Tone-of-voice insights from music and the organization of the Tonnetz

Varieties of Tone of Voice and Engagement with Global Strategy (Part #10)

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Musical accompaniment of a Global Reset? It is unclear how justified it may be to explore the relation between tone in music and that in tone-of-voice. Clearly a degree of association is recognized and exploited in practice, especially in use of musical metaphors to distinguish tones-of-voice. It may well be seen as appropriate to accompany or introduce any presentation of a strategy with the use of music as a means of branding the strategy through associating music with however the tone-of-voice is understood (as described above).

Of concern is necessarily whether the choice of music should be governed by the criteria conventionally associated with an advertising jingle -- or martial music, according to a different tradition.

Geometry of music: The merit in exploring further is the sense in which the organization of tones as experienced in music has progressed far beyond what might be readily associated with scales and musical notation as conventionally represented. This suggests that there are new ways of comprehending the coherence of music which may be of value to comprehension and use of tones-of-voice -- and to the articulation of strategy.

The research of particular interest is notably associated with the manner in which the human brain appears to organize the patterns of tones that is appreciated, as clarified by the work of Dmitri Tymoczko relating music and geometry (A Geometry of Music: harmony and counterpoint in the extended common practice, 2011; The Geometry of Musical Chords, Science, 313, 2006). The manner in which such organization of tones in musical tuning and harmony has long been been explored in terms of the Tonnetz (a tone-network), namely a conceptual lattice diagram representing tonal space. Tymoczko has developed his understanding through its generalization (The Generalized Tonnetz, Journal of Music Theory, 56, 2012. 1).

Musical theory has notably developed further through a loose collection of ideas termed Neo-Riemannian theory. These are characterized by a central commitment to relating harmonies directly to each other, without necessary reference to a tonic.  Extended progressions of harmonies are characteristically displayed on a geometric plane, or map, which portrays the entire system of harmonic relations. Where consensus is lacking is on the question of what is most central to the theory: smooth voice leading, transformations, or the system of relations that is mapped by the geometries.

Of potential relevance to any organization of human values, Neo-Riemannian transformations can be modeled with several interrelated geometric structures. The Riemannian  (tonal grid, left below) is a planar array of pitches along three simplicial axes, corresponding to the three consonant intervals. Major and minor triads are represented by triangles which tile the plane of the Tonnetz. Edge-adjacent triads share two common pitches, and so the principal transformations are expressed as minimal motion of the Tonnetz. Pitches in the Tonnetz are connected by lines if they are separated by minor third, major third, or perfect fifth. Interpreted as a torus the Tonnetz has 12 nodes (pitches) and 24 triangles (triads).

Various experimental graphical representations of the Tonnetz have been presented:

• a toroidal view (below centre) has been proposed by David Bulger, and revised subsequently (David Bulger and Richard Cohn, Constrained voice-leading spaces, Journal of Mathematics and Music, 2015)
• an embedding of the Tonnetz on a 4D Hypersphere (below right) based on graph theory, as proposed by Gilles Baroin (The planet-4D model: An original hypersymmetric music space based on graph theory, 2011).

A valuable discussion of such possibilities is provided by Jason Yust (Organized Time: rhythm, tonality, and form, 2018; Geometric Generalizations of the Tonnetz and their Relation to Fourier Phases Spaces, 2017)

Selection of graphical representation of the Tonnetz
Planar organization of the Tonnetz	Toroidal view of the neo-Riemannian Tonnetz	Planet-4D embedding theTonnetz on a 4D hypersphere
Watchduck (a.k.a. Tilman Piesk), CC0, via Wikimedia Commons	Adaptation of Davidwbulger, Public domain, via Wikimedia Commons	Gilles Baroin PHD, Public domain, via Wikimedia Commons

It is intriguing that the focus on the Tonnetz offers explanations articulated by triangular modelling in various configurations, some lending themselves readily to 2D visualizations with tonal accompaniment. Nothing appears to be said about how the articulation is experienced, even though it could be said to be about the potential experience of an organized array of tones -- although an exception is the argument with regard to the Cognitive Interpretation of the Tonnetz (Robert Lieck, et al., The Tonal Diffusion Model, Transactions of the International Society for Music Information Retrieval, 3, 2020, 1). This then helps to frame the question of the nature of the cognitive engagement with the potentiality of a global array of tones-of-voice or of human values. The question acquires greater focus if the experiencer is recognized as located at the centre of a spherically symmetrical polyhedron -- especially one whose faces are triangulated. Aspects of the challenge of triadic thinking with respect to any mapping are discussed separately (Triangulation of Incommensurable Concepts for Global Configuration, 2011).

The models above are all suggestive of an imaginative approach which could be considered appropriate to the visualization of tones-of-voice and human values of relevance to representation of global strategy. In contrast to the "flatland" of checklists, the cognitive implications of the torus as a design metaphor, are discussed separately (Imagining Toroidal Life as a Sustainable Alternative: from globalization to toroidization or back to flatland? 2019).

Of relevance to this supposition is the language and preoccupation of a recent study (A. Milne, D. Bulger and S. A. Herff, Exploring the space of perfectly balanced rhythms and scales,  Journal of Mathematics and Music, 11, 2017, 2-3):

Periodic scales and meters typically embody "organizational principles" -- their pitches and onset times are not randomly distributed, but structured by rules or constraints. Identifying such principles is useful for understanding existing music and for generating novel music. In this article, we identify and discuss a novel organizational principle for scales and rhythms that we feel is of both theoretical interest and practical utility: perfect balance. When distributed around the circle, perfectly balanced rhythms and scales have their "centre of gravity" at the centre of the circle. The present article serves as a repository of the theorems and definitions crucial to perfect balance. It also further explores its mathematical ramifications by linking the existing theorems to algebraic number theory and computational optimizations.

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