The Global Tonnetz Free

The 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 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 Scholars have offered what could be called “synchronic” and “diachronic” explanations to elucidate how and why. The synchronic explanation starts from the observation that the lattice appears to have been optimized for triadic harmony, the signature feature of Western tonal practice. For that reason, the logic goes, it is chauvinistic to suggest that the Tonnetz could be used to reveal insights into non-Western forms of music that do not traditionally feature triadic harmony, such as Japanese gagaku (雅楽) or a Hindustani rāga. To do so would seem to entail rejecting the norms of cultural relativism, or upholding the presumption that the West held privileged access to a universal truth that the rest of the world lacked. Theorists like David Lewin, in his discussion of the Tonnetz among other “musical spaces pertinent to theories of Western tonality,” thus stress that the harmonic “intuitions” the diagram represents rely heavily upon “cultural conditioning.”¹⁰ Candace Brower takes care not to “suggest that we can extrapolate from the experiences of Western subjects to those of non-Western listeners” in accounting for modes of interaction with her own Tonnetze.¹¹ Fred Lerdahl proposes that the structures of “Western tonality” he models in his own twist on the Tonnetz are grounded in the principles of psychoacoustics, but he also makes sure to state that there is nothing “obligatory” about the way they have been developed; other systems are at liberty to “not pursue this route.”¹² Each of these scholars approaches the Tonnetz differently, but they all concur that whatever conclusions they draw are mostly, if not exclusively, pertinent to Western music theory alone.

The diachronic explanation, championed by historians of theory, involves tracing the various appearances of the Tonnetz over the past three centuries of musical discourse. Edward Gollin, for instance, presents the stemmatic diagram shown in figure 2 in order to summarize his narrative about how Carl Ernst Naumann’s 1858 synthesis of notational conventions and proto-Tonnetze by Moritz Wilhelm Drobisch and Moritz Hauptmann eventually led (through Arthur von Oettingen) to Riemann’s better-known version of the diagram in figure 1.¹³ Other scholars have provided historical accounts that allow for a more extended genealogical tree. Kevin Mooney locates the origins of the diagram a hundred years earlier in Leonhard Euler’s Tentamen novae theoriae musicae, written in 1739 while the theorist was at the Imperial Russian Academy of Sciences.¹⁴ Richard Cohn and Suzannah Clark draw attention to the important roles that theorists from Bohemia and France played in its dissemination, while Ellen Lockhart has brought previously overlooked English sources back to light.¹⁵ Taken as a whole, this body of scholarship depicts a network of communication and exchange that spans from St. Petersburg to London. Nevertheless, despite efforts to extend its breadth, that network remains confined to European capitals until the twentieth century, when neo-Riemannian theorists in North America began to revive interest in the diagram’s varied affordances. The overarching conclusion one inevitably draws from this scholarship is that the genesis and evolution of the Tonnetz took place within the boundaries of what is generally called the West.

Look closer, however, at the caption beneath figure 1. This reproduction of the Tonnetz is not taken from one of Riemann’s texts, but from an article by the Japanese music theorist, inventor, and physicist Tanaka Shōhei (田中正平) published in 1890—years before Riemann had published a Tonnetz like it.¹⁶ Tanaka had adapted this Tonnetz from a diagram in Otakar Hostinský’s 1879 treatise, Die Lehre von den musikalischen Klänge, which similarly locates fifth-chains on a horizontal axis and major-/minor-third-chains on diagonal axes, but is flipped upside-down.¹⁷ Tanaka’s new configuration quickly became the standard for German-language scholarship, reappearing in works by Carl Eitz and Arthur von Oettingen, and eventually in Riemann’s “Ideas,” from which text Brian Hyer and others eventually derived the version that took hold in neo-Riemannian scholarship.¹⁸ But Tanaka’s personal ambitions for the Tonnetz extended well beyond the reshaping of its constituent parts. One of his goals was to demonstrate that the tonal relations it represented were illustrative of a music-theoretical discourse that had been co-produced by Germany and Japan, rather than Germany alone. Later in his career, he also sought to show how the diagram could be used to represent Japanese, Thai, and Indian tonal systems. The synchronic and diachronic explanations of the Tonnetz are deficient in describing his various renditions of the diagram, designed to demonstrate that so-called “Oriental” musical systems were represented within its geometries and that its evolution had involved Asia alongside Europe in its making.

Figure 3 reproduces another Tonnetz that cannot simply be identified as Western. Drawn in 1918 by the theorist G. S. Khare from Pune, this Tonnetz superficially resembles Naumann’s but was likely inspired by the duodenes of the English theorist Alexander John Ellis. Khare’s Tonnetz seems to have first appeared in a private letter about Sanskrit music theory addressed to Abraham Pandither of Tanjore (now Thanjavur), who then republished it alongside a scathing critique of its reasoning.¹⁹ Khare had used the diagram to represent a tuning system that he attributed to a Sanskrit text by the seventeenth-century theorist Ahobala. That tuning system, he contended, was derived from an even more ancient Indic approach that involved dividing the octave into twenty-two quanta called śrutis. The standard synchronic and diachronic models thus fall short of Khare’s Tonnetz as well. How, then, might these two Tonnetze complicate the understanding of the diagram that has emerged in recent academic scholarship? And what might the absence of both diagrams from previous historical accounts reveal about shortcomings in our current approaches to the history of theory?

To answer those questions, I will apply methods from the field of global history to the Tonnetz, drawing on primary sources and archival documents—including previously unexamined manuscripts, letters, photographs, and instruments—that allow us to retrace the history of its exchange among various contexts in Germany, India, and Japan. In taking this approach, I follow my colleagues in musicology, music theory, and ethnomusicology who have already drawn on global historiography in their work. Daniel Chua has recently called for a global music history that constitutes “the study of all music everywhere in any way and every way; it explores the interconnectivity of everything,” while Olivia Bloechl writes of the need for historians to respond to the “growing mountain of evidence that interconnection across borders mattered, in some cases fundamentally.”²⁰ Such manifestos have fostered a vibrant body of scholarship drawing on approaches as varied as the history of cultural transfer, microhistory, and the study of “contact zones” to explore cross-cultural networks of contact that produce (and are produced by) the traffic of musicians, musical works, theories, instruments, and listening practices.²¹ Bloechl argues that these approaches carry enormous potential: namely, “a sense of the possibility that ‘globalizing’ music history may have a democratizing effect on our fields,” by “repudiat[ing] the premises of European exceptionalism and white supremacy that lurked beneath many nineteenth-century historiographies of music, particularly nationalist ones.”²² Global history, by such accounts, seems to offer more than a new perspective on past events of musical culture: it may provide a set of disciplinary correctives that would assist the development of a new and more open scholarly community.

I respond to these calls in the next four sections, which are structured as a series of four microhistories that trace a tale of two Tonnetze. I first illustrate how Tanaka presented the Tonnetz of figure 1 as evidence of a “mutually civilizing” network of transnational exchange between Germany and Japan that was required to facilitate the continued development of tonality.²³ Next, I turn to the response of Riemann, who used the same Tonnetz to re-center European musical logic and reaffirm that its unilinear development was the product of dynamics internal to Europe. I then shift to North India, to show how Khare and some of his associates in the so-called Philharmonic Society of Western India drew on Indian nationalist logic in using the Tonnetz in figure 3 to bolster claims of Indian “priority” in the development of musical modernity. I close by examining how the work of the Philharmonic Society inspired Tanaka to revise his stance on the Tonnetz in later years, leading him to offer an alternative Pan-Asianist account of history that cast Japan as the last bastion of an uncolonized Asian spirit. The Tonnetz thus came to play a set of contradictory roles: in some contexts it was deployed to reify imperial schemes to control musical culture, while in others it was used in acts of cultural self-fashioning that supported arguments for sovereignty. Occasionally, the objectives of these authors backfired, and the diagram functioned in a manner opposite to what was intended.

These microhistories uncover networks of scholarly connection that stretched well beyond the confines of Europe. I am skeptical, however, that tracing such networks is sufficient on its own for global history to deliver on its promise to “repudiate” the specters of Eurocentric historiography. For that reason, my method diverges in the final section, where I chart a course around certain methodological pitfalls. I draw on Sebastian Conrad’s distinction between three types of global history: as a “history of everything”; as a “history of connections”; and as a “history based on the concept of integration.”²⁴ The first two approaches, I argue, are the primary models for most recent manifestos for global music history: Chua’s study of “all music everywhere in any way and every way” fits the model of Conrad’s type one, while Bloechl’s surveys of “interconnection” appear to evoke type two. But if the goal of global music history is to break free from the grip of Eurocentrism, I argue that such approaches must be supplemented by Conrad’s third type focused on “integration.” To that end, I clarify in the final section how the claims about the Tonnetz presented in these microhistories, divergent as they appear, are linked by their concerns for modernity. I couple Conrad’s theories with insights from historian Sanjay Subrahmanyam in showing how their ideas about the distributed origins of modernity are best captured by the concept of “emergence” as described in philosophy, science, and systems theory. I argue that emergent historical phenomena, including modernity, do not come from a particular place of origin or follow a clear chain of causation, but rather surface from the integration of social structures located in various places across the globe. As a constitutive feature of an emergent musical modernity, the Tonnetze of Tanaka, Riemann, and Khare that I discuss here are not predominantly or finally Western, but always already global, both in their origins and effects. For this reason, each of their Tonnetze constitutes a “global Tonnetz.” I finally close with broader reflections on the methodologies of global music history, suggesting that we must learn to entertain multiple historiographic approaches at once—even when they appear paradoxical—if we are to counterbalance the centripetal forces that persist in haunting our narratives.

The tale begins in Stuttgart on November 6, 1891, with a lecture presented by Tanaka—shown from around that time in figure 4—before the Württemberg Society for Commercial Geography and the Promotion of German Interests Abroad. His subject was not music theory, but the history of transnational exchange between Japan and foreign countries.²⁵ To begin, Tanaka related how foreign exchange played an integral role in the region’s development. The seeds of Japanese civilization were planted in ancient times by China, he explained, while the first contact with European missionaries in the sixteenth century brought further scientific, medical, and technical knowledge. Not long thereafter, however, the increasingly “provocative behavior” of those missionaries led the Tokugawa shogunate to implement a seclusionist policy, severly restricting trade, foreign relations, and migration both in and out of the country. Japan then found itself closed off to the rest of the world for 265 years, living “as if trapped in their own cage.”²⁶ Bans were placed on Western literature and knowledge, and the absence of “vital stimuli and international competition caused the stagnation of science and technology,” leaving Japan with “poor transportation and undeveloped industry and trade.”²⁷ A few “pioneers” however continued studying Western texts in secret, and the “drops of new culture” they introduced slowly “united in a mighty stream breaking through all obstacles.”²⁸ The seclusionist policy finally ended in 1853, and the restoration of Emperor Meiji in 1868 constituted an event of “world-historical significance in that it caused the rupturing of all artificial dams, thereby directing the flood of Western cultural currents into the quiet stagnating waters of our country, purifying and invigorating them, and driving them into the general stream of human progress.”²⁹

Tanaka’s lecture might seem little more than a rehash of Europhilic tropes. In his telling, Japan stagnated when left to its own devices, but the reintroduction of Western knowledge had enabled it to transform into a modern nation-state. But his framing included an important nuance. As he explained at the outset of his lecture, “one of the most pronounced features of modern culture is surely the united striving of all mankind toward a great common goal,” and no nation could attain that telos on its own:

Tanaka’s term for that mutually civilizing force was geistiger Verkehr, reflecting his belief that the benefits that had emerged from the traffic between Japan and other countries was not only intellectual, but spiritual as well: “stimulated by external inspiration and pressed ahead by robust competition, the fresh, restless spirit of Progress needed only be awakened, as if spontaneously lit, to work miracles everywhere.”³¹ The implication was clear: Europe had much to offer Japan, but Japan had just as much to offer Europe.

For Tanaka, all the nations of the world had progressed along the same universal timeline, fueled by “robust competition” and collaboration. Europe was currently ahead of Japan because it had been more open to exchange with other countries, he suggested, not because it was essentially superior. His perspective harmonizes closely with much turn-of-the-century Japanese historiography, which, as the historian Stefan Tanaka has shown, tended to accept one of the key historiographical premises of contemporary Western historiography: that a set of universal laws governed the progress of societies, and that progress was measured by the advancement of science and technology.³² The advantages of this narrative for Japanese historians at the turn of the twentieth century were two-fold. First, it provided a rationale for the reopening of Japan to the world during the Bakumatsu and the Meiji Restoration. Second, by drawing on the Social Darwinism of the English philosopher Herbert Spencer, while at the same time decoupling his scheme of social progress through “survival of the fittest” from the assumption that one nation was intrinsically more fit than any other, many Meiji-era historians believed they could recast the West as simply another competitor on the universal path toward progress, and upend the notion that their nation constituted a “perpetually incomplete version of the West.”³³ Once it had adapted to the contingencies of the new global environment, Japan would establish itself as Europe’s equal and release its own westward-flowing “purifying streams.” Indeed, Tanaka would suggest elsewhere that the moment had already arrived, at least in the field of music theory: for thanks to his own research, Japan was in the position to help the West realize the sounds of the Tonnetz on keyboard instruments for the very first time.

The pathway that led Tanaka to Stuttgart was far from straightforward. Born in a semi-rural village on Awaji Island in 1862, Tanaka had just reached school-going age when the Meiji Empire initiated a flurry of measures aimed at modernizing the government, military, economy, and education. He was, as a result, among the first generation to receive tutelage in Western subjects under government-appointed European and American educators.³⁴ At sixteen, he entered Tokyo University and studied under the supervision of physicist Thomas Corwin Mendenhall, who lauded Hermann von Helmholtz’s acoustical research as “one of the most splendid contributions to modern science,” and encouraged Tanaka’s interests in acoustics and music theory.³⁵ Mendenhall also directed Tanaka to study voice, violin, and organ, and held weekly musical soirées that combined violin recitals with discussions of the logic behind “pure harmony [junsei wasei 純正和声].”³⁶ At twenty, Tanaka received his master’s degree in physics granted by Tokyo University, and two years later, he received a scholarship from the Meiji government to complete his doctorate at the University of Berlin.³⁷ While he was there, he learned acoustics from Helmholtz himself, as well as counterpoint and musicology with Heinrich Bellermann and Philipp Spitta; thereafter, he also studied harmony and musical form with Ludwig Bussler at Stern’sches Konservatorium (now the Berlin University of the Arts). He sang with a choral society and played violin in the conservatory orchestra.³⁸ Tanaka immersed himself profoundly in German cultural life. At that time, most Japanese students abroad returned after four years, but he remained in Europe for sixteen.

The primary hope of the Meiji government for their students abroad was that they would acquire the tools needed for assisting with Japan’s ongoing modernization. During his studies, however, Tanaka came to believe that Germany, not just Japan, needed assistance with its musical progress. Helmholtz, his mentor, had instructed that musicians from around the world had naturally gravitated toward the inclusion of the just octave (2:1), fifth (3:2), and fourth (4:3) in their tonal gamuts on account of the correspondences between these intervals and the overtone series.³⁹ Soon after, he had posited, some musicians had developed a new principle for determining even smaller intervals, which consisted “in an endeavor to distinguish equal intervals by ear, and thus make the differences of pitch perceptibly uniform.”⁴⁰ This principle, he claimed, was “still employed among the less civilised nations.”⁴¹ Others, including European musicians, had continued to draw inspiration from the overtone series, incorporating the major third (5:4), minor third (6:5), and major sixth (5:3). Eventually, however, in modern times, European musicians stopped using perfectly tuned intervals in performance, and opted for twelve-tone equal temperament instead.⁴² At first this switch had seemed favorable for musical progress, he argued, as it was far more practical for fixed-tone instruments, and had little effect on the tuning of fifths and fourths. Equal-tempered thirds and sixths, however, were less than ideal.⁴³ Triads were eventually drained of their consonance, forcing composers searching for color and contrast to deploy even harsher dissonances and modulate excessively. Ultimately, Helmholtz maintained, equal temperament “threaten[ed] entirely to destroy the feeling for tonality,” the “principle upon which modern musical art is founded.”⁴⁴

Building on these arguments, Tanaka concluded that German music was in a profound spiritual and technological crisis, beholden to a tuning system that would inevitably cause the tonal system represented in the Tonnetz to collapse. “Great” composers like Mozart had somehow managed to listen past the tempered intervals of their instruments, and were “led by their keen perception of tones” to “create their works in just harmony.” One could tell as much by systematic analysis of the harmonic logic of the progressions their music contained.⁴⁵ But in the present day, most musicians were misled by the equal temperament they heard daily, and were beginning to mistake it for “genuine harmony”: “on authority, as well as shere [sic] custom, they are not only satisfied, but even consider it as a sound basis to their musical practise [sic].”⁴⁶ To avoid a future marred by excessive chromaticism—if not the complete disintegration of tonality—Tanaka believed that Germany had to restore the practice of just intonation. To that end, he set about developing a new fixed-tone instrument that could preserve just consonances while still facilitating modulations and transpositions with relative ease. By 1890, following the completion of his doctorate, Tanaka published an article in Guido Adler’s Vierteljahrsschrift für Musikwissenschaft announcing his solution: the “enharmonium” shown in figure 5, equipped to translate the ideal tonal geometries of the Tonnetz into real, audible sound.

Tanaka’s article, entitled “Studien im Gebiete der reinen Stimmung [Studies in the Field of Just Intonation],” is in four parts.⁴⁷ Part one begins with the claim that the tonal imagination is based on the properties of the just fifth and the just third, which form the major and minor triad when harmonized. He discusses his configuration of the Tonnetz, illustrated in figure 1, and explains its various features and attributes. He begins by showing that, to locate a major triad above the C situated at the center of the diagram, one must select the fifth to its immediate right, and the third that sits on the upward-right-leaning diagonal. (One cannot take the third from the same horizontal axis as the initial tone, as this forms a Pythagorean ditone instead of a just major third, which is roughly one-fifth of a tone too high.) After explaining how to locate other types of chords and collections, he eventually arrives at the topic of how one might realize the Tonnetz on a keyboard instrument. In doing so, Tanaka navigated a contradiction: the Tonnetz represents an infinite continuum, but a keyboard instrument has a finite number of keys. His solution, shown in figure 6, consisted of twenty keys per octave and featured double- and triple-split sharps, an extra key for an E#, a transposing mechanism, and a knee lever that automatically swapped sharps into flats.⁴⁸ He then explains, in granular detail, how this configuration grants the performer the ability to play almost any work in the Western classical repertoire in a temperament that realizes the intervallic relationships represented in the Tonnetz with near perfect fidelity.⁴⁹ Parts three and four of his article finally situate his Tonnetz and the keyboard in a broader historical context: first, within the development of harmonic theory from ancient Greece to the common-practice era, and then in relation to the genealogy of alternative enharmonic keyboard designs from the Renaissance to the present day.⁵⁰

As complex as the keyboard initially seems, Tanaka explains, the advantage of his configuration is that it manages to preserve as much of the original layout of the keyboard as possible, given the number of tones required. The pianist can thus use many of the same hand fingerings for various scales or chords: to play a C major triad, for instance, one can hold the hand exactly as one might at a conventional piano. Notably, as figure 7 shows, the triangular disposition of the thumb, middle finger, and fifth finger that reaches for this C major triad on Tanaka’s enharmonium (or the modern keyboard) corresponds to the arrangement of elements that forms the same triad on the Tonnetz. Tanaka thus ensures that the gap between the diagrammatic abstractions of the Tonnetz and the haptic experience of playing the keyboard remains small, guaranteeing that his reformation of Hostinský’s version of the diagram and the keyboard interface go hand in hand.

Following the publication of “Studien,” Tanaka embarked on an extensive tour of musical institutions across German-speaking lands to introduce his instrument. Newspapers reported that reception was favorable, and by April 1890, Tanaka had secured an audience with Emperor Wilhelm II, who agreed to finance Tanaka’s plans for a full-size just-intonation organ.⁵¹ Tanaka consistently presented his enharmonium as developed for the benefit of German music; he even claimed to have considered coloring its keys red, white, and black after the flag of the German Empire, dissuaded only by the difficulty in obtaining a suitable dye.⁵² At the same time, however, Tanaka framed his invention as a gift from Japan to Germany: an act of geistiger Verkehr. Indeed, the initial prototype had been funded by the Meiji Government and Governor of Tokyo, based on Tanaka’s promise that the instrument would confer “not only individual honor, but honor to Japan,” and his presentations were regularly attended by Japanese ministers and ambassadors in a show of their national support.⁵³

His efforts seem to have paid off, as Tanaka’s German audience appears to have recognized the enharmonium as transnational gift. The composer Heinrich von Herzogenberg saluted the enharmonium as a “transfusion of Japanese keenness of intellect and technical skill into the arteries of an eminently occidental question, and to the true blessing of the same, for its solution has been placed on a practical basis and has succeeded in [sic] such an eminent degree.”⁵⁴ Gustav Engel, a prominent tenor and music theorist, marveled that “a Japanese man should successfully assist in the resolution of the most subtle difficulties in the area of musical acoustics.”⁵⁵ Hans von Bülow—whom Tanaka also credited with naming the instrument the “enharmonium”—appeared to revel in this notion:

Tanaka looked past the condescension and Orientalism of these tributes, publicizing many of them widely and even compiling selections into two albums he published in German and English.⁵⁷ For him, such testimonials served to illustrate his success in communicating that Japan had lessons for Germany on how to surmount its own inability to realize the tonal imagination represented in the Tonnetz.

A detailed account of his demonstration of the enharmonium to Anton Bruckner, recorded by the composer’s amanuensis Friedrich Eckstein, provides further insight into how Tanaka represented the instrument as a token of geistiger Verkehr. On July 26, 1890, Tanaka arrived unannounced at Bruckner’s door and introduced himself as a member of the Japanese Legation and a former pupil of Helmholtz. After handing over a copy of his monograph, he cajoled Bruckner and Eckstein into joining him in the carriage of the Japanese ambassador on a short ride to the Japanese embassy. Upon arrival, Tanaka toured them through “a series of staterooms decorated with the most splendid treasures of East Asian art” before reaching the private boudoir of the ambassador’s wife, where an ordinary harmonium sat next to the “strange [merkwürdig]”-seeming enharmonium.⁵⁸ Seated at his instrument, Tanaka struck up the opening bars of Wagner’s Lohengrin, annotated in a “peculiar [eigentümlichen], complicated [verwickelten], and altogether incomprehensible [unverständlichen] notation” for the enharmonium—most likely a one-piano version of what is shown in example 1.⁵⁹ He then moved to the ordinary harmonium and played the same passage, which according to Eckstein now sounded “unbearably out of tune.”⁶⁰ Bruckner found the effect “most surprising and overwhelming,” especially because he had assumed the ear could not tolerate the purity of intervals in just intonation.⁶¹ Tanaka’s performance, Eckstein wrote, thus demonstrated “the singular pleasure of having all those otherwise only ideal, purely cerebral reinterpretations demonstrated in real life by clearly discernible tone steps and key exchanges!”⁶²

It appears from Eckstein’s account that Tanaka had managed to communicate his message: that just intonation was meant to be heard, not just seen on the printed page. Even if analysis alone was capable of revealing that the music of “great” composers like Wagner (and Mozart) was rooted in the just-intonation logic of the Tonnetz, its true meaning would remain concealed until it was realized in practice on instruments that could give voice (and tactile grip) to the Tonnetz’s full harmonic glory. One might also deduce from Eckstein’s account that Tanaka deliberately staged a defamiliarizing encounter for his guests, one in which the strains of the German tonal imagination would appear to emerge from a Japanese invention. The journey in the ambassador’s carriage, the tour of the Japanese art collection, and the presentation of the instrument in the boudoir of the ambassador’s wife had primed Tanaka’s visitors to expect the sounds of the Far East. But instead, Tanaka played something quintessentially German: Wagner. The exotic setting and “strange” appearance of his instrument belied its ability to render the Tonnetz in its true acoustic magnificence, closer to the quick of Wagner’s musical ideals. The West may have conceived the Tonnetz, and Wagner and Mozart unfurled its potential, but it required Japan to actually sound out its truth. In this way, Tanaka demonstrated that the continued progress of modern musical culture would depend upon geistiger Verkehr: it would be a co-production of Germany and Japan.

Tanaka won Bruckner over, but was less successful with Riemann. Shortly after the enharmonium was debuted, the German theorist touted the “ingenious construction” of its interface as “the most perfect for the demonstration of the advantages of just intonation over our strongly tempered twelve semitone system.”⁶³ But he also raised doubts, as he speculated about “whether or not [equal temperament] will resist all efforts to introduce just intonation into musical practice remains to be seen”:

Even if just-intonation instruments like Tanaka’s did not “fail due to practical unfeasibility,” Riemann wrote, they “entail a tremendous restriction in chordal and modulatory processes. And what would be gained from it: some slight sensual euphonious quality of the individual harmonies—at the expense of more profound expression, seizing and captivating the soul!”⁶⁵ By 1914, Riemann’s position had hardened: “the ‘Alpha and Omega’ of musical artistry is not found in the actual, sounding music, but rather exists in the mental image of musical relationships that occurs in the creative artist’s imagination—a mental image that lives before it is transformed into notation and re-emerges in the imagination of the hearer.”⁶⁶ In his view, there was no need to realize the Tonnetz because the essence of music resides on the imaginative rather than the acoustical plane. An informed listener of Mozart can continue to apperceive the logic of the Tonnetz at the core of his harmonic syntax, notwithstanding its equal-tempered exterior. Tanaka’s efforts, however impressive, were pointless.

Riemann grounded his written arguments in music-theoretical claims, but it also didn’t hurt, politically speaking, to claim that Tanaka had been wasting his time. In the years after Tanaka presented his instrument, Germany’s relations with Japan had shifted, precipitated by Japan’s victories in the Sino-Japanese War and the Invasion of Taiwan in 1895. Emperor Wilhelm II, perceiving the ascension of Japanese military might as a threat to Germany’s own colonial ambitions in China, began fueling racist campaigns against the people of East Asia, whom he framed as a “Yellow Peril.” One notorious example appeared in the lithograph shown in figure 8, illustrated by Hermann Knackfuss but based on a sketch by Wilhelm II himself, entitled “Peoples of Europe, Guard Your Most Sacred Possessions! [Völker Europas, wahrt eure heiligsten Güter]” and depicting Germany as the Archangel Michael mustering allegorical representations of Europe against a Buddha surrounded by flames. The language of Wilhelm II’s rallying cry eventually infiltrated musical polemics. Not long after, Riemann’s associate Felix Draeseke invoked this language in calling on German musicians to “preserve your most precious possessions,” and not be “blinded by subversives who do not want progress, but only a revolution!”⁶⁷

Indeed, just a few years before Draeseke issued his invocation, Riemann began writing histories of music theory that depicted Germany as the primary driver of musical progress along a universal developmental pathway.⁶⁸ One such example appeared in a lecture about Japanese music that he delivered on March 16, 1902, before an audience of three hundred at the Museum für Völkerkunde in Leipzig.⁶⁹ Riemann had been invited by the museum to introduce a performance of traditional Japanese music by the twenty-four-year-old musician Kōda Kō (幸田幸) on the museum’s prized thirteen-string koto, an instrument previously displayed at the 1889 Paris World Fair, but now gathering dust in a display case.⁷⁰ Despite acknowledging that Japanese music was a gap in his expertise, and that he would have to rely on secondary literature, Riemann used this event as an opportunity to present his own narration of the development of Japanese music theory from its origins to the present, contextualized by references to what he believed to be the universal pathway of musical progress.⁷¹

Both East Asian and European traditions, he claimed, could trace their lineage back to a “primordial time [Urzeit]” when the only consonance that was recognized and understood was the fifth.⁷² Early on, musicians discovered that chains of fifths could be formed symmetrically above and below that fifth, and that the tones from these chains could then be reorganized into a melodic gamut. First came the recognition that a chain of five fifths that stretched symmetrically outwards from a single pitch, identified by Riemann as “A” (e.g. G–D–A–E–B), could be rearranged to produce the anhemitonic pentatonic collection (G–A–B–D–E).⁷³ Next, by extending the chain one more fifth in either direction, musicians derived heptatonic collections (such as C–G–D–A–E–B–F#) that they then rearranged into scales (G–A–B–C–D–E–F#–G). Various civilizations continued to explore gamuts comprising fifth-related tones, until the potential for the major and minor third to unlock a new axis of tonal relations was finally discovered.⁷⁴ That discovery, he asserted, eventually led to the achievement of modern tonality. It was what made the Tonnetz possible.

According to Riemann’s narrative, Japan had been diverted along the way. While it did progress from pentatonicism to heptatonicism, it failed to discover the consonance of the third as other nations had. Japanese music thus remained trapped on a one-dimensional plane of fifth-relations. In fact, Riemann argued, it had even regressed by veering back from heptatonicism to pentatonicism by omitting two scale degrees—the central A and the D a fifth below—“not out of inner necessity, but out of arbitrariness” to develop the pentatonic gamut E–F#–G–B–C–E.⁷⁵ This pitch collection sounded like it was based on a minor scale to what Riemann called the “modern” (i.e., Western) ear, but technically, as it was tuned by fifths alone, it was better known as the “artificial modern minor pentatonic [die künstliche moderne Moll-Pentatonik].”⁷⁶ This was the system, he concluded, that Kōda would present in her demonstration of three Japanese compositions for the instrument: “Rokudan no shirabe (六段の調),” “Shōchikubai (松竹梅),” and “Azumajishi (吾妻獅子).”⁷⁷

In characterizing Japan as regressive, if not pre-modern, Riemann was deploying the customary tropes of what Johannes Fabian dubs “allochronism”: the notion, recurrent in colonial discourse and persistent in anthropology, that non-European spaces exist in another past time that Europe had left behind in its developmental progress into the present.⁷⁸ Not all of Riemann’s audience could be persuaded, however, especially as a sizeable contingent of “Japanese Leipzigers,” mostly diplomats and students abroad, were in attendance. According to one audience member, the novelist Iwaya Sazanami (巌谷小波), the German audience “had no true understanding of the wonderful koto that had not been played for ten years,” even if they were “completely absorbed” by its sound.⁷⁹ Following Kōda’s demonstration, Iwaya also noted, Riemann had programmed two harmonizations of Japanese melodies for violin and piano that he had written himself: “Haru no uta (春の歌),” and “Kimi to wakarete (君と別れて),” shown in example 2.⁸⁰ These harmonizations were far from positively received by his Japanese audience, Iwaya recorded: “it was as if they were ridiculing both us and the wonderful tones elicited by the koto that soothe the ears by giving us earaches instead.”⁸¹

Displeasure may however have been precisely what Riemann hoped to obtain. In a later comment, Riemann admitted that the harmonizations were “strange [sonderbaren] . . . this I know well myself.”⁸² That strangeness, he wrote, was however to “compel audiences to perceive the melodic tones as the exclusive product of fifth relations.”⁸³ In his arrangements, he avoided all harmonies other than the fifth-related minor tonic, minor subdominant, and minor dominant, despite the presence of options that would have sounded more idiomatic in tonal harmony. In a conventional minor-mode work, for instance, the downbeat of m. 14 would typically feature the major dominant instead of the minor dominant, in order to heighten the resolution of the leading tone to the first scale degree.⁸⁴ A move to the major mediant may also have been suitable. But for Riemann’s purposes, both options were off the table—the former as it involved degrees outside the “artificial modern minor pentatonic” collection, the latter as it involved third relations. Riemann’s purpose was to demonstrate that the tonal imagination that had conceived these exotic melodies was mired in the one-dimensional plane of fifth relations, and could not be successfully “modernized” by triadic syntax. If progress culminated in the Tonnetz, Japan’s incapacity to reach that end goal ensured it would be shut off from triadic harmony.⁸⁵

Yet Riemann’s arguments were called into question by the mere presence in the hall of Japanese musicians who had integrated into German musical life. Kōda, for instance, was in Germany because she was a highly accomplished violinist studying with Joseph Joachim at the Universität der Künste.⁸⁶ Many audience members were doubtless aware that she could easily have played Riemann’s harmonizations herself, undercutting his efforts to silo German and Japanese musical practices. And had Riemann acknowledged Tanaka’s influence over his own conception of the Tonnetz, he would have been forced to admit that Japan was helping to reshape the conceptual foundations of triadic tonality—proving that Japan was not mired in the past, but as modern as Germany.

Let us shift to the ballroom of the Lakshmi Vilas Palace, residence of Maharaja Sayajiro Gaekwad III of Baroda State (Vadodara), and the site of the First All-India Music Conference held between March 20 and 25, 1916. Sponsored by the Maharaja as a “scheme of social progress,” the event summoned a group of leading theorists from across India, shown in figure 9, to assist with the goal of “systematising Indian music and placing it on a scientific basis.”⁸⁷ Doing so, the organizers believed, would foster “Indian unity” and the “Soul of the Nation.”⁸⁸ Differences of opinion were encouraged, in the spirit of fostering a modern debate: for as the local Minister of Education explained in his opening address, “it is the trend of modern civilization to make a united effort to bring about the solution of difficult and debatable subjects by organizing Conferences and Congresses, where all shades of opinion are fully represented.”⁸⁹ As it turned out, serious divisions emerged straightaway on the subject of how Indian music should be tuned.

The central provocation of the event consisted of a pair of speeches by two theorists: Krishnaji Ballal Deval, a retiree of the Revenue Department of the Bombay Presidency, and Ernest Charles Clements, the District Judge of Belgaum (Belagavi). Both were founding members of the Philharmonic Society of Western India, an organization dedicated to “the interpretation of ancient Sanskrit works on music,” and the promotion of musical practices that conformed—supposedly at least—to the theoretical principles they contained.⁹⁰ Deval and Clements used their allotted time to set forth their theories about the tuning systems described or alluded to in ancient sources, and to examine their correspondences to the current practices of Hindustani musicians. Their arguments proved particularly divisive, and the disagreements that followed their presentations soon spiraled into broader arguments concerning the nature of Indian science, the status of the West in the constitution of Indian music, and the role music theory should play in the nationalist movement that would shape Indian music theory for years to come. In the years that followed, G. S. Khare, a former engineer at the Kurundvad (Kurundwad) Public Works Department and a member of the Philharmonic Society, would invoke the Tonnetz to defend the Society’s claims about musical tuning, thus dragging the diagram into the contentious debates over the status of musical modernity in India.⁹¹

Deval’s paper for the All-India conference began with a close analysis of passages from the Rāgavibodha by the seventeenth-century theorist Somanātha that dealt with musical tuning. Most of Deval’s talk fixated on the idea that the author was aware of two foundational principles of modern acoustics: artificial harmonics and the overtone series.⁹² According to Deval’s interpretation, Somanātha’s description that one must lightly touch specific frets on the vina in order to check the tuning confirmed that the author must have been listening for artificial harmonics, produced when a string was partially depressed at one of its nodes. This could be confirmed, he claimed, by restaging the process that Somanātha described, as shown in figure 10. The fact that Somanātha was aware of harmonics confirmed to him that Sanskrit authorities must have used the harmonic series—namely the third, fifth, and seventh partials that produced the just fifth, the just third, and natural seventh—to tune their instruments in just intonation. But it also showed, he claimed, that Sanskrit authors had already conducted the same scientific experiments that modern Western scientists were just beginning to undertake. Somanātha was therefore not only as empirically minded and exacting as modern European scientists, but far ahead of them, having articulated the same principles centuries earlier.⁹³ As Deval previously asserted, there was “no difference of opinion as regards the important laws and principles followed in explaining the theory of Music and of Musical Scales by the scientists of the West and by our authors”—and since India had determined those laws and principles first, it was India alone that could claim the “palm of priority in the science of the Musical Scale.”⁹⁴

Deval’s analysis ran contrary to the view of most English musicologists—and many Indian ones as well—that Sanskrit musical sources “sedulously avoided applying mathematics to their scales.”⁹⁵ Exact calculations for the scale of each rāga, Deval maintained, could indeed be obtained from Somanātha’s description. To illustrate his claims, he unleashed a deluge of commentaries and calculations. He explained how Somanātha had followed both Bharata, the author of Nāṭyaśāstra (circa second century BCE and second century CE), and Śārṅgadeva, the author of Saṅgītaratnākara (circa thirteenth century CE), in dividing the octave into twenty-two smaller segments of unequal size called śrutis.⁹⁶ These three śrutis came in at least three different sizes: the pramāṇaśruti, equal to the syntonic comma at 81:80, as well as two others falling at 25:24 (the chromatic semitone) and 256:243 (the limma).⁹⁷ By clustering these three types of śrutis in groups of two, three, and four in the manner shown in figure 11, Deval claimed, Somanātha had managed to derive the just-intonation intervals of 16:15 (equivalent to the diatonic semitone), 10:9 (equivalent to the minor tone), and 9:8 (major tone). He then used those to locate the seven śuddha svaras, or pure musical tones: sa re ga ma pa dha and ni.⁹⁸ He also noted that some rāgas required the tunings of the śuddha svaras to be adjusted to new ratios that involved the seventh overtone, creating what was called a septimal harmony. For Deval and his colleagues at the Philharmonic Society, this was a significant feat, especially as Western music only made use of intervals tuned by chains of fifths and thirds. Their incorporation of septimal harmonies thus showed “that their ears had reached the highest developments that man is capable of [. . . and] places the music of India in the first rank of intellectual developments of the Musical Arts.”⁹⁹ If the musical scale was indeed “one of the most important and concise means of judging of the musical state of a nation,” as Deval had previously written, the musicians at the conference searching for the tools to express the voice of modern India needed to look no further than the region’s first music-theoretical documents.¹⁰⁰

In his own lecture, Clements showcased a keyboard instrument that the Philharmonic Society had developed to realize the tunings that Deval was describing. He began by claiming that, according to Śārṅgadeva and a close analysis of the best Hindustani performers, the most “common” rāgas used in modern practice required at least sixteen different śrutis in just intonation—that is, at least sixteen different tones per octave. This meant that the twelve-key portable harmonium, which had become a standard feature of the Hindustani musical ensembles since its integration in the mid-nineteenth century, was in his opinion entirely unsuitable for Indian music.¹⁰¹ The Society had therefore collaborated on the development of the “Indian Harmonium,” shown in figure 12, that featured twenty-two keys per octave, one for each of the śrutis. He concluded by illustrating how the instrument could be used for recital and pedagogy, as a tool for ensuring that Hindustani music was “accurately and scientifically taught in schools,” and performed in a manner consistent with what was described in Sanskrit texts.¹⁰² In an article published just the previous year, however, Clements had gone even further in maintaining that the instrument could also help Western musicians reverse the “disintegrating tendencies” that had plagued European music ever since it took up equal temperament.¹⁰³ His beliefs thus synchronized with those of Deval, who embraced the idea that Europe and India shared a common “Aryan” ancestry, and thus that a harmonium would prove beneficial for both regions as “every effort should be made to arouse an interest in Hindu music amongst the civilized and rich nations of the West, so that they will have the benefit of having Aryan quarter or one-third tones (the want of which is so keenly felt in Europe), and we will get their help and sympathy thus assisting each other in the interest of this dying art.”¹⁰⁴

At first glance, Deval and Clements seem to have settled on an arrangement akin to Tanaka’s geistiger Verkehr. They similarly urged that Western nations, having stumbled into a soulless materialism while becoming modern, seek assistance from the East in regaining their footing. Like Tanaka, they envisioned a mutually civilizing collaboration in the field of enharmonic musical instrument design. But there were crucial distinctions between their proposals, borne from their different positionalities. India, unlike Japan, was a colony of a European nation. When Deval and Clements were writing, political tensions were particularly rife as British colonial authorities pursued increasingly drastic actions to quash Indian calls for home rule. The coercive military power of the British state created pressures that ultimately split the two theorists on the status of modernity and the question of Indian autonomy.

Clements, for instance, did not believe that Sanskrit natural philosophy followed the models of a modern science: on this point, he told the conference, “Mr. Deval was wrong.”¹⁰⁵ In his view, Hindu music theory was a pre-modern discourse, involving an original form of music purer than anything the ancient Greeks had proposed.¹⁰⁶ This purity, he argued, was an alternative to the corruptions of modernity, and this was the source of its appeal. He explained: “we moderns under cover of equal temperament chop and change [enharmonic intervals] without the slightest discrimination or thought” and treat dissonance in a manner that “finds no warrant.”¹⁰⁷ The European system had “outlived its utility,” and it was time to adopt an approach that would be “truer to nature.”¹⁰⁸ European musicians thus had to return to their primordial beginnings, as Indian musicians already had:

Because most Indian musicians had continued to use these ancient tuning systems, he argued—excepting the unfortunate artists who had taken up the tempered harmonium—India was out of sync with the modern world. Clements thus reiterated the tropes of allochronism, but instead of deploying them to encourage the “development” of India into a modern British state, he called for the return of a unified Britain and India to a more noble past.¹¹⁰ For ultimately, in his view, India did not need Britain to teach it how to be modern; Britain needed India to learn how to be ancient.¹¹¹

For Deval, however, it was essential that “Hindu science” be recognized as modern. As the work of the historian Gyan Prakash shows, arguments like his were commonplace in turn-of-the-century Indian discourse, with similar claims offered about the modernity of ancient Hindu chemistry, Ayurveda, and physics.¹¹² “Smoldering underneath” these claims for locating modern scientific reason in ancient Sanskrit texts, he writes, was the “explosive” desire to “make their own what was associated with colonial rulers.”¹¹³ The equation of ancient Hindu philosophy with modern Western science demonstrated that India was “‘always already’ a construct of modernity,” and any practices in India that were un-modern—such as the use of equal temperament—were to be attributed to the debasements of colonial invasion.¹¹⁴ Building further on this point of view, Prakash writes, the restoration of modern India was believed possible only through nationalization.¹¹⁵ In his earliest musical studies, Deval had drawn on the chronological research of Bal Gangadhar Tilak, an early leader of the Indian nationalist movement, in devising the historical chronologies that would support awarding the “palm of priority” to India for discovering key acoustical principles first, including just intonation. He also prefaced his first study of Hindustani tunings with the epigram “no nation can prevail in the struggle for existence which is not scientifically equipped on all sides.”¹¹⁶ Deval’s rhetoric thus appears to place him among the rank of Hindu intellectuals who sought to claim, in Prakash’s words, “nothing less than the right of Indians to the autonomy, authority, and universality of their national culture.”¹¹⁷ Whereas for Clements, just intonation represented a purification of modernity via a return to glorious origins, for Deval it was the system whereby, as Prakash puts it, “India could be modern without being Western.”¹¹⁸

For Deval to simply claim that India was already modern would already have constituted a rejection of Clements’s allochronicism. But for him to contend that it was, in fact, always already modern introduced a new layer of complexity. As Prakash notes, the premise that modernity in the Indian context should involve the recovery of ancient “Hindu science” necessarily reverses the common assumption that modernity constitutes the endpoint of forward-seeking progress. It suggests that Indian modernity exists not only in the present, but also in the past, “crack[ing] the image of cohesion and synchronicity.” Thus “the past irrupts, it does not evolve, into the present”; it is no longer configured as an origin, as in the unilinear developmental scheme, but an “anteriority,” while the present is constituted as a “palimpsest of the past, where the modern nation [is] seen as a belated realization of the ‘before.’”¹¹⁹ This entanglement of two temporal schemes in continual friction with one another was inevitably refracted in the Philharmonic Society’s schemes for representing just intonation—as local past and shared present overlaid, the Hindu psyche on the one hand, and universal musical consciousness on the other.

Fragmented and contradictory as they were, the presentations by Deval and Clements sparked vehement debates that swallowed up much of the symposium. The most prominent Indian musicologist at the conference, Vishnu Narayan Bhatkhande, took Clements’s side in rejecting Deval’s hypothesis that Somanātha had used artificial harmonics to tune the strings. Sanskrit experts summoned to check Deval’s reading consequently agreed with Bhatkhande. Bhatkhande then continued to point out so many errors in Deval’s charts and calculations that the audience “laughed at the very absurdity of the suggestion involved in Mr. Deval’s theory.”¹²⁰ But then he turned against Clements as well, once musicians were summoned to evaluate whether the intervals of his Indian Harmonium actually corresponded to those used in modern performance, as he had claimed. Zakiruddin Khan of Udaipur, a widely respected specialist in the ancient dhrupad style discussed in Bharata’s text, demonstrated that the tunings of the Kāfi, Khammāja, and Bilāvala rāgas were incorrect, compelling Clements to concede that some of the tones of the Philharmonic Society’s instrument “did not tally with those sung by the practical artists as going by those names in current practice.”¹²¹ Another prominent musicologist, Abraham Pandither, then offered an alternative proposition: that the tuning systems described in Sanskrit texts were actually bastardized versions of an even older tuning system recounted in the Old Tamil epic Cilappatikāram (Iḷaṅkō Aṭikaḷ), which he claimed described an octave that was divided not into twenty-two but twenty-four śrutis, all equal in size. Equal temperament, not just intonation, was thus the true “modern” system that ancient India had discovered prior to the West. After Zakiruddin Khan testified that Pandither’s tunings were correct, most participants of the conference, including Bhatkhande, took his side. Nevertheless, the members of the Philharmonic Society continued to vigorously defend their positions afterwards, through publications, editorials, and private letters sent far and wide.

It was in one of those letters, addressed to Pandither, that Khare introduced the Tonnetz of figure 3 to defend the Philharmonic’s claims about the modernity of the just-intonation schemes they discerned in Sanskrit sources. In a short accompanying text, Khare explained that the diagram represented a scheme for calculating the correct locations of the twenty-two śrutis described in a passage of Ahobala’s seventeenth-century text Saṅgītapārijāta. In this passage, Khare wrote, Ahobala provided a system for obtaining the twenty-two śrutis through sa-pa relations alone—that is, through fifths.¹²² Those fifths, however, came in two sizes, both derived from the sa-pa relations embedded within the two ancient grāmas, or scales: the ṣaḍjagrāma, for which the distance between sa-pa was a just fifth (3:2), and the madhyamagrāma, in which the distance between sa-pa formed a “false fifth” that was slightly flatter, at a just fifth minus a syntonic comma (3:2 – 81:80 = 40:27). By alternating between these two types of fifths, according to Khare’s scheme illustrated in figure 13, Ahobala had supposedly obtained the twenty-two śrutis represented in the Tonnetz of figure 3.¹²³ This, for Khare, confirmed the hypothesis of the Philharmonic Society that the śrutis were originally in just intonation, supporting the underlying claims of Deval and Clements.

In fact, the tuning scheme represented in the Tonnetz of figure 3 did not correspond to any of the systems that the members of the Philharmonic Society had proposed at the conference.¹²⁴ The advantage for Khare in ignoring these discrepancies, however, was that he could extend Deval’s assertion that Indian musicians had deserved the “palm of priority” for discovering just intonation to include the claim that Indian musicians had discovered the Tonnetz as well. Moreover, he was able to contend that the topography of the diagram should be read in an entirely different manner than that to which European theorists were accustomed. By suggesting that the Tonnetz had been constructed by just fifths and false fifths—instead of just fifths and just thirds—the Western triad no longer appeared to be a constitutive factor behind its design; rather, it was little more than an incidental byproduct of an alternative generative principle first ascertained in ancient India. Second, by taking the unusual step of using ratios instead of note names to identify each interconnected element of the diagram, Khare not only reinforced the notion that the diagram was borne from a rational “Hindu science,” but also encouraged the reader to view each element as the product of a relation to the central tonic, akin to how a Hindustani ensemble might tune itself in performance by reference to the constant drone of the tanpura. Just as Tanaka had brought forward the isomorphism between the shape of the major triad on the Tonnetz and the hand position for a major triad on the keyboard, Khare introduced an alteration to the diagram that highlighted an audile technique associated with Hindustani listening practices. Thus Khare induced a shift in the temporal and spatial scheme of the Tonnetz, so that it now represented not only India’s contemporaneity and co-presence, but also its ancient heritage as an irruption of ancient Indian logic onto the rational schemes of the global, interconnected present. Borne from the frictions and contingencies of the late British Raj, Khare’s Tonnetz became a palimpsest through which the temporal and spatial schemes of global modernity and ancient India merged.

The arguments of the Philharmonic Society of Western India never achieved much traction in India. Several decades later, however, they gained a second wind in Japan amid the scholarly circle surrounding Tanaka Shōhei. Much had changed for the theorist in the intervening years. He had remained in Germany for a while longer, but called his research into just intonation to an abrupt halt around 1895, shortly after Japan entered the Sino-Japanese War. Observing the Meiji Empire’s rush to develop heavy industry and the army, Tanaka had decided to switch careers and become a mechanical engineer, and after returning to Japan in 1900, shuttled between positions in the railway system, army, government, and private industry. In his spare time, however, he began researching hōgaku (邦楽), or traditional Japanese music. Eventually he attracted such a large coterie of academic researchers and traditional musicians that his house began to serve, as the musicologist Shinohara Moriyoshi has put it, as the “de facto Institute of Traditional Japanese Music.”¹²⁵

In the 1930s, Tanaka decided to return to musical studies full-time. But by that point, the political and ideological landscape of Japan had been radically transformed. Following the weakening of Anglo-American power after the Great Crash of 1929, nationalist factions had begun to champion the idea that the world had reached a “great turning point,” and pushed the government to strategize for a “new order” (shinchitsujo 新秩序) that would place Japan at the center of international affairs.¹²⁶ The Japanese empire expanded ever more rapidly, establishing colonies, puppet regimes, and occupations that would eventually encompass over three million square miles across Asia and the Pacific. In the 1940s, the government announced that the end goal of this campaign was to establish the so-called Greater East Asia Co-Prosperity Sphere—a new regional order premised on “co-existence” and shared wealth, coordinated by Japan—which would involve “the liberation of the peoples of the Orient from the shackles of Western Europe” and “let all the nations and peoples of Greater East Asia take their proper place.”¹²⁷

Whether Tanaka agreed with this new political enterprise, or like many of his colleagues simply felt the need to conform, his language started to drift toward the new rhetoric of these times. While the Tonnetz and the enharmonium remained fundamental to his work, their significance shifted as he began drawing on two interconnected trends in Japanese historiography that played an important role in the hardening of the nation’s imperial strategy: Pan-Asianism and tōyōshi (東洋史). As described by the influential Japanese scholar Okakura Kakuzō (岡倉覚三), Pan-Asianism was premised on the idea “that the Asiatic races form a single mighty web” and should thus stand together against Western imperialism.¹²⁸ In his 1903 publication, The Ideals of the East with Special Reference to the Art of Japan, Okakura proposed that all Asians, and some Pacific Islanders, could trace their racial lineage back to the populations of the Near East, and that their cultural development could be attributed to two sources: “Indian religion,” or Buddhism, and “Chinese ethics.”¹²⁹ Asia was thus “a united living organism, each part dependent on all the others, the whole breathing a single complex life.”¹³⁰ But that organism, he averred, was under attack from the West. British imperialism had rendered India “barren of that religious life which is the essence of her nationality,” transforming it into “a worshipper of the mean, the false, and the new.” China had been “hurled upon the problems of a material instead of a moral civilization,” and found itself “writh[ing] in the death-agony of that ancient dignity and ethics” that had once made it great.¹³¹ Japan alone had resisted the allure and aggression of the West, thanks to “the unique blessing of unbroken sovereignty, the proud self-reliance of an unconquered race, and the insular isolation which protected ancestral ideals and instincts at the cost of expansion.”¹³² It was the “museum of Asiatic civilization,” where Asia’s cultural “unity-in-complexity” had been protected, and thus it was the only nation powerful enough to restore the “Asiatic ideal, replete with grand visions of the universal sweeping through the concrete and particular.”¹³³ Okakura’s rallying cry was ready-made for Japanese imperialism: “victory from within, or a mighty death without.”¹³⁴

The promotion of Pan-Asianism involved dismantling Western authority by consigning classical Greece to “her proper place, as but a province of that ancient Asia to which scholars have long been looking,” while settling “degrading disputes about priority.”¹³⁵ Both these strategies frequently recurred in the historiographical discipline of tōyōshi, or “Oriental history.”¹³⁶ As Stefan Tanaka explains, tōyōshi emerged in the 1890s among Japanese scholars who took umbrage at Western narratives of world history that portrayed Asia in allochronic terms. The central argument of its principal architect, Shiratori Kurakichi (白鳥庫吉), was that Japan had developed from the same origins as Europe in Central Asia, and was just as modern as the West, even if it had arrived at that point by an alternative developmental pathway.¹³⁷ Shiratori and his associates attempted to prove these arguments by mimicking the Orientalist model: Japan took on the role of the West as the society that measured its progress against Others, while each of Japan’s less powerful Asian neighbors became “another Other,” to borrow Rumi Sakamoto’s turn of phrase.¹³⁸ Thus tōyōshi boosted an image of the Orient (tōyō 東洋) as a “competing totality—a new sequence and order—to the universal of the West,” while helping to stabilize the status of Japan as equally modern to Europe, even if it was Oriental.¹³⁹ Just as Orientalist knowledge was used to consolidate authority and power over its object, the knowledge of tōyōshi was pressed into service by Japanese authorities eager to expand their imperial holdings throughout the neighboring region. It to “described the Orient, authorized a particular view of it, taught it, and ruled it, all at once.”¹⁴⁰

The formation of tōyōshi after the refashioning of Orientalist models also led Shiratori to develop a new temporal paradigm that could compete with the authority of Western unilinear developmental frameworks and support representations of Japan as equivalently or superiorly modern. That model was similarly progressive, portraying Japan as shuttling ever forward toward greater stages of social, cultural, and political advancement. Drawing liberally on the arguments of Leopold von Ranke, Shiratori began by arguing that the progress of nations was anchored by universal spiritual principles that existed outside of time. Those spiritual principles manifested differently in each nation, dependent on the era and geography. Each nation throughout world history thus possessed its own version of the spiritual idea that guided development and carried a distinctive essence. In Japan, Shiratori wrote, that universal spirit had manifested in the religious form of government called matsurigoto (政), premised on the notion that national affairs were determined by the gods. For millennia, matsurigoto had ensured the progress of the nation while protecting its essence. The “particularistic potential” of Shiratori’s theory however came to the fore in his analysis of Japan’s Others.¹⁴¹ China, in his telling, evinced a fundamentally conservative essence that impeded its progress. European nations also displayed a tendency toward spiritual weakness and materialism, eventually dragging the peoples they colonized down to their level. Tōyōshi thus sanctified the progress of Japan as more advanced than that of its Asian neighbors, yet also more spiritually sound than that of its Western foes.

The effects of these disciplines on Tanaka’s Japanese-language scholarship appear in articles he penned for the inaugural edition of the journal of the Tōyō Ongaku Gakkai (東洋音楽学会), known today in English as the Society for Research in Asiatic Music. Founded in 1936 by a group that included Tanaka’s student Tanabe Hisao (田辺尚雄), and supported by an advisory committee that included Tanaka and Shiratori, the founding purpose of the Society was to redress the failure of Western scholarship to recognize the true nature of the “Oriental spirit [tōyō seishin (東洋精神)].”¹⁴² Tanaka’s first contribution was a single-page salutation in the opening pages of the issue, welcoming a new readership and setting expectations for what sort of research the journal would present. He began with a genealogy of world music that traced everything back to a single Eastern source:

The differences between these local colors could impede cross-cultural understanding, which is why non-Asian scholars failed so often to understand Asian music:

The burden thus fell on Asian scholars to explain “Oriental culture”:

In his own introduction to the journal, Tanabe picked up on Tanaka’s metaphor to clarify that this brilliant light would benefit not only Asia, but the whole world:

Japan alone was equipped to carry the torch that could illuminate the path forward for modern humanity.

In many ways, Tanaka was echoing the same themes he had expressed forty-seven years earlier during his lecture at Stuttgart. As before, Tanaka related how isolation had previously forced cultures to progress mostly independently, and that in the present day, exchange—geistiger Verkehr or kōtsū oyobi shisō kōkan—was more common.¹⁴⁷ Japan also provided a pathway to the next stage of global modernity in both texts, although this time it was by shedding brilliant light beams instead of releasing purifying streams, in a manner more befitting of the “land of the rising sun.” In his article for the Tōyō Ongaku Gakkai, however, new conditions had been imposed upon geistiger Vekehr/kōtsū oyobi shisō kōkan, hampering its operations as a mutually civilizing force. The East still had something beneficial for the West: ex oriente lux. But the West had little to offer the East by this point, because it was too prejudiced to grasp the essence of the Orient.¹⁴⁸ The trajectory was now more one-sided, directed from Japan outwards to the rest of the world.

Tanaka’s second article for the journal, “The Substance of Indian Temperament [Indo gakuritsu no hontai 印度楽律の本体],” drew on the scholarship of the Philharmonic Society in order to lay out a thesis for what kindled that light.¹⁴⁹ European and Asian music, he began, had originated “in the central plains of Asia.” It was from there, “in ancient times, along with the sudden rise in the prosperity of humanity,” that “music gradually flowed east into China, west into Egypt, Arabia, and Greece, and south into India.”¹⁵⁰ For a while, each culture developed independently, yet along similar lines. At some point in the Middle Ages, however, East and West began to diverge. Europe began to cultivate harmony, eliminating modes that were “inconvenient to this harmonic activity,” until only the major and minor scales were left; melodies accordingly “changed into simple, artless things with few ornaments,” while temperaments emerged as practical expedients for these developments.¹⁵¹ India however shunned harmony, in favor of developing melodic modes that featured pitches attuned to the overtones of a drone. Building on the arguments of the Philharmonic Society of Western India, Tanaka reiterated the claim that Indian theorists had divided the octave into twenty-two śrutis in order to locate the svaras at just-intonation intervals.¹⁵² Eventually, he wrote, ancient Indian musicians obtained “complete mastery over the natural seventh.”¹⁵³ Since the British introduction of keyboard instruments, however, “the problem of equal temperament, which arose five hundred years ago in Western Europe, is now invading the entire nation.”¹⁵⁴ India had started to “lose the correctness of tuning and its refinement and the danger of music being generally contaminated and losing its light has become a deep worry for thoughtful Indian musicians”¹⁵⁵ The only way to save it was to undertake “thorough relief measures” based on a detailed study of Indian music.”¹⁵⁶

The Philharmonic Society had started to make progress, but Tanaka believed he was uniquely poised to carry their work forward, and accordingly felt “the joy of attaining a light in the darkness.”¹⁵⁷ His tasks included locating the exact placements of each of the twenty-two śrutis, inventing a new system of musical notation for representing intonation correctly, systematizing the classifications of the rāgas, and calculating the correct placement of the frets on the vina. In the conclusion, Tanaka even outlined the location of each of the twenty-two śrutis on a new version of his enharmonium, shown in figure 14, which had been manufactured for the past decade or so by Nippon Gakki Co. (now the Yamaha Corporation), and was suitably equipped with extra keys and a new knee lever that could shift four keys to produce septimal intervals.¹⁵⁸

In hewing to the strategies of tōyōshi, Tanaka had compelled a shift in what just intonation represented. No longer was it construed as a structural principle that had been invented by a particular group at a particular time; rather, it was a timeless spiritual principle anchoring musical development across the entirety of Eurasia, from before recorded history to the present day. Even if it was true that Europe had opted to develop harmony and Asia to develop melody, Tanaka argued, the progress of music in both domains was stabilized by these same principles. Nevertheless, Europe and Asia did not enjoy equal claims on just intonation. The research of the Philharmonic Society, Tanaka argued, proved that Asian musicians had discovered its principles and articulated them in writing long before those in the West. It also showed that Asian musicians had pushed just intonation further by incorporating just sevenths into the tonal network formed by just fifths and thirds. Finally, it illustrated how the West had forgotten these principles while focusing on developing harmony, whereas Asian musicians had managed to maintain them through the present day.¹⁵⁹ Europe was on an errant path away from musical progress; its recent colonial incursions in Asia threatened to drag the rest of the world down with it. What was needed was a powerful Asian counterforce to stop those incursions, restore and realize the spiritual principles of just intonation, and continue to cultivate its development through the addition of natural sevenths. For Tanaka, Japan was the main candidate.

Tanaka expanded his arguments in his 1940 book The Foundations of a Japanese Harmony (Nihon wasei no kiso 日本和声の基礎). There, he argued that Japanese music should in fact begin to incorporate harmony, not so much because the West requires it, but because of the intricacy of the times: “the complexity of modern life has led to the demand for more complex methods of expression.”¹⁶⁰ Japan should do it, however, on its own terms. Although Japanese musicians had spent at least thirty years trying to harmonize traditional music, they had done so through an “adaptation of Western harmony” that was fundamentally unsuitable for Japanese melodies.¹⁶¹ Accordingly, “the urgent task of the day is to break free from the shackles of Western harmony [yōshiki wasei no kihan 洋式和聲の羈絆] and reconsider our own national style of harmony with a spirit of independence and self-reliance.”¹⁶² The best way to accomplish this, he proposed, would be to base those harmonies on the principles latent within the ryo (呂) and ritsu (律) scales used in gagaku, a form of imperial court music that drew on heterogeneous sources and had by Tanaka’s time come to represent the ultimate Pan-Asianist synthesis.¹⁶³ The primary reason to use gagaku scales was that these modes featured seven instead of five tones, and could thus afford the pitches required to derive fifth-related triads that could function as tonic, dominant, and subdominant. But Tanaka’s analysis of the tuning of ryo and ritsu scales also indicated the presence of septimal harmonies, as shown in figure 15. To harmonize the combination of just fifths, thirds, and sevenths in their design would thus technically require nothing less than the unfolding of the Tonnetz from two into three dimensions: an expansion from “original just intonation [honkaku no junseichō 本格の純正調]” to a new “broadly defined just intonation [kōgi no junseichō 広義の純正調].”¹⁶⁴ From this three-dimensional lattice, whose form starts to appear in diagrams from his final publications, such as figure 16, a new triad would emerge, featuring a new class of major and minor “septimal thirds” tuned at 9:7 and 7:6 in lieu of the familiar major and minor thirds at 5:4 and 6:5.¹⁶⁵ This new system, Tanaka explained, could only be realized on his Japanese enharmoniums, equipped as they were with new keys and knee levers. His instruments were suitable for the production of Western and Japanese harmony all at once: one single interface, capable of producing the modern music of the entire Eurasian continent.

Tanaka thus suggested that Western theorists like Riemann had it completely backward. Japan did not lag behind the West, stuck on a single dimension of the Tonnetz. Instead, it had surpassed it by unfolding into a third dimension beyond the two that Europe had managed to grasp.¹⁶⁶ The three-dimensional harmony of modern Japan would ultimately displace its two-dimensional Western analogue by pointing toward a musical progress more closely attuned to the timeless spirit of musical nature. It would also bring about the invention of a new type of Eastern triad that would be different from the Western triad, but equivalent in status, dislodging its global centrality. Under the protective shield of Japan, the musical Orient would finally establish itself as the equal of the musical Occident, overriding its perpetual representation as an inferior Other.

Through the twists and turns of these microhistories, the Tonnetze of figures 1 and 3 have come to host a wide array of contradictory political claims: imperial and anti-imperial, Eurocentric and Asiacentric, national, transnational, and pan-national. How then might one interpret these focused snapshots within the broader history of the Tonnetz? One approach may be to expand the stemma of figure 2 to include Tanaka and Khare, while also adding the other Tonnetze discussed by Kevin Mooney, Suzannah Clark, Richard Cohn, and Ellen Lockhart addressed in the introduction. This tactic seems consistent with Conrad’s second type of global history, focused on connection as stemmas represent networks and transfer.

Figure 17 shows the sort of diagram that might result: a complex web of entanglements, with direct lines indicating citations, and dotted lines indicating potential yet unconfirmed influences.¹⁶⁷ From Euler, four threads stretch forth. The one on the left heads to France (Vial) while the one on the right leads to England (Stanhope and Farey). Both then terminate abruptly. The two in the middle extend through German-speaking lands, and quickly lead to a dense tangle of connections that draws in authors from Bohemia (Hostinský), the Governate of Livonia (von Oettingen), England (Ellis), and eventually Japan (Tanaka). Tanaka’s own line of influence passes through Eitz and von Oettingen before returning to Riemann. Meanwhile Khare’s thread picks up from Ellis, but ends there, as no other scholars in India or elsewhere seem to have picked up on his diagrammatic techniques. The stemmatic diagram of figure 17 thus succinctly represents the interwoven threads of influence described in the microhistories above.

As I see it, however, figure 17 also reveals the limitations of Conrad’s second type of global historiography focused on traceable connection. It represents a sparse amount of trans-linguistic exchange prior to 1874: scholars writing in German, English, and French appear to have little interest in citing each other’s work. More importantly, the sheer dominance of German-language sources from before and after Tanaka and Khare risks “incorporating Asia into the realm of German history for the sake of displaying the global dimension of German history,” to paraphrase the historian Hoi-Eun Kim.¹⁶⁸ Europe appears to speak first, with Asia responding. It conjures an authorial chain that weaves backward, like a red thread, to a single European progenitor, tugging the Tonnetze from Asia into a narrative dominated by the West. The focus on connection represented in figure 17 also risks reiterating a Western developmental timeline that undermines the effectiveness of any arguments intended to recuperate Asian agency in the design of the Tonnetz. It raises concerns that global historiography can have a far less “democratizing effect,” to return to Bloechl’s turn of phrase, than initially hoped for.

Perhaps this is simply what the evidence tells us. So might argue the global historian Jürgen Osterhammel, who observes that the records of global history suggest that the objects of European art music appear to exhibit “a stability and consistency that is rare in modern cultural history.”¹⁶⁹ Scholars may recognize that any claims European art music “was of virgin cultural birth and owes little to other civilizations must be viewed with suspicion,” but still find it difficult to incorporate the rest of the world into the history of nineteenth-century European art music in anything more than a secondary, peripheral, or passive role.¹⁷⁰ Osterhammel points out that few European composers had roots in the colonial world; their engagement with non-Western music typically took place through the prism of Orientalism, which ensured it “rarely revolutionized their entire artistic identity.”¹⁷¹ Trade, conquest, travel, and missionary activity did “little” to change the “hermeticism” and “self-centeredness” of European music prior to the mid-twentieth century.¹⁷² One is forced to conclude, he claims, that “European music is not derivative of external cultural models,” and that it “has ‘colonized’ the world without changing very much.”¹⁷³ We must wait “until we are presented with enough empirical evidence to the contrary”—which, given the colonial architecture of the archive, may be a pipe dream.¹⁷⁴

Of course, one might draw on any number of counter-examples of non-Western theorists, composers, or musicians who exercised a measurable influence on the development of Western music to illustrate the fallacy of Osterhammel’s sweeping claims. How about the history of the Creole virtuoso Joseph Boulogne, Chevalier de Saint-Georges? Or the unidentified Arabic inventor of theʿūd, inspiration for the European lute? But Osterhammel’s argument, notwithstanding its shortcomings, reveals an important point: any global history that focuses exclusively on recorded moments of intercultural connection is doomed, on its own standing, to reiterate a more or less familiar story about its origins. More often that not, Saint-Georges and the lute function as exceptions that prove the familiar rule. Do we really need to wait for even more empirical evidence to tip the scales? Or is there already evidence to tell us that figure 17, in its attempt to represent the “red thread,” operates as a red herring?

I believe there is. Let me start by pointing out one consistent theme across each of these narratives. In each case, the Tonnetz plays a crucial role in supporting claims about what it means to be musically, and thus culturally, “modern.” For Tanaka, the demonstration that Japan possessed the technical skill to realize the Tonnetz constituted proof of its ability to participate with Germany in the “united striving of all mankind toward a great common goal” that constituted the defining features of “modern culture.”¹⁷⁵ Riemann, on the other hand, was intent on demonstrating that the musical scales of Japan were examples of an “artificial” and “arbitrary” modernity, as they were not based on the natural principles of the Tonnetz that Western musicians had discovered and long exploited. Khare and Deval focused on defending the argument that ancient Indian theorists had discovered the science behind the Tonnetz long before European theorists––thus, “Hindu science” was always already modern, and Hindustani musicians deserved the coveted “palm of priority.” Finally, later in his career, Tanaka picked up on the claims of Deval and Clements in contending that Japan was the only Asian nation that could defend the integrity of the Tonnetz from degradation by European temperaments. To invent, write about, or sound out the Tonnetz was for each of these authors an essential part of contending with what it meant to be modern.

This observation inevitably leads to a thorny question: what does it mean to “be modern?” One perspective that has traditionally held sway in the fields of history and philosophy is that modernity, as Timothy Mitchell has put it, is a “process begun and finished in Europe, from where it has been exported across ever-expanding regions of the non-West.”¹⁷⁶ The intellectuals Max Weber, Jürgen Habermas, Theodor Adorno, and Max Horkheimer (among others) locate the beginnings of modernity in the European Enlightenment, when rationalism and disenchantment with religion came to the fore.¹⁷⁷ Walter Mignolo and Madina V. Tlostanova have located its “reference point” further back in the European Renaissance, which saw the complementary “colonization of time and the invention of the Middle Ages, and the colonization of space and the invention of America that became integrated into a Christian tripartite geo-political order: Asia, Africa, and Europe.”¹⁷⁸ For them, the essence of modernity is European and colonial: “it was from and in Europe” that it emerged, “not in and from Asia, Africa, or America,” and could never have been sustained without its “darker and constitutive side: the logic of coloniality.”¹⁷⁹ Postcolonial scholarship has offered counterproposals that the origins of modernity should be resituated in the colonial margins,¹⁸⁰ the dynamics of intercultural sociogenesis,¹⁸¹ or the “Third Space” of cultural hybridity co-produced by non-West and West.¹⁸² Nevertheless, these models also generally unfold from the assumption that there has always been a West and its exterior, even before the conditions were present for this neat European-centered dualism. Even the model of cultural hybridity presupposes some degree of original or essential difference between whichever opposing sides are to be hybridized.¹⁸³

Within the field of global history, however, some scholars have advocated for an alternative hypothesis that distributes the origins of modernity across an interconnected geography that spills beyond the imagined borders of the West. Sanjay Subrahmanyam identifies an imperative in “delink[ing] the notion of ‘modernity’ from a particular European trajectory,” advocating that we instead “argue that it represents a more-or-less global shift, with many different sources and roots, and—inevitably—many different forms and meanings depending on which society we look at it from.”¹⁸⁴ This claim, he continues, requires a new model for understanding not only where modernity comes from, but also the processes by which it came to be around the world. Modernity is thus best understood as “historically a global and conjunctural phenomenon, not a virus that spreads from one place to another,” that is “located in historical processes that brought hitherto relatively isolated societies into contact.”¹⁸⁵ Others have followed suit. Conrad is thus building on Subrahmanyam’s work in calling for his “third” approach focused on “integration”—that is, historical “patterns of exchange that were regular and sustained, and thus able to shape societies in profound ways”—and accordingly argues for a reinterpretation of modernity as a “co-production” of global cultures.¹⁸⁶ Music historians have offered similar expressions, including Gary Tomlinson, who urges scholars to inquire into the “peculiar deafness that at once constitutes our modernity and conceals the global forces in it.”¹⁸⁷ Each of these exhortations challenges historians of modernity to set aside their search for the “palm of priority.” They call for modernity to be recognized as the product of a complex polyphony, not a single voice.¹⁸⁸

Such claims push us to rethink more than just the causes of modernity, because these arguments challenge us to develop new models for understanding causality itself. One such model that I believe can be helpful is that of “emergence” as it has been articulated by philosophy and systems theory.¹⁸⁹ In these fields, “emergence” is used to describe how phenomena arise, seemingly without an identifiable cause, from the collaborative functioning of the constitutive parts of an entire interconnected system.¹⁹⁰ Emergent phenomena are thus “higher-level phenomena,” which cannot be explained or predicted by reference to any singular constitutive element or “lower-level entity”; they defy reductionist models that attempt to locate their origins or cause in individual parts. To approach the history of modernity from an emergentist perspective would thus involve rejecting the reductionist hypothesis that ways of being modern originate from a particular group located at a particular place in the world (such as the West or the East) and accepting that they instead emerge from a network stretching between various groups located around the world. It is, in essence, to recognize the truth contained within Tanaka’s premise that a mutually transformative geistiger Verkehr constitutes the motivating factor behind modernity, without accepting the teleological framework he ascribes to it. It also means recognizing that being modern involves the self-fashioning of one’s identity on the global stage, for it is only and always in reference to the global that local expressions of modernity can happen. If we accept the arguments of Tanaka, Riemann, and Khare that the Tonnetz is quintessentially part of what it means to “be modern,” then it follows that their Tonnetze derive in some way from the drive to participate in a mode of existence and a code of behaviors whose origins are not located in any singular region, but distributed across the globe.

Each of these Tonnetze is thus a “global Tonnetz,” in the sense that they map out positions for Japan, Germany, or India on the musical stage of global modernity. To argue that they are all global Tonnetze would not however amount to saying that they are all the same. As Subrahmanyam argues, we have erred in “identifying ‘modernization’ with the growth of a certain type of uniformity”; it is the result of “uneven processes,” which are inevitably shaped by “strong local roots and colors.”¹⁹¹ But it is also to argue that it would be counterproductive to speak of the diagrams by Tanaka, Khare, and Riemann as “alternative Tonnetze,” in the same mode as Shmuel N. Eisenstadt or his followers speak of “alternative modernities.”¹⁹² A discourse focused on modernity’s “alternatives” necessarily posits the existence of an original modernity that is subsequently modified by local contingencies into various forms; it gets us back to a search for original, locatable causes instead of emergent phenomena. Mitchell encourages us to recognize modernity as a singular concept characterized by “uniqueness, unity, and universality,” but at the same time to understand that “this always remains an impossible unity, an incomplete universal,” as each representation of modernity introduces “the possibility of a discrepancy, that returns to undermine its unity and identity.”¹⁹³ It is akin to what the anthropologist Anna Tsing calls an “engaged universal”––an ideal, like environmentalism or human rights, that aspires to be universal but encounters friction as it manifests around the globe and engages with local contingencies that ultimately endow it with discrepant charges and configurations.¹⁹⁴ There is only one modernity, and so there is only one global Tonnetz, notwithstanding the divergent forms it may have taken in the hands of Tanaka, Khare, or Riemann as they engaged with the contingencies of local politics. For underlying each of their representations was the same goal: to demonstrate how their nation participated in the construction of the engaged universal of musical modernity on the global stage.

And so, what seemed to start as a tale of two Tonnetze, figures 1 and 3, has turned out to be two tales of one Tonnetz: the global Tonnetz. The first historical account I related in the previous sections focused on the microlevel of interpersonal interactions within the local politics of nationalism. The second account, presented in this section, focuses on the macrolevel of phenomena emergent from the global forces of modernization. Both may be “global histories” according to Conrad’s schema—although I believe it might be more accurate to call the second account “history of the global,” in that it focuses on accounting for the ways in which the Tonnetz has played a constitutive role in shaping how the notion and perception of the “global” in music theory has historically been produced. What these two tales nevertheless tell us about the origins of the Tonnetz does not readily connect. On the one hand, a microhistorical focus on details of interpersonal connection and contact can make it all too easy to slip into the logic of priority and causation that Tanaka, Riemann, and Deval deploy, conjuring a red thread that weaves back to a single author. On the other, a macrohistory that draws on the concept of emergence provides us with a framework for distributing authorship by locating it within processes of representation performed by multiple actors concerned with fashioning their place on the globe. Must we place our stock in one version of history over the other?

I believe we need both versions, discontinuous as they may be. The first approach arcs toward the latter, insofar as microhistory offers essential and granular details about the lower-level dynamics of interconnection that create the conditions of possibility for higher-level phenomena to manifest. Nevertheless, a gap remains between them: one cannot “build up” from the former, with its emphasis on transfer and exchange, into the latter, with its emphasis on emergence.¹⁹⁵ These microhistories tell us that Tanaka and Khare may have played a role in the distribution and development of the Tonnetz, while drawing out its latent haptic and acoustic properties, but ultimately the diagram originated with Leonhard Euler; macrohistory tells us that its configurations are similar but distinct representations of the emergent phenomenon of modernity. Both tales are true. Just as physicists have come to accept the concept of particle-wave duality––recognizing that certain quantum entities demonstrate either particle or wave properties depending on experimental circumstances––historians need to learn to accept that certain phenomena from the past exhibit a form of historical duality. No singular approach to them will be able to fully capture their internal contradictions and multiple behaviors.

As it turns out, the risks of overlooking this fact are discernible within the Tonnetz itself, for the diagram has repeatedly doubled as a visual scheme of the global history of music theory, representing its development through its peculiar arrangement of tonal elements. Tanaka and Riemann use the Tonnetz to overlay sequential moments of music-theoretical breakthrough—the primordial discovery of a fifth above the tonic, its subsequent expansion in Eurasia into an axis of fifth-relations, and the medieval European discovery of the third—in composing a tonal network that extends into two dimensions. For Khare, that sequence was slightly different: first came the Sanskrit realization that śrutis could be arranged in fifths and false fifths, followed by the later discovery in the West that one could also derive triads from their scheme. For Tanaka toward the end of his career, new overlays were being added in the present day by Asian musicians who were pushing to realize the diagram’s full potential by extending it into the third dimension. Each author thus used the diagram to capture successive moments in the unfolding of a music-theoretical principle through networks of exchange across time and space, in a manner not dissimilar to how early modernist photographers have overlaid multiple exposures to represent the motion of an object as it traverses space in time. See, for instance, Étienne-Jules Marey’s “chronophotographic” study from around 1890 of a pole vaulter shown in figure 18. The Tonnetze of Riemann, Tanaka, and Khare similarly capture the dynamic process by which a tonal network gradually unfurled from a central node to extend across the entirety of tonal space through chains of fifths, false fifths, thirds, or sevenths that brought increasingly more elements in contact. In the critical apparatus that accompanies the diagrams, that process is then linked metaphorically to the process by which knowledge of the Tonnetz diffuses throughout the world, starting out from a central location—Germany, Japan, or India, etc.—from where it spread outward through chains of contact and exchange to extend across the modern globe.

Focusing on the history of how networks of contact and exchange unfold in linear time, without also accounting for emergent global phenomena, helped the authors of these Tonnetze to support claims for priority, origins, or authorship. Today, historians who continue to confine themselves to that approach will likely find themselves in Osterhammel’s trap: bound, despite any inclination to insist otherwise, to present European music as “hermetic,” “self-centered,” and having “colonized the world without changing very much.”¹⁹⁶ If we overlook the principle of historical duality, we may find ourselves reanimating a musical geography that conforms with the topology of the Tonnetz. It will look like a global network of connections, ensnaring as much as it can within its network, generated from and centralized around a single localized entity: Europe, or one of its Others, as the tonic.