Fétis’s Racial Frame of Tonality (Part II)

Thomas Christensen

[…] Continuation of: Part I

I have to admit it was a rude shock when I learned of Fétis’s virulent racist views. It was especially deflating as I had invested so much time working on my study of his theory of tonality and its reception only to belatedly discover that my protagonist ended his life as an out and out racist. To be sure, there were unsavory moments in many of his earlier writings. But for reasons that I try to outline in my book, his earlier writings were tempered by Kantian liberalism. His full-frontal embrace of the most aggressive tenets of biological racism in his final writings was truly an alarming change of heart.

So what was I to do? I was not ready to give up my book project and simply “cancel” Fétis (as the expression seems to be today). My book was a historical sketch of an undisputedly important music scholar from the 19th century, even as the picture I was drawing became increasingly more complex and disconcerting. And I did not want to turn my story of Fétis’s Faustian choice to embrace the darkest corners of racial biological thinking into a didactic morality tale for our times. Anyway, as far as I know there are no statues of Fétis to take down. (Well, check that; there is a bronze bust of him to greet visitors in the lobby of the Brussels Conservatory of Music, which he helped to found.)


But what about his key concept of tonality? That is something that certainly still lives on in our discipline of music theory, even if we may not have always recognized its burdensome legacy. To be sure, few of us use the term tonality in the aboriginal way in which Fétis conceived it. Nonetheless, it may be that the term to this day contains something of its 19th-century DNA even if we may not be aware of it. Certainly, traces of its biological “frame” often seeps into the metaphors with which we sometimes speak of tonality.

Think of how we often trot out clichés about tonality using images of ontogenetic growth and decline: the 17th century saw the “birth” of modern tonality, the 18th century witnessed its full “maturation,” while in the late 19th and  early 20th century we see the “decay, ” “crisis,” or perhaps even the “death” of tonality. (Much of this rhetoric has been brilliantly analyzed by Brian Hyer in his New Grove entry on “Tonality.”) Fétis seemed to be ever vigil about policing the boundaries of tonalité moderne. It was one reason he was so obsessed with finding its precise birthdate (he arrived ultimately at 1605 for this happy moment); but he also worried about its degeneration with the increasing encroachment of the ordre omnitonique  and all its many affective affinities and vertiginous modulations. Even more objectionable, in his view, were the attempts of some modern composers to write hybrid varieties of tonality that combined features of modern tonality with antiquated modal resources, or worse, tonalities of the Orient. It was a sin of tonal miscegenation. It’s no surprise, of course, that in the 20th century, music critics within Nazi Germany condemned almost any deviation from the most conservative models of harmonic tonality as degenerate art (while at the same time claiming tonality as the exclusive musical patrimony of the Aryan race). Yet anxieties of the “borders” of tonality continue to be raised in theoretical literature. Schoenberg may have mocked the question of “Tonal oder nicht Tonal?” with a canon; but arguments over its answer became a veritable killing field in the world of 20th-century music aesthetics and analysis.

It’s surely unrealistic to say we should just jettison the term. If we didn’t have the word “tonality” we’d have to invent something as a synonym to describe a tonal practice that covers a huge swath and timespan of  Western music in which most of us intuit common salient features, even if we might disagree on the precise physiognomy of these features. No, we are probably stuck with the term.

But that doesn’t mean we need be insensitive to how we think about and use the concept of tonality in our theories and analyses. On the contrary, our increased awareness of its fraught origins in Fétis’s writings could inspire us to greater care in how we go about distinguishing  norms and varieties of tonality as well as defining their boundaries (if that is indeed what we want to do). For that matter, maybe the whole regime of our taxonomic-centered discipline might be due a good rethinking. In any case, as historians of music theory, perhaps we can play a salutary role in teaching our colleagues a little of the history in which some of the terms by which we analyze and teach music originated—and have become our own.  That kind of historical and cultural positioning is a frame that we all could use.

Fétis’s Racial Frame of Tonality (Part I)

Thomas Christensen

It is hardly an exaggeration to say that Philip Ewell’s SMT plenary talk last year on music theory’s “White racial frame”—and also including the expanded essay he published this past summer and the recent issue of the Journal of Schenkerian Studies (devoted as a response to Ewell’s SMT talk)– has ignited a debate within the music-theoretical community that is unprecedented, a debate that has spilled over the boundaries of our publications and chat lists into the wider public sphere. (When was the last time the name of Heinrich Schenker or a mention of the Society for Music Theory has appeared in the news section of the New York Times or the Economist?)

As we know, the aspect of Ewell’s argument that has provoked the most impassioned response—both in support and in opposition—was his searing indictment of our discipline for harboring deeply embedded racial biases in the repertoire, tools, and language with which we study and teach music.  So ingrained are these racial contaminations, Ewell argues, that most of us are scarcely aware of their presence. Whether one agrees or not with his diagnosis of this condition and his prescription for cure, I think all of us should be grateful to Philip for the prod he has given us to face some uncomfortable questions honestly and fearlessly. 

It is not surprising that Ewell focused much of his essay on the writings of Heinrich Schenker, since Schenker’s writings have long been known to harbor some notorious moments of racial and nationalist commentary. But I think Ewell is right when he suggests that the problem goes beyond Schenker, as obvious a target as he is. And this is where the history of music theory might be useful to test that hunch. As scholars of historical music  theory, we have a particularly good perch to survey the history of our field and see what other skeletons might be hiding in our discipline’s closet. Thus we might have a useful role to play in confirming—or perhaps qualifying—Ewell’s indictment. For me, it is a charge that hit home personally.

In my own recently-published book on François Joseph Fétis (Stories of Tonality in the Age of François Joseph Fétis; University of Chicago Press, 2019), I stumbled upon a few of these skeletons as I read many of Fétis’s later—and lesser known—publications. And what I found was that it was not simply that Fétis expressed strong racial prejudices. (For a Belgian national writing at the dawn of Belgium’s infamous colonialization of the Congo, such prejudices were hardly unusual; on the contrary, they were ubiquitous and quotidian.) But what was truly startling, I found, was how these prejudices seeped into a key theoretical concept with which he is identified today: tonalité. As tonality is one of those “frames” by which music theorists have long used in their studies and analyses, it may come as something of a surprise to discover its deeply racist genealogy. 

I won’t try to rehearse the entire history here. (For those readers who are interested, I’d refer them to the fifth chapter of my book, where the whole sordid story is told.) But I can reduce Fétis’s essential conclusions to a series of bullet points: 

  • For Fétis, tonality was a patchwork of scale systems that have varied over time and space.  
  • Each civilization, people, and race have had a tonality that was uniquely suited to their character and needs based on the pitch content and tuning of its chosen scale system (the ancient Greek modes, Indian rags, Arabic Maqam, Chinese and Celtic pentatonic scales, and so on).   
  • The tonality used by a given people determines—but also limits—the kinds of music that can be created with it.
  • Only the European white race, with its advanced system of the diatonic scales of the 12 transposable major and minor keys, possesses a tonality that allows for true musical art. 

