Fishing for complements with chord, scale, and rhythm nets

Abstract

The aim of this paper is to argue that complementation is an operation similarly fundamental to music theory as transposition and inversion. We focus on studying the chromatic complement mapping that translates diatonic seventh chords into 8-note scales which can also be interpreted as rhythmic beat patterns. Such complements of diatonic seventh chords are of particular importance since they correspond to the scales popularized by the Jazz theorist Barry Harris, as well as to rhythms used in African drum music and Steve Reich's Clapping Music. Our approach enables a systematic study of these scales and rhythms using established theories of efficient voice leading and generalized diatonic scales and chords, in particular the theory of second-order maximally even sets. The main contributions of this research are (1) to explicate the correspondence between voice leadings and rhythmic transformations, (2) to systematize the family of Barry Harris scales, and (3) to describe classes of voice leadings between chords of different cardinality that are invariant under complementation.

Keywords:

• Clapping music
• rhythm
• Barry Harris
• voice-leading efficiency
• swap distance
• complementary chords
• betweenness graph

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplemental data

Supplemental data for this article can be accessed online at http://dx.doi.org/10.1080/17459737.2022.2164627http://dx.doi.org/10.1080/17459737.2022.2164627.

Notes

1 Inspired by this observation, Tymoczko (2011, Section 4) leaves an exercise to his readers in which the structural similarity of “lattices” representing complementary chords is to be explained: “Consider any single-step voice leading between two chords belonging to the same scale. This voice leading defines a unique single-step voice leading between the complements of those chords. […] This seems to show that the lattices representing complementary chords are structurally similar, even though they have different dimension.” One possible motivation (among others) of the present article is to accept the challenge behind this exercise and to explicate this observation from a more general point of view.

2 The original paper is based on an abstract metric space instead of our concrete choice of ${\mathbb{Z}}_n$.

3 Jacob Collier's term negative harmony for the mirroring of given “positive” pitch structures became a new-fashioned synonym for the old-fashioned Oettingen-Riemann dualism.

4 Personal communication between the authors and Francisco Gómez.

5 “J” perhaps standing for “John” and “Jack”, the first names of Clough and Douthett (1991).

6 Douthett (2008) developed a dynamical model of concentric rotating circles, whose $28\times 84$ configurations correspond to (the 28 inversions of) all diatonic seventh chords within each of the 84 diatonic modes whose configuration space can be represented with the help of the Douthett graph.

7 The Power Tower graph is also embedded into a chromatically saturated version of the Douthett graph for ${\mathrm{C}\mathrm{M}}_{12,7,4}$ (Harasim, Noll, and Rohrmeier 2019). However, its characterization as a subgraph is not trivial in that setting.

8 Notice that there might exist more than one relational voice leading which minimizes the distance between two chords: for instance, $R=\left\{\right(0,1),(2,3\left)\right\}$ and $R^{\mathrm{\prime }}=\left\{\right(0,3),(2,1\left)\right\}$ are voice leadings of distance 2 between the chords $X=\{0,2\}$ and $Y=\{1,3\}$ from ${\mathbb{Z}}_4$.

9 An interactive 3D model can be found here: https://www.geogebra.org/m/hqdbt6kj

10 In view of our footnote 1 about exercise 8 in Tymoczko (2011, p. 431), this result extends the findings in Harasim, Schmidt, and Rohrmeier (2016).

11 Yust mentions Colannino, Gómez, and Toussaint (2009) only in passing and leaves the rest to the reader.

12 Colannino, Gómez, and Toussaint (2009) spare no effort to motivate the concept of the directed swap-distance as an extension of the usual swap distance, but they do not use proper directed swap-distances to compare 8-beat rhythms with 7-beat rhythms.

13 Jason Yust pointed out (e.g. see Yust 2015), that adding an onset at the center of inversional balance means that all coefficients lie on the same line through the origin, meaning that they are either equal in phase (3, 4, 5), opposite (1), or have null coefficients (2, 6).