Jack Douthett and mathematical music theory

Abstract

Jack Douthett's work over three decades was central to defining an era in mathematical theory. The present special issue attests to his abiding influence over the field, as well as the energy he brought to research in all areas of mathematical music theory through his collaborations, correspondence, and relationships.

Keywords:

• Jack Douthett
• maximal evenness
• scale theory
• voice leading
• filtered point-symmetry

2010 Mathematics Subject Classifications:

• 00A65

When Jack Douthett had passed away in May of 2021, it was a given that this Journal would do something to honour his legacy. He was one of the founders of the Journal and the Society for Mathematics and Computation in Music in 2007, and his work was crucial to growing the field to the point where it could support an independent society and journal. My hope for this issue is to highlight the importance of his work, and to give a picture of his dedication and the inexhaustible energy he brought to breaking new ground in mathematical music theory.

I find it striking how Douthett's body work is at once large, spanning many years and many articles, and broad in its influence, but also remarkably unified. The ideas that appeared in his entrée into mathematical music theory, his article with John Clough, “Maximally Even Sets” (Clough and Douthett 1991), thread through his entire output. Specifically, two concepts, evenness and voice leading, are present across the span of Douthett's oeuvre and essential to his influence on twenty-first-century music theory, evenness and voice leading, and have their source in the way Clough and Douthett define maximal evenness. Another important aspect of this article has to do not so much with its content but how it came to be, establishing a collaborative spirit that I believe guided Douthett throughout his career.

The concept of evenness and its importance in defining basic objects of music theory, chords and scales, is the one most foregrounded in Clough and Douthett 1991. Douthett would continue to explore mathematical representations of evenness throughout his life. While maximal evenness effectively described the most common scalar collections of music theory, the pentatonic, whole-tone, diatonic, and octatonic, the common chords were only maximally even as subsets of the diatonic. The idea of second-order maximal evenness neatly accounts for major, minor, and diminished triads, and typical varieties of seventh chord (major, minor, dominant, half-diminished). He and Clough applied this idea to the ancient system of Indian scales in Clough et al. 1993 and Clough, Cuciurean, and Douthett 1997. In collaboration with Richard Krantz (Douthett and Krantz 1996; Krantz, Douthett, and Doty 1998) Douthett showed that maximal evenness had applications outside of music in physics. The concept as defined by Clough and Douthett (1991), however, did not immediately generalize. In a collaboration with Steven Block (Block and Douthett 1994) he explored some possible ways one could quantify the evenness of a set more generally. This work leads directly to his final article, published in this issue (Douthett et al. 2022), which addresses the evenness of partitions. A set might be seen as a simple case of a partition of the 12-tone aggregate, separating it into two complementary sets. Douthett et al. (2022) extend concepts of evenness to partitions into three or more sets. This article is a fitting culmination of a lifetime of work in music theory, and we at the Journal of Mathematics and Music are honoured to be able to publish it.

A less obvious feature of Clough and Douthett 1991 is its importance to the concept of voice leading that would emerge as a central contribution of mathematical music theory in the 1990s. It is typical to formalize voice leading as a mapping from one set to another, as in the influential theories of Lewin (1998) and Straus (2003). Beyond the formalization, however, it is clear that concepts of proximity are essential to the idea of voice leading. This makes sense given the common meaning of the term in music theory, where pitch proximity is often essential to defining voices or lines. The strong group-theoretic foundations of earlier mathematical music theory, in which the integers modulo 12 typically functioned as a first principle, led earlier theorists away from incorporating concepts of distance, which are not natural to groups (a point made forcefully by Tymoczko 2009). Nonetheless, Richard Cohn, in early work that he discusses in his introduction to Douthett's letters (Cohn 2022), focused on a concept of voice-leading distance (“parsimony”) defined as a relation (“P-relation”).

Douthett's work on voice-leading networks, which began in letters building on Cohn's ideas, some of which are transcribed and published in this issue (Douthett 2022a, 2022b), is some of his most influential. Networks of chords are eminently practical tools for theorists and analysts, and single-semitonal voice leading is a useful relation for many applications. When one constructs such a lattice for a limited number of the most even chords of a given cardinality the result is typically some kind of cubic lattice, with dimension corresponding to the number of voices that can move up or down at a given time. Douthett and Steinbach (1998) constructed such lattices for major, minor, and augmented triads (“Cube Dance”) and minor, dominant, half-diminished, and diminished sevenths (“Power Towers”). These became important referential structures for the voice-leading-based theories of Cohn (2012) and Tymoczko (2011), among many others. The idea of cubic lattices continued to yield fresh music-theoretic fruit years later, as Douthett, Steinbach, and Hermann (2018) demonstrate.

It may not have been initially obvious that Clough and Douthett 1991 had any direct relationship to these ideas about parsimony and voice leading. However, Clough and Douthett's way of defining maximal evenness, their J-function, was a crucial departure from the integers modulo 12 as the basis of mathematical music theory. This function, which maps one equal division of the octave (e.g. equiheptatonic) into another (e.g.12-tone) using a floor function is defined on real numbers modulo the octave, not just integers. Concepts of distance are natural to the real numbers, so it is easy to see how a diatonic scale is close to an even division of the octave by 7. The floor function is specifically premised on the idea of proximity.

The significance of the concept of distance underlying the J-function to voice leading would not be entirely clear until later when Douthett published his chapter on filtered point-symmetry (Douthett 2008). In a sense, filtered point-symmetry was nothing more than a visualization of nth-order maximal evenness and the underlying mechanism of J-functions. This visualization had the important effect of more clearly reifying the voices defined by the process, as “beacons” passing through multiple filters. As Douthett then subjected the filter configurations to dynamic processes, the fixed identity of the beacons became voices. The process restricts the motion of voices to small distances as the filters rotate continuously, and through this mechanism Douthett could derive all the important progressions highlighted by neo-Riemannian theory. His work on filtered point-symmetry was further developed by Richard Plotkin in his dissertation (Plotkin 2010) and in collaboration with Douthett (Plotkin and Douthett 2013). A contribution to this issue by Asensi Arranz and Noll (2022) further expands upon filtered point-symmetry, discussing historical precedents for its circular model, and eliciting further mathematical and music-theoretic ramifications of the idea.

Another notable aspect of Douthett's body of work, evident in that first paper with John Clough through to Douthett et al. 2022, is how collaborative it is. This is an unmistakable aspect of his character as a thinker, and crucial to the quality and influence of his work. Remembering the story Jack told of how he initially came to collaborate with John Clough, beginning with Clough's enthusiastic response to a letter from Jack, a complete stranger to him, I like to think that this initial collaboration with Clough inspired Jack's lifelong capacious intellectual generosity towards anyone with an interesting idea, regardless of their status as music theorists or lack thereof. Probably it was mostly just his character, though. The number of co-authors he published with, and also the volume of unpublished correspondence he produced, attests to his collaborative nature. The small sample of these that we publish in this issue (Douthett 2022a, 2022b) give a taste of the mathematical and music-theoretic richness of these letters, some of which found its way into published articles, while much of it has yet to do so.

Many of the tributes in Krantz 2022 mention this aspect of Douthett as a thinker and collaborator. The number of people that he shared the gifts of his insights with in this way is impressive; while he certainly corresponded heavily with his many co-authors, he also had quite substantial correspondence with theorists who were not co-authors. I can personally attest: Jack sent me significant letters (alas, electronically and not hand-written!) on multiple topics such as higher-order maximal evenness, filtered point-symmetry, and the discrete Fourier transform, which influenced my thinking about all of those topics.

Acknowledgments

Thanks to Leah Kier who has done so much to honour Jack since his passing, and thanks to B.C. Nowlin studios who has generously allowed us to use the following image, Jack's favourite painting, by B.C. Nowlin, entitled “Victory.”

Disclosure statement

No potential conflict of interest was reported by the authors.