The following quotation from the first volume of his Histoire générale de la musique of 1869 illustrates Fétis’s racist views bluntly:

The true history of music begins only with the general history of this privileged [white] race, one which never has known the state of savagery, and who, upon making their first appearance in the world, showed themselves relatively advanced, cultivated, and of such great superiority over all other races that no comparison between them can be made. The white race alone is endowed with the faculty to modify itself perpetually, to present itself in history in a thousand differing ways. Contrary to the other races, one of which [the black race] remains in servitude and stays in a permanent state of social infancy, and the other [yellow race] which has attained a certain degree of civilization but one that it can never surpass,  the white race has developed over time all the consequences of its moral organization. It perpetually adds to the knowledge it has already acquired. It possesses a sentiment of beauty, of grandeur, and it is to it that we owe the creation of pure art and the progress of science.

(Histoire générale de la musique depuis les temps les plus anciens jusqu’à nos jours.  5 vols. Paris: Librairie de Firmin Didot Frères, 1869-1876. Vol. I, 108) 

And what is the reason for the superior development of the white race and consequently the music that it has produced over time? It turns out it can be attributed entirely to biology. The very opening sentence of his Histoire makes this clear: “The history of music is inseparable from the degree to which the special faculties of the races were cultivated” (HGM, Vol. I, i). If there was any doubt as to what Fétis meant by those ”special faculties,” he cut to the phrenological core in the opening sentences of Volume 2: 

The sentiment for music, among nations as well as individuals, is due to the conformation of the brain. . . . The relations of sounds do not affect people of races in the same way; what charms one displeases the other, precisely because the organs of the brain are not of the same dimensions.

(HGM II, i)

And let there be no mistake as to which race has won the prize in the lottery of brain size: it is the race blanche—the white race, “which alone has produced music that may be elevated to the dignity of art” (HGM I, vi). 

This last point should turn some heads with its obvious nod to the discredited “science” of phrenology. But Fétis was actually relying here upon the fashionable writings of Joseph Arthur, Comte de Gobineau, whose four-volume book “Essai sur l’inégalité des races humaines” (Essay on the Inequality of the Human Races, 1853-55) he had devoured—and annotated–with enthusiasm. (Again, details and references to this information may be found with some further contextualization and elaborations in my book in Chapter 5.)    

(to be continued)

Continuation: Part II

Can the History of Theory Be Decentered? Part V – Some FAQs (and not-so-FAQs)

Alexander Rehding

[…] Continuation of: IV. Pros and cons […]

What are the dangers of this plan? What if it is taught badly?

This is undoubtedly unchartered territory for most of us, and yes, there are some vagaries that can seem terrifying even to seasoned teachers. To a certain extent, we all have to be ready to plunge into the unknown. Most of us are not trained to do this. I think it’s completely legitimate to seek help where we can. Invite an expert over to Skype into your class. Explore new territory together. Even working through difficult passages together, and exploring these difficulties together, is an opportunity for a teaching moment. Learn to formulate good questions and seek answers together. As the old adage has it, the path may be the goal.

And yes, there may well be a few treacherous cliffs ahead. Here’s an example that I encountered in planning this course. For the longest time my draft syllabus ended with the American experimentalist Henry Cowell—I was hoping to include at least one representative of the LGBTQIA+ community, and I had long been interested in Cowell’s theoretical ideas. Cowell lived at a time, not too long ago, when deviations from heteronormativity were criminalized. In 1936 he was arrested on charges of Section 288a of the California Penal Code.[1] His loyal wife, Sidney Robertson Cowell, whom he married soon after his release from prison and who maintained his legacy after his death, was reluctant to comment on this aspect of his life.

But the more I tried to imagine what this particular discussion would look like in a teaching situation, the more I came to the conclusion that Cowell was not the right figure in this scenario. The terms we use nowadays don’t map cleanly onto the legal situation in the early twentieth century—the specific crime Cowell was charged with was oral sex; the other party was a seventeen-year-old male. A discussion dissecting the distinctions between a modern perspective on gay identity and what would now be considered statutory rape might be interesting, but it just harbored too many risks in this context. If the discussion somehow ended up reinscribing the false prejudice that homosexuality and pederasty were somehow connected, then this lesson would have spectacularly backfired. I considered other music theorists, who might provide a less conflicted picture. But in the end I decided with regret that my syllabus would have to remain without an aspect of LGBTQIA+ history for the time being. Perhaps others will be more successful.

Who exactly is this course for?

Or, asking more pointedly, are the criteria for inclusion in the course based on INS guidelines? The answer is—all hyperbole apart—yes, to a certain extent that is the case. The task of representation that is front and center here is shaped by the particular makeup of contemporary US society. There’s an undeniable civics component here. (The issues surrounding Cowell in the previous question similarly indicated that the issues are determined by twenty-first-century debates.)

The make-up of the syllabus may not be an exact match, but it’s useful to think through the history of music theory within different parameters. This is the subtle difference between the purposes of a global outlook and a diverse syllabus that I am proposing here. It is less relevant that the syllabus cover all world cultures than that diverse groups of students see themselves reflected in these figures—or, turned around, that the history of music theory is not an exclusive white male affair.

Follow-up question: if this comes down to civics lessons, why not go further down the road of actual representation? Where are the African Americans and the Indigenous Americans?

This is where the syllabus, like most syllabi, ends up being a compromise. Most syllabi try to tell a more or less coherent story. I just haven’t found materials—and I probably have to plead ignorance to some extent—that would bring in those cultures in a way that would allow us to tell a more or less coherent story. As far as I can tell, there are probably other ways in which these particular issues that arise from African American or Indigenous Americans can be better addressed. I am eager to hear what others come up with. After all, this should ideally be a collective project.

Isn’t five cultures in one semester a bit much? Won’t this be superficial?

Yes, that’s probably true. There is definitely an It’s-Tuesday-today-so-this-must-be-Paris aspect to the journey taken in this syllabus. But then again, I would argue that this is a broader problem of the American educational system. For one, it seems to me that this criticism would equally apply to most History of Theory surveys, whether culturally diverse or not.

It’s worth bearing in mind that the Five-Classics structure I wound up with in my thought experiment is the result of the particular constraints of our program. If we had two (or even four) semesters at our disposal, the syllabus would look quite different. Other programs may have more flexibility in this regard.

In any case, we should definitely make use of any help that we can get. If this were to become a teaching reality, I would absolutely invite specialists from other departments in my institution, or invite guest lecturers from elsewhere to talk to us on Skype or Zoom, to help us with specific figures or contexts.

This is quite a responsibility. Are we ready for this?

To begin by stating the obvious: diversity and inclusivity cannot just be proclaimed, and then everybody moves on with their regular business. That would be a sure way to fail. The responsibility comes with the specific ways in which such ideas are implemented. The next answer makes a few further suggestions on this topic.

Must we do this? What other models might work?

To answer along similar lines to the previous question, no one should feel forced to do this. This blueprint is, after all, nothing more than a thought experiment at the moment. I believe this syllabus, or something like it, would quite work well for our program. But that’s not to say that this would be true for other graduate programs. There’s nothing sacrosanct, for instance, about the focus on “Five Classics” (unless you’re an orthodox Confucian). This was simply a canvas on which this alternative plan could be designed.

