox at the heart of Western music. The paradox is that no tuning system exists for keyboard instruments such that all the harmonious intervals are perfectly tuned. Here we propose a radical solution based on the notion of complex tonality. We begin with an overview of the just system of intonation, which originated in antiquity and was much studied in our era during the 16th, 17th and 18th centuries, reaching a high point in the work of Euler (1739, 1774).

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We can define the just system as the totality of the various ways of representing tones by the vibrational frequencies of strings of various lengths, each such frequency being given by products of powers of the primes 2, 3, and 5 times some fixed frequency, which we call the tonal centre. Over time, many different ways of choosing the twelve notes of the chromatic scale in the just system have been proposed. Our goal is to construct a complex analogue of the just system, with broad applications, based on the ratios of certain Gaussian integers, in such a way that some of the insufficiencies of the just system can be resolved.

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Among the various diatonic scales that have been considered in the just system, the so-called intense diatonic scale of Claudius Ptolemeus is especially notable (Solomon 2000). Ptolemy’s scale, expressed in units of time such that the frequency of middle C is unity, takes the form

One can then ask how the Ptolemaic scale can be embedded in a chromatic scale. There are many possibilities, but one good example is the following, due to Johannes Kepler, which is known as Kepler’s Monochord No. 2 (Barbour 1953), here transposed down a fifth:

In Section 2 we look at the just system in some detail and we sketch the arguments leading to Ptolemy’s diatonic scale and to Kepler’s chromatic scale. Among the defects of the just system, clearly visible in Kepler’s scale, are the multiplicities of primes required for the tritone interval from C to F. Also troubling, when one looks further, are the multiple tunings for the notes D and B, and the unpleasant “wolf” tunings of certain intervals. As a first step forward, we propose an alternative approach to building up scales in the just system by use of an analogy with chemistry. In Section 3, Atoms, molecules and particles, we generate the just scale by the aggregation of a small number of whole tone and semitone intervals, which we call atoms. We show that one recovers the seven modal scales (in their modern form) by permutations of the aggregation sequence of the given whole tones and semitones. This leads to a consistent assignment of all the tones of the chromatic scale with the exception of F. The equal temperament system, which evolved as a practical response to the difficulties of the just system, can also be cast into an “atomic” framework by use of the chemical analogy, but one loses the mathematically pleasant representation of harmonious intervals as products of powers of primes. One is thus motivated to delve more deeply back into the just system, with a view to generalizing it in some way.

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Written by Jeffrey R. Boland & Lane P. Hughston