The Global Tonnetz

The Global Tonnetz DANIEL K. S. WALDEN Introduction: The Rorschach Test T he Tonnetz is like the Rorschach inkblot of music theory: what you see within it reveals how you make sense of tonal structures, our perceptual abilities, and the musical imagination. Commonly represented as shown in figure 1, the basic function of this diagram is to map out the tonal continuum as a web of relations between discrete and interlocking chains of fifths and thirds, conjuring a topography of pitches that can be used to orient the ear and calibrate tonal distances. Some interpret the Tonnetz in acoustical terms, as a representation of tuning relationships.1 Others describe it in psychoacoustical terms, as the rubric through which the mind converts acoustical stimuli into mental percepts.2 It has been deployed as a diagram for examining the group-theoretic possibilities of various forms of tonal organization,3 as a map for the disposition of buttons on fixed-tone instruments like the concertina,4 as a system for representing the distribution of pitch classes within a given composition,5 and as an array linked to the complexes of neurons in the medial prefrontal Research for this article was pursued with support from Yale University, Durham University, the Harvard Music Department, The Queen’s College (Oxford), the America-Japan Society, the Reischauer Institute for Japanese Studies, the British Academy, and the Leverhulme Trust. I am profoundly grateful to the personnel of the archives and libraries I visited over the years, and especially to Dr. Tanaka Tasuku and Mrs. Kumiko Tanaka for sharing materials from their family collections. I could not have written this article without Giulia Accornero, who helped me shape these ideas over years of conversation and careful reading of dozens of drafts. Thank you also to Alexander Rehding, Suzannah Clark, Katherine Butler Schofield, Liam Hynes-Tawa, Matthew Rahaim, Nathan Martin, Richard Cohn, Katherine Hambridge, Amanda Hsieh, Tuomas Eerola, Ian Dickson, Daniel Joseph, Judith Schelly, and Michael Walden for their invaluable comments and assistance. All diagrams and translations are my own, unless otherwise noted. 1. See Euler, Tentamen; and Farey, “On Mr. Liston’s.” 2. von Oettingen, Harmoniesystem; Hostinský, Die Lehre; Riemann, “Ideen.” 3. See Cohn, Audacious Euphony. 4. See examples in Gawboy, “The Wheatstone Concertina.” 5. Lieck, Moss, and Rohrmeier, “The Tonal Diffusion Model.” Journal of the American Musicological Society, Vol. 77, Number 2, pp. 447–510 ISSN 0003-0139, electronic ISSN 1547-3848. © 2024 by the American Musicological Society. All rights reserved. Please direct all requests for permission to photocopy or reproduce article content through the University of California Press’s Reprints and Permissions web page, https://online.ucpress.edu/journals/pages/reprintspermissions. DOI: https://doi.org/10.1525/jams. 2024.77.2.447. 448 Journal of the American Musicological Society Figure 1 The Tonnetz. From Tanaka, “Studien,” 5. In this version, the horizontal axis is tuned in perfect fifths (3:2), while the upwards diagonal axis is tuned in major thirds (5:4). Minor thirds (6:5) run along the downwards diagonal axis. The underlines indicate a pitch has been lowered by a syntonic comma, while the overlines indicate it has been raised by the same amount. Two under-/overlines indicate two syntonic commas, and so forth. cortex that are said to process tonal relationships.6 Some theorists shuttle between these varied perspectives in pursuit of interdisciplinary objectives. All told, by observing how scholars approach the Tonnetz, we can learn as much about their epistemological and ontological frameworks as we can about their theories. There are also multiple techniques for interpreting the discrete tonal elements that the Tonnetz represents—that is, the letters connected by the lines. These are often read as individual tones, but just as frequently, as the roots of harmonies, keys, or tonal regions. Nor is there much consistency concerning how the Tonnetz should be oriented. The arrangement shown in figure 1 is the most well-known following its appearance in the foundational article “Ideas for a Study ‘On the Imagination of Tone’” (1914) by the German music theorist Hugo Riemann.7 But sometimes theorists place the axis of major thirds at a perpendicular angle to the axis of fifths; in other cases, the thirds descend as one scans the page upward, or the horizontal and vertical axes are swapped. The Tonnetz’s geometry also changes depending on the tuning between its elements. When construed in just intonation, as in figure 1, the fifths are tuned to the ratio 3:2 and the thirds to the ratio 5:4, and it takes the form of an infinite two-dimensional plane. But when it is tuned in equal temperament, it bends into a three-dimensional torus.8 The Tonnetz has also gone by multiple names in different languages: a “duodene” or a “Table of Intervals” in English, a Tonverwandtschaftstabelle or Tongewebes in German, and a junseionkeimō 純正音係網 in Japanese, to name a few of its many aliases. A consensus nevertheless seems to have emerged in the last fifty years or so that the Tonnetz is a phenomenon specific to “Western” culture.9 6. Janata et al., “The Cortical Topography.” 7. Riemann, “Ideen,” 20. 8. See Hyer, “Reimag(in)ing Riemann.” 9. I use “Western” and “non-Western” in this article, not to affirm the value or utility of these categories, but because I cannot critique the epistemological frameworks I discuss