There are various ways in which a more diverse HoT could be implemented, and I know that there are some exciting syllabi in existence that include elements of diversity. Some people might feel, for instance, that focusing on five marginalized figures might be confusing or overwhelming. They may decide that the inclusion of one non-European theorist offers enough contrast. And they may be right. Even one figure who interrupts the white-male predominance will go a long way in inviting diverse thinking. Or they could focus the entire semester on certain specific aspects of HoT that invite cross-cultural thinking. The question of tuning in different cultures and at different times may be such an angle.

The most important aspect in all this is that this is a collaborative and open-ended project. My perspective, like that of any one person, is limited, and others will likely have had ideas that haven’t occurred to me. I share this draft with the wider community in the hope that it might spark some further ideas with others, and to launch a wider discussion on how ideas about diversity and inclusiveness can be implemented in our teaching of HoT.


[1] See Michael Hicks, “The Imprisonment of Henry Cowell,” Journal of the American Musicological Society 44, no. 1 (Spring 1991): 92–119. Available, free of charge, here.

Can the History of Theory Be Decentered? Part IV – Pros and cons

Alexander Rehding

[…] Continuation of: III. Some Consequences […]

One possible criticism of the design I sketch out here is that it provides an outline that is still largely oriented by the concerns of a traditional western HoT, even though it populates its syllabus with women and POCs. This is a legitimate point: the selection of figures still allows a coherent story, but this story is only possible—more or less—against the tacit background of the figures from the white male syllabus. In studying Ptolemais, for instance, we have to study the broader harmonicist tradition, in studying Johanna Kinkel we also have to study broader notions of chromatic relations. In the story that the new syllabus outlines, then, students would still learn about modes and tetrachords, about the need for and problems of temperament, the questions of harmony and tonality, and so on.[1]

A syllabus that curates the figures that represent certain predetermined questions—isn’t this the definition of tokenism? Pedagogy should not be reduced to a diversity checklist, which would simply reproduce the hierarchies and hegemonies lurking under the narrative surface.[2] But there is more going on in this syllabus: even if we change our story by choosing different people through which we tell a similar story, we do actually change quite a bit about the story itself: we position the issues between different cultural nodes and open up new connections that had previously not been accessible.

The tale of equal temperament, for instance, was told in my old syllabus through Daniel Chua’s highwire act of bringing together Vicenzo Galilei and Max Weber. But there is no reason that Zhu Zaiyu couldn’t also be used to explain the calculation of equal temperament, and to do so in a musical context that does not deal with the problem of triadic harmonies. This approach would raise all sorts of interesting questions: How do specific concepts relate to different musical contexts? What problem exactly did Zhu Zaiyu’s equal temperament solve in the context of Ming-dynasty yayue?

If the task is to open the students to other possibilities, then teaching other cultural and musical contexts is a plus. When the students learn how Arabic and Persian music theory utilizes the oud, they don’t only learn how the Greek tradition was absorbed by the Islamic world, but they also establish a point of contact on which later studies into South-Western Asian music[3] may build in a different context. When the students learn, with Johanna Kinkel, about Chopin through enharmonic microtones, they learn about a way of listening through new ears (and new politics) to a repertory that may have seemed all too familiar. When the students learn about Julián Carrillo’s vision of the future of music, they learn about changing conceptions of consonance and dissonance, and conceptions of musical progress. Schoenberg’s story is an important one here, too, but—and this is the critical point—it’s not the only one.

This is one way in which diversity manifests itself and spreads outwards. Far from just paying lip service to academic fashions, we will have actually created a structure that offers old and new points of contact, in ways that the old syllabus did not support.

Is it a problem that the cultures represented here are mostly drawn from the most notable cultures of the world, as even the crustiest European cultural mandarin would acknowledge? While the syllabus includes an important Mexican composer-theorist, there is no representation of several central minority groups of contemporary American culture: Indigenous Americans and African Americans. (This point is what I flagged earlier when I noted that global music theory is not the same as diverse theory.) This is a more serious concern and should be considered. If the list were longer, and if the semester had more weeks, there is no reason why both could not be included. I should also work harder to include more African American scholars—a problem that Phil Ewell discusses at length. Indigenous Americans present, as so often, a separate and possibly more protracted issue that absolutely deserves attention. So far I have not found a solution that would satisfy me, and I would be interested to hear from others who have tried.[4]

Perhaps the most important aspect of this thought experiment, certainly to me, is the thought that we need to rethink the relationship between people and ideas in the history of music theory. I sometimes joke that I teach music theory because I am not terribly interested in people. This usually gets a laugh, and there may be a certain kernel of truth to it. But I realize that this joke is not totally harmless, because the professed lack of interest in people is tantamount to accepting that our established histories continue to be peopled by heteronormative white males, which can somehow pass as neutral. As Phil Ewell reminds us, this indifference amounts in practice to a tacit diversity quota, and it’s currently set at 0%. Not every music theory student has the luxury of blithely ignoring ethnic and gender makeup.

One incentive that I hope students would take away from such a diversified syllabus is that they also recognize its opportunities. I am specifically thinking here of unrealized potential. The English secondary literature, say, on Zhu Zaiyu is still pretty sketchy, to say nothing of Johanna Kinkel, who is virtually unknown even in specialist circles. There is plenty of scope for exciting dissertation topics. The additional benefit is that as the research becomes more robust, these figures will automatically become more central. New nodes and points of contacts emerge. In a word, our story is beginning to change.

[…] Continuation: V. And finally, some FAQs (and not-so-FAQs) […]


[1] My colleague David Damrosch would call this a “countercanon.” He points out that the major figures of traditional canons are relatively unaffected by the diversification of the field and in fact form a “hypercanon.” (This would be our Schoenberg–Boethius syllabus.) Rather, he continues, it is the “shadowcanon” of the minor figures of traditional canons (the Marchettos, Morleys, and Marpurgs) that lose cultural capital in this process. See his “World Literature in a Postcanonical, Hypercanonical Age,” in Haun Saussy, ed., Comparative Literature in an Age of Globalization (Baltimore: Johns Hopkins University Press, 2004), 43–53.

[2] Or, to make this point more emphatically, there is always the danger of the McDonaldization, of constructing a canon from the perspective of American hegemony, an act of colonization rather than globalization. See Djelal Kadir, “Comparative Literature in an Age of Terrorism,” in Saussy, ed., Comparative Literature, 68–77. Thanks go to Lester Hu for pointing me to this thought-provoking article.

[3] Thanks go to Siavash Sabetrohani for pointing out the problems with the standard label “Middle Eastern.” I have followed his advice in using a more adequate geographic term, but since this label is not (yet) very common, it seems useful to add this short explanatory note.

[4] Ongoing or very recent work, such as Danny Walden’s provocative work on the early generation of comparative musicologists [paywall] working with Indigenous Americans, or Jennifer Bain’s work on chant transmission among the Mi’kmaw, offer promising starting points here.

Can the History of Theory Be Decentered? Part III – Some Consequences

Alexander Rehding

[…] Continuation of: II. Five Different Classics […]

Writing a diverse HoT will require an embrace of the limits of what one person can do as a teacher. In conventional HoTs, the standard languages German, French, Italian, Latin, and ancient Greek can be expected; they would—ideally—be needed to follow my old syllabus. It’s remarkable that for a number of the texts we look at—despite their much-invoked centrality to the field—there are no published English translations.[1] For practical purposes, I let my students work on translations as a group. There is usually at least one person with decent French and another with good German, occasionally someone with Latin or Greek, and among all of them, they can figure out most of the texts in the original language. Sometimes we work through originals in parallel with translations to tease out nuances wherever possible.

Would something equivalent even be possible for our diverse syllabus? Realistically speaking, we have to kiss goodbye to the expectation of comprehensive language skills.[2] Of course, my syllabus is still pretty moderate and focused on themes that resonate well with some central themes of European discourses.[3] What if we were to add, say, Sanskrit, Zulu, Russian, Nahuatl, Japanese, Omahan, Turkish, Javanese, and Tuvan? (Phil Ewell’s blog has much more to say on language skills.)

Some might argue that good historical work cannot be done without thorough language skills. These ambitions are understandable, but if we want to be serious about the global demands of music theory, or rather its diversity tasks—the two are not quite the same thing—then we have to be realistic. As historian Jürgen Osterhammel, a leading authority in global history and the author of several monumental volumes on world history, argues at length, it is simply not possible to do close-reading work with primary sources in multiple languages at the same time. Reliance on dependable translations is a precondition of a more diverse HoT. A global and/or diverse approach must have the courage to let go of the old language requirements. It doesn’t mean that the work is inferior. It just means that we place our priorities elsewhere.

Languages aside, I’d be deluding myself if I pretended that a diverse syllabus can be taught in the same way as the old syllabus. We know far less about Ptolemais than we do about Boethius; material about Johanna Kinkel is much harder to find than material about Helmholtz. Zhu Zaiyu and Al-Fārābī have amassed considerable secondary scholarship, but mostly within their own language groups. (There are no translations of Zhu Zaiyu or Al-Fārābī available at the moment, though Alison Laywine has been working on a translation of al-Kitāb.) Teaching these figures would require a lot more groundwork, a lot more contextual work, a lot more reliance on the existing secondary literature (some of which is, how to put it, less than up-to-date), and frankly, a lot more informed speculation. In all likelihood, it will require a certain amount of pedagogical finesse, as the students will have to synthesize a vast amount of material. It will also require putting my cards on the table and explaining that I am not in full control of the material, and rely—as the students do—on whatever material is available to me. We will, quite literally, all be learning this material together.[4]

In fact, this teaching model might well provide an alternative to Osterhammel’s individual synthesis of vast amounts of (secondary) material. His model, which retains the individual authorship of traditional humanistic scholarship, runs the risk of solipsism. The alternative is to harness the many diverse specialisms of scholarly collectives, each bringing their own thoughts to bear on a project.[5] Publications might well come to resemble seminar discussions.

[…]Continuation: IV. Pros and cons […]


[1] While I’m at it: Calvin Bower’s translation of Boethius has been out of print for a long time, and goes for about $500 on the second-hand market. Yale University Press, if you read this, please consider reissuing this book, perhaps as print on demand.

[2] Lest I be misunderstood: what I am talking about here is specifically a teaching scenario (or even more specifically, the qualifier “comprehensive”). In fact, this situation makes the need for published translations of the highest scholarly caliber even greater. In parallel, I hope we will usher in a new golden age of scholarly translations of music-theoretical texts, as we saw in the 1990s.

[3] One aspect of its moderation is the extent to which notions such as “History,” “Theory,” or “Music” are left unchallenged.

[4] In 2012, I offered a class that was structurally similar, which explored the Enlightenment reception of Chinese music and culminated in a group exhibition—in this I was as much a student as the graduates, in that before the start of the semester I didn’t know where exactly we would end up. It was a valuable experience for all involved.

[5] I thank Lester Hu for pushing me further along in this direction. There is plenty more to say here. Our current academic institutions, for instance, are typically geared toward incentivizing and rewarding individual scholarly contributions over group work.

Can the History of Theory Be Decentered? Part II – Five Different Classics

Alexander Rehding

[…] Continuation of: I. Prequel: Five Classics […]

But there was no resting on laurels. No sooner had I completed the syllabus than I started reading some recent inquiries into critical race theory and music theory—especially Phil Ewell’s series of blog posts “Confronting Racism and Sexism in Music Theory” and Vijay Iyer’s analysis of recent theorizing on improvisation [paywall]—and what I read gave me pause. What our syllabus didn’t do was to step outside of the heteronormative white-male framework. It reinscribed the same canon of sanctified figures that is apparently gender- and race-neutral, but is in fact simply white and male. Forget omitting Fux, Koch, and Marx—what this syllabus didn’t have was any women, any people of color, or any LGBTQIA+ persons.

Why this is noteworthy:

  1. Representation matters. (As a gay white male, I confess that I sometimes forget this point. It’s easy to argue that we should be interested in the ideas behind the people. But if all the theorists look like me, it’s only fair to acknowledge I am plenty represented, and that others might not feel that way.)
  2. If we don’t work toward representation ourselves, then it will never happen. The pretext that we cannot change history—that certain figures were simply more central than others, that certain figures brought new ideas to the discussion—is irrelevant. If we select five figures from a timeframe that spans one and a half millennia, we make massive omissions however we slice it.

Would such a goal even be achievable? What would an equivalent syllabus look like if it tried to add this level of diversity? What if it included no white males and instead elevated a canon made up entirely of women and POC persons? To plan this out, I decided to stick with our five-figure structure. Here is what I came up with in my thought experiment:

  1. Ptolemais of Cyrene (3rd century bce?)
  2. Al-Fārābī (c. 872–950/51)
  3. Zhu Zaiyu (1536–1611)
  4. Johanna Kinkel (1810–1858)[1]
  5. Julián Carrillo (1875–1965)

There is a fairly coherent story around these figures as well. It is a story of tuning and consonance, and the various larger units that can be created out of our decisions to divide tones in particular ways and not in others. (For what it’s worth, other figures I considered but ultimately did not include are the Japanese scholar Shohe Tanaka, on whom Danny Walden has written in his dissertation, and the American experimentalist Henry Cowell, on whom I will write more later. There are doubtless plenty others who are not even on my radar.)

In putting together this alternative “Five Classics,” I was looking for a number of different factors at the same time, and it’s worth laying out clearly the kinds of calculations that went into this.

  • What are the topics worth covering? Specifically, what key concepts will students be able to further explore in other contexts?
  • What historical timeline will provide us with a relatively coherent narrative that could, ideally, span the entire semester?
  • How can we cover a reasonable range of different historical and cultural constellations and periods? What arguments and issues from any one debate can we build on in future topics?

As a side note, I have been quite interested in the kinds of instruments that music theorists use in order to demonstrate their ideas, such as the monochord and the piano. The affordances and constraints of these instruments can tell us quite a lot about the musical universe in which the theories operate. The diverse theorists also introduce a new range of music-theoretical instruments: the oud of the Arabic world and the Chinese twelve lü.

Most of the concerns and considerations here are not fundamentally different from the earlier syllabus. In our effort to cover a large timespan, we aimed to find connections and coherence wherever possible.

Of course, certain things that we assume in conventional (let’s just call it heteronormative white male) histories of music theory cannot be carried over into this conglomerate. In this conventional world it is possible to write history as a series of dialogs, in which later theorists critique earlier theorists. In my old syllabus, we can read Helmholtz picking up ideas from Rameau, Rameau taking issue with Zarlino, or Zarlino referencing Boethius. We can be fairly sure that each theorist on the old syllabus was familiar with the works of the earlier ones, at the very least certain works of the previous generation. This syllabus can easily be read as a meeting of minds in timeless space in which heteronormative white male theorists discuss music-theoretical issues and illuminate them from all sides.

Such a meeting of minds is more difficult to construe for a more diverse group of theorists. There is no evidence that, say, Johanna Kinkel read Zhu Zaiyu, or was even aware of his groundbreaking work in Ming-dynasty China. Al-Fārābī pays, as far as I am aware, no particular heed to Ptolemais’ work, just as Julián Carrillo, while trained in Germany, does not shed any light on Johanna Kinkel. Any connections between these figures must remain imaginary and imaginative. But this is not to say they cannot coexist—in theory.

We will simply have to try a little harder to uncover such connections. It may not always be possible to make links on a person-to-person basis, but the links exist on a discursive level, in which Ptolemais—whose work we only know obliquely, through Porphyry—becomes the figure through whom we explore the world of ancient Greek music theory. Once we broaden out in this way, it is no problem to link Ptolemais’s work with that of Al-Fārābī, who expanded Greek principles into the Islamicate[2] context of the tenth century ce—or rather, the third century ah in the Islamic calendar, to further decenter our sense of chronology. The famous story of Zhu Zaiyu, who calculated equal temperament in China in 1584, some years before Simon Stevin made the equivalent calculation in Europe, using his own methods and technologies, offers its own global context.

[…] Continuation: III. Some Consequences […]


[1] Thanks go to Danny Walden for introducing me to Johanna Kinkel, the female composer and music theorist in the circle around Karl Marx exiled in London, and her remarkable call to Emancipate the Quartertone!

[2] With thanks to Siavash Sabetrohani, I adopt this term from Marshall Hodgson, The Venture of Islam: 1. The Classical Age of Islam (University of Chicago Press, 1975), 59. “I have been driven to invent a term, ‘Islamicate’. It has a double adjectival ending on the analogy of ‘Italianate’, ‘in the Italian style’, which refers not to Italy itself directly, not to just whatever is to be called properly Italian, but to something associated typically with Italian style and with the Italian manner. One speaks of ‘Italianate’ architecture even in England or Turkey. Rather similarly (though I shift the relation a bit), ‘Islamicate’ would refer not directly to the religion, Islam, itself, but to the social and cultural complex historically associated with Islam and the Muslims, both among Muslims themselves and even when found among non-Muslims.”

Can the History of Theory Be Decentered? Part I – Prequel: Five Classics

Alexander Rehding


Until fairly recently, Harvard’s History of Music Theory exam was rather free-wheeling.[1] It allowed our graduate students to focus on any period or topic in the history of theory. But my colleagues and I eventually came to see this model as unwieldy. An overhaul was needed. To make the exam more manageable for all sides, we decided that the questions should be drawn from a sequence of five central music theorists from across the ages.[2]

In general, our qualifying exam tries to strike a balance between teaching the things that “belong in the music theorist’s toolkit” (i.e. are important for the job market) and letting students explore their personal interests. While keeping requirements to a minimum, we aim to fill this toolkit with skills and research areas that future employers tend to expect. By being canonical in some areas, the exam structure opened up greater freedom to explore in others.

The “Five Classics” (to go full-on Confucian[3] for a moment) of our HoT exam were one of those canonical areas. It covers a range of different periods, topics, and approaches that would give students a basis for further exploration. The current list includes:

  1. Boethius
  2. Zarlino
  3. Rameau
  4. Helmholtz
  5. Schoenberg

So far so good. Gradually, though, it dawned on us that now we have to somehow teach all these figures. Taking into account other required courses, we realized that we could only afford to dedicate one semester to HoT. And this semester would need to cover all fifteen hundred years between Boethius and Schoenberg.

Every challenge is an opportunity. After some experiments trying to do these five theorists justice in one semester-long syllabus, I realized that none of my schemes were working. There was always too much to know—too much musical repertoire, too many texts, too much intellectual history, too much cultural background, too much historiography. In no way could all this be squeezed into one semester. I was about to give up when I finally hit on a solution that I thought would work: instead of starting with the profound alienness of Boethius, it would make much more sense to start our story from the end. Why not begin with Schoenberg, whose musical world is much closer to most of our students, rather than with the war-ravaged Late Antiquity of Boethius?

By working in reverse chronological order, starting with the trope of the “end of tonality,” we could capitalize on the knowledge of music history that many students would bring to the course. This way we could start by delving into the trajectories of musical evolution that were so commonly invoked in the late nineteenth century, and that were retrofitted[4] so as to explain why Schoenberg’s atonality was either a historical necessity, or a terrible aberration that led to the irrecoverable loss of a once-flowering tradition. These discussions would then allow us to introduce various historiographic topics, and would lead, almost imperceptibly, to the next topic on our list: the issue of tonality, and the challenges that Helmholtz’s scientific example offered to traditional music theory.

It did not take long to realize that the actual problem I was facing was less about the material itself, but about the ways in which a coherent story could be told in one semester. (At the other extreme, I had taught seminars before that focused exclusively on a single historical figure, such as Rameau or Riemann, or on a selection of topics within a given century.) Going backwards in time was not really a problem as long as we thought of it as a succession of topics rather than a succession of people. While the exam structure still focused on the works of five central figures, there was no problem if the seminar told the story of theoretical concepts. And, in the service of one coherent red thread, we could definitely do worse than to choose the old story of pitch relations, talking about changing notions of key concepts such as consonance, scales, counterpoint, harmony, and tonality. A reverse chronological order would, at least, ensure a different perspective that does not replicate the well-worn tropes of progressive complexity leading toward the present age. By moving backward in time, we would have to confront—in what I was hoping would be a striking realization—that the numerical structuralism of the Pythagorean tradition has never fully disappeared, but merely faded into the background because of our reliance on equal temperament. So complex is the numerical reality of our music that we have taken a collective shrug and directed our attention to other issues.

To tell the full story, I ended up modifying the syllabus in one major regard: We moved backward to Rameau, and then inserted a hard break. The second (shorter) half of the semester began with Boethius and his adaptation of Greek modes, and moved forward to Zarlino. This had the advantage of avoiding the complexity of introducing the concept of modes within a contrapuntal framework. It had the additional benefit that we could close our overview with further historiographic questions, circling back to the issue of tonality, into whose service Zarlino had been pressed by the likes of later commentators such as Riemann and Matthew Shirlaw. By putting Zarlino at the end we could close the circle of the semester, and connect the broad historiographic questions that we had raised at the beginning of our journey.

I was pretty happy with the structure of the semester. Sure, it was not perfect. There was no Formenlehre, no Affektenlehre, no rhythmic theory. There was no Koch, no Kirnberger, no Fux, no Mersenne, no Gaffurius, no Marchetto, no Guido, no [insert your favorite theorist here]. As with any syllabus, mine had holes. Yet it offered cohesion and reasonable coverage. Moreover, the warped chronology did a pretty good job of reframing canonical figures in a way that would nudge them (and the students) out of their comfort zones.

[…] Continuation: II. Five Different Classics […]


[1] This blogpost would not have been possible without the many invaluable conversations and exchanges that preceded and accompanied its writing. Thanks go particularly to Will Cheng, Phil Ewell, Roger Grant, Drew Hicks, Lester Hu, Nathan Martin, Carmel Raz, Siavash Sabetrohani, and Danny Walden, as well as the students in my current Music 220: History of Music Theory.

[2] A tip of the hat goes to Henry Klumpenhouwer here, who first suggested a five-book structure to me.

[3] Thanks go to Lester Hu for pointing out the irony that a sixth Confucian classic, now lost, was a volume on music.

[4] Historians have come up with the useful term “backshadowing” here. See Michael Bernstein, Foregone Conclusions: Against Apocalyptic History (Berkeley: University of California Press, 1994).

The Influence of Riemann (and Richter) on Music Theory in Scandinavia

Thomas Jul Kirkegaard-Larsen

Even though it is well known that Hugo Riemann’s function theory traveled far and wide beyond the boundaries of German-speaking Musikwissenschaft, the exact impact it has had in other linguistic and national communities—the ways in which it has been adapted, transformed, and combined with local idiosyncrasies—has been subject to limited scholarly scrutiny. Lately, however, the surge of interest in global histories of music theory (see Raz et al. 2019) has given the international music theoretical community new insights into function theory’s many different national adaptations. Recent examples include Gesine Schröder’s entry on this blog and the research project on Chinese reception of European theory to which it refers (see Cheong and Hong 2017; 2018; Ying and Komatović 2017), as well as the 2018 special issue of Zeitschrift der Gesellschaft für Musiktheorie devoted to research in the global reception of European music theories, in which one article focuses on Riemann’s (as well as Schenker’s and Schoenberg’s) dissemination in Brazil (Almada et al. 2018).

It may come as no surprise, then, that Riemann’s function theory also found its way up north, to Scandinavia. More surprising, perhaps, is the fact that even in Scandinavia, in which German was considered the musicological lingua franca for much of the 20th century before it gradually turned towards English, the direct influence of German sources is very limited. Moreover, even though the Swedish, Danish, and Norwegian languages are similar enough to enable cross-national communication—and even though Scandinavian theorists demonstrably have influenced each other across national borders—the function theories of the three countries are remarkably different from each other. This blog entry gives a brief introduction to some of the ways that function theory was received and transformed in Sweden, Denmark, and Norway, and makes reference to some of my own research published in forthcoming articles.



Apart from an article in the Swedish Journal of Music Research from 1931 which primarily discusses Hugo Riemann’s idea of Scheinkonsonanz (Svensson 1931), the first Swedish text on function theory was Sven E. Svensson and Carl-Allan Moberg’s Harmonilära from 1933. Here, Riemann’s direct influence is palpable: the authors adhere to his much criticized theory of harmonic dualism and structure the textbook in the style of Riemann’s Handbuch der Harmonielehre, which—from the third edition and onward—presents the theory of functions in tandem with his earlier theory of Harmonieschritte (Riemann 1898). But though the textbook of Svensson and Moberg has certainly been influential insofar as it introduced function theory to Sweden, until then dominated largely by Aron Bergenson’s Harmonilära (1899) (which was itself modelled on Ernst Richter’s Lehrbuch der Harmonie [1853]), none of Svensson and Moberg’s dualistic aspects are to be found in later publications. Instead, a monistic theory evolved in publications by Ingemar Liljefors (1937; 1951), Harald Göransson (1947; 1950), Valdemar Söderholm (1959), and Henry Lindroth (1960) to mention but a few of the most formative theorists. Unlike the case in Denmark and Norway, there are several Swedish publications clearly inspired by Austro-German function theories, especially those of Hermann Grabner (a main inspiration in Göransson 1947; 1950) and Wilhelm Maler (influential in Tegen 1974 and Smedeby 1978). Thus, present-day Swedish function theory is quite similar to what one finds in German publications, but with many unique characteristics nonetheless. For instance, several Swedish publications group chromatic mediants into a matrix of categorizations along three axes: a particular mediant is either upper or lower; primary or secondary; and even has a subdominant or dominant function—a peculiar addition to David Kopp’s 2002 book on historic mediant theories worthy a study of its own.


Function theory enters Denmark the same year as it enters Sweden: in 1933 with Finn Høffding’s Harmonilære. Høffding’s account of function theory could not have been more different from that of its contemporaneous Swedish counterpart: there are no traces of harmonic dualism, or any reference to Harmonieschritte. In fact, Høffding altogether avoids the term “function,” replacing it with the idea of “affinity” between chords. This term, Høffding explains, is “borrowed from chemistry, in which it designates a substance’s tendency to connect with a certain other” (Høffding 1933, 4; my translation). In effect, there is a change of focus from function residing in the chords to affinity between chords—though in practice, Høffding’s analyses apply function letters in a conventional way, albeit with some terminological innovations. For instance, Høffding presents a new function symbol for the chord on the sixth scale degree as it appears in deceptive cadences: A tilted T symbolizing that it functions as a tonic substitution, conceptually different from Riemann’s Tp, tonic Parallel (later Danish theories write Ts or Tst as shorthands for the tonic substitution). Høffding thus makes way for a new breed of function theory—only fully realized in Svend Westergaard (1961)—in which chords in paradigmatic progressions receive distinct functional suffixes; third related chords are not always a Parallel, a Leittonwechsel, or a Gegenklang (Grabner 1944), but may be analyzed as a substitution of a main function, a derivation, a prolongation, and more. With Teresa Waskowska Larsen and Jan Maegaard’s Indføring i romantisk harmonik (1981), this system evolved into an analytical approach that to a certain extent anticipates later Anglo-American transformational and neo-Riemannain approaches, a point that I discuss furtherly in my article “Transformational Attitudes in Scandinavian Function Theories,” forthcoming in Theory and Practice.


Function theory gets a late start in Norway compared to its neighboring Scandinavian countries. The first publication with traces of function theory is Thorleif Eken’s Harmonilære from 1948. Eken refers to the Danish Harmonilære of Povl Hamburger and Hakob Godske-Nielsen (1939) as his main inspiration; though adding a few function theoretical aspects, Hamburger and Godske-Nielsen’s textbook is closely modelled on that of the Danish J. D. Bondesen (1897), which is in turn strongly influenced by Richter’s Lehrbuch (1853), a book that Bondesen had previously translated to Danish (Richter 1883). In other words, Richter-inspired textbooks such as these, as well as the Norwegian Praktisk harmonilære by Gustav Fredrik Lange (1897), continued to exert influence in much of 20th century Scandinavian theory, especially in Norway. In Eken’s function theory, this influence is still noticeable. Eken uses Roman numerals rather than function letters, but he refers to his numerals as “function numbers.” It was only with Anfinn Øien’s publications (1971; 1975) that function theory seriously entered Norway. Like Eken, Øien looks to the writings of the Danish Povl Hamburger and presents a function theory with the same unique terminology that Hamburger presents in his Harmonisk analyse (1951). In effect, Norwegian theory largely emanates from Hamburger and his analytical practice where the Parallel-term is reserved for instances where there is an actual tonal indication of the Parallel key; for third-related chords that do not imply a change of key, one uses the functional suffixes submediant and mediant instead. This practice continues in Norwegian theory to this day—though the usefulness of function theory has been questioned in the Robert Gauldin-inspired publication by Petter Stigar (Gauldin 1997; Stigar 2004).

Ramifications of Riemann and Richter

The influence of Riemann and Richter in Scandinavia is in itself not the interesting point here. Rather, it is the fascinating and unpredictable ways that the theories have been received, remodelled, and ramified in directions more or less segregated from their origins. There are many reception histories to be written here.

Thomas Jul Kirkegaard-LarsenThomas Jul Kirkegaard-Larsen is a PhD student in Musicology at Aarhus University, Denmark. He holds MA and BA degrees from Aarhus University in Musicology and Scandinavian Languages and Literature. His dissertation focuses on the traditions of Anglo-American Schenkerian theory and Continental-European Funktionstheorie for the purpose of reassessing and bridging the perceived gap between these traditions. Thomas is the winner of the Patricia Carpenter Emerging Scholar Award 2018 as well as a Fulbright Scholarship 2017–18 which supported his stay as a Visiting Research Scholar at the CUNY Graduate Center.


Almada, Carlos de Lemos, Guilherme Sauerbronn de Barros, Rodolfo Coelho de Souza, Cristina Capparelli Gerling, and Ilza Nogueira. 2018. “The Reception and Dissemination of European Music Theories in Brazil: Riemann, Schenker, and Schoenberg.” Zeitschrift der Gesellschaft für Musiktheorie 15 (2): 129–154.

Bergenson, Aron. 1899. Harmonilära. Stockholm: Abraham Lundquist Musikförlag.

Bondesen, Jørgen Ditleff. 1897. Harmonilære. Copenhagen: Det Nordiske Forlag.

Cheong, Wai Ling and Ding Hong. 2017. “Eine Altlast – das Brigaden-Lehrbuch in China.” Zeitschrift ästhetische Bildung 9.

———. 2018. “A Soviet Harmony Textbook’s Twisted Fate in China.” Zeitschrift der Gesellschaft für Musiktheorie 15 (2): 45–77.

Eken, Thorleif. 1948. Harmonilære. Oslo: Norsk Notestik & Forlag.

Gauldin, Robert. 1997. Harmonic Practice in Tonal Music. New York: W. W. Norton & Company, Inc.

Grabner, Hermann. 1944. Handbuch der Harmoielehre. Berlin: Max Hesses Verlag.

Göransson, Harald. 1947. Funktionell harmonilära, 2nd ed. Stockholm: no publisher.

———. 1950. Lyssnarens harmonilära. Stockholm: Forum.

Hamburger, Povl. 1951. Harmonisk analyse. Copenhagen: Aschehoug.

Hamburger, Povl, and Hakon Godske-Nielsen. 1939. Harmonilære. Copenhagen: Aschehoug.

Høffding, Finn. 1933. Harmonilære. Copenhagen: Wilhelm Hansen.

Kopp, David. 2002. Chromatic Transformations in Nineteenth-Century Music. New York: Cambridge University Press.

Lange, Gustav Fredrik. 1897. Praktisk harmonilære. Kristiania: Warmuths Musikforlag. [2nd ed. in 1905; 3rd and 4th ed. undated.]

Larsen, Teresa Waskowska, and Jan Maegaard. 1981. Indføring i romantisk harmonik. Copenhagen: Engstrøm & Sødring.

Liljefors, Ingemar. 1937. Harmonilärans grunder med ackordanalys enligt funktionsteorien. Stockholm: C. A. V. Lundholms Aktiebolag.

———. 1951. Harmonisk analys enligt funktionsteorien. Stockholm: C. A. V. Lundholms Aktiebolag.

Lindroth, Henry. 1960. Musikalisk satslära. Malmö: Allhems Förlag.

Raz, Carmen, David E. Cohen, Roger Mathew Grant, Andrew Hicks, Nathan John Martin, Caleb Mutch, Melanie Wald-Fuhrmann, Felix Wörner, and Anna Zayaruznaya. 2019. “Going Global, In Theory.” Musicological Brainfood 3 (1).

Richter, Ernst Friedrich. 1853. Lehrbuch der Harmonie. Leipzig: Breitkopf und Härtel.

­­———. 1883. Harmonilære. Translated by J. D. Bondesen. Copenhagen: Thorvald Petersens Bog- & Musikhandel.

Riemann, Hugo. 1898. Handbuch der Harmonielehre, 3rd ed. Leipzig: Breitkopf und Härtel.

Smedeby, Sune. 1978. Från treklang till nonackord: Harmonilära. Stockholm: Eriks Förlaget.

Stigar, Petter. 2004. Elementær harmonilære. Koralharmonisering, kontrapunkt, generalbass og variasonssatser. Bergen: Fakbokforlaget.

Svensson, Sven E. 1931. “Till förståelsen av dissonansbegreppet.” Svensk tidskrift fïr musikforskning 13: 163–170.

Svensson, Sven E., and Carl-Allan Moberg. 1933. Harmonilära. Stockholm: Carl gehrmans Musikförlag.

Söderholm, Valdemar. 1959. Harmonilära. Stockholm: Nordiska Musikförlaget.

Tegen, Martin. 1974. Musikteori I. Stockholm: Stockholms Universitet – Musikvetenskap.

Westergaard, Svend. 1961. Harmonilære. 2 vols. Copenhagen: Wilhelm Hansen.

Ying, Wang, and Nikola Komatović. 2017. “Eine prächtiges Wachstum! Vorbilder, Originalität und Auswirkungen von Igor Sposobins Harmonielehrbuch, unter Berücksichtigung der Situation in China und in Serbien.” Zeitschrift ästetische Bildung 9.

Øien, Anfinn. 1971. Grunnbok i funksjonell harmonilære. Oslo: Universitetsforlaget.

Øien, Anfinn. 1975. Harmonilære: Funksjonell harmonik i homofon sats. Oslo: Norsk Musikforlag.

A Princely Manuscript at the National Library of China — Part I: Guido’s Hexachords and the 18th-century Chinese Opera Reform

Zhuqing (Lester) Hu

Originally from the court library of the Qing Empire (1636-1912), National Library of China Putong Guji 15251 (c. 1707) is a manuscript in Chinese of 127 folios, roughly 24*12 cm in size, organized in four separate stitched wrapped-back fascicles (ce 冊) in one cloth encasement.Many scholars have perused its first 51 folios, which contain a copy of the earliest Chinese-language treatise on Western music, Elements of Pitch Pipes (c. 1690);[1] see Image 1, for example, for an illustration of the Guidonian hand in f. 16r.


Image 1: An illustration in f. 16r of the Guidonian hand, featuring exquisite fingernails and shrouded by mystifying clouds at the wrists — both were distinct features of hand diagrams used in guqin 古琴 or Chinese zither manuals of the time. © 中国国家图书馆 National Library of China

Virtually no consideration, however, has been accorded to the latter 76 folios, which feature a commonplace book on Chinese music theory in ff. 52-83 and a collection of preparatory notes and reckonings for a treatise on Chinese musical tuning in ff. 84-127. The reason for this one-sided attention is easy to gauge. Whereas the commonplace book and the collection of notes in ff. 52-127 are but two among hundreds of early modern Chinese writings on “indigenous” music, the copy of Elements in ff. 1-51 arose directly from musical encounters between China and Europe that otherwise left few treatises in the early days. The musica practica content of hexachords and staff notation in Elements developed from the music lessons that the Portuguese Jesuit Tomás Pereira (1645-1708) gave to the Kangxi Emperor (r. 1662-1722) and his sons between the 1670s and 90s. Many of its passages also came straight from Books V and VII of Athanasius Kircher’s (1620-1680) Musurgia universalis (1650), of which the Belgian Jesuit Ferdinand Verbiest (1623-1688) produced a partial Chinese translation in 1685.[2]

After spending two months last year copying the latter 76 folios of Putong Guji 15251 into my laptop word by word, however, I realized that their “Chinese” contents were as much a product of trans-Eurasian musical exchanges as was the “Western” content of its first 51 folios. Granted, unlike the explicit references to xiyang 西洋 or Western Europe in Elements, the commonplace book and collection of notes in ff. 52-127 do not declare any foreign connections. Still, these latter folios constitute a case of what art historian Jonathan Hay has termed “hidden cosmopolitanism,” where cross-cultural engagements were not narrated as such by their historical participants and thus become “hidden” in their sources.[3] This hiddenness, Hay argues, does not imply a lack of depth: on the exact contrary, hidden cosmopolitanisms often arise when once-foreign influences have been “assimilated to the point of invisibility.” Such was the case of ff. 52-127 in Putong Guji 15251. As I will show in this blogpost, the apparently parochial contents of these folios epitomize the global embeddedness of the Qing court in its production of musical knowledge. Even without a single reference to Guido’s syllables, ff. 52-127 reveal the musical, political, and epistemological reasons for which a treatise on musica practica was copied into ff. 1-51 in the first place.

My post will divide into two parts. The current part focuses on the commonplace book in ff. 52-83, showing how the Qing used Guido’s hexachords to reform the modal theory and pedagogy of Chinese opera. A future part will focus on the collection of notes in ff. 84-127, showing how musica practica facilitated the Empire’s proposal of a fourteen-tone temperament in 1714. Besides shedding light on these overlooked traces of Sino-European musical exchanges, I also hope to draw the attention of historians of music theory to cases of hidden cosmopolitanisms similar to the latter folios of Putong Guji 15251. In our current endeavors to globalize the study of the history of music theory, scholars have mainly focused on texts that openly invoke their connections with a cultural “Other.” These “explicit cosmopolitanisms,” however, confine our narratives to the surface of cross-cultural encounters and neglect the deep transformative effects of transregional integrations. While sources like the copy of Elements of Pitch Pipes in ff. 1-51 of Putong Guji 15251 may pique our multicultural sensibilities, it is only by heeding the less obvious globalities such as those in ff. 52-127 that we may grasp the scope and depth of musical interconnectivities across the early modern world.

A Preview of Qing Imperial Music Theory

Before diving into the commonplace book in ff. 52-83, it is important to understand the context in which the entire Putong Guji 15251 was compiled. While no preface or postscript explains how its three constitutive sections relate to one another, two critical pieces of paratextual evidence illuminate how the continuously foliated manuscript came together as a whole. While the three texts were written by several different hands in black ink, a single hand annotated the entire manuscript in vermilion, a restricted color at the Qing court. A stamp impression on f. 1r further discloses the identity of this annotator: “Conferred by His Imperial Highness Prince Cheng of the First Rank, Third Son of the Emperor.” This was the full title of Aisin Gioro In-c’i (1677-1732), who was also the de facto eldest son of the Kangxi Emperor after his two elder brothers were politically disinherited in 1712.[4] Though he ultimately failed to prevail in the ongoing succession strife, In-c’i garnered significant clout from editing his father’s 100-volume (juan 卷) Origins of Cosmological Sciences,[5] which comprises one treatise each on astronomy, music theory, and mathematics. While much of their content derived from Jesuit lectures to the Emperor, the treatises themselves were written by a team of Manchu, Mongol, and Chinese scholars under In-c’i, who regularly reported to his father with drafts starting in 1711.[6] It was in this process that In-c’i adapted the Western content of Elements in ff. 1-51 of Putong Guji 15251 into the last volume of Orthodox Meaning of Pitch Pipes (1714), the Qing’s official music theory treatise that also makes up volumes 43-47 of Origins.[7]

In-c’i’s end goal of drafting a comprehensive music treatise for the Qing Empire continues to inform the entire Putong Guji 15251 after f. 51. In fact, the commonplace book in ff. 52-83 laid out the course for all Qing-sponsored research on music in the next century, of which Orthodox was but an initial step. According to a brief foreword in f. 52v, In-c’i himself ordered digests of Chinese writings on music to be compiled, with the focus on comprehensiveness rather than depth. The first three chapters of the resultant commonplace book amass materials on lülü 律呂 (Because lülü “pitch pipes” were the primary tools for studying tuning in the Chinese tradition, the term was a metonym for tuning and music theory in general—ergo the title of Elements of Pitch Pipes, even though pitch pipes are never mentioned in this treatise). As I will show in the second installment of this two-part blogpost, Chapters 1-3 of the commonplace book perfectly parallel the first volume of Orthodox. Particularly, they set the stage for a series of experiments on tuning whose results would be recorded into ff. 84-127 later in Putong Guji 15251 and lead to the proposal of a fourteen-tone temperament.

Moving beyond tuning, Chapters 4-6 spell out the ultimate agenda of Qing-imperial music theory: a reform to opera, particularly regarding modes or diao 調. Since the genre matured in the 14th century, most Chinese operatic traditions have been deriving their melodic materials from preexisting qupai 曲牌 “fixed tunes.” Comparable to contrafacta, each fixed tune features a characteristic metric and rhyme scheme for fitting new lyrics and a melody malleable to the tones of different words. The theoretical backbone for these thousands of fixed tunes used in various singing genres was a system of eighty-four modes, in which the seven types of diatonic scales are transposed to start on each pitch of the twelve-tone octave. Thus, after Chapters 1-3 of the commonplace book introduce the twelve pitch pipes (lülü) and the seven-note diatonic scale, Chapter 4 enumerates their combinations with two slide charts, each comprising one fixed wheel labeled with the twelve pitches on the outside and one movable wheel labeled with the seven notes of the scale on the inside (see Image 2). Out of all these theoretically possible transpositions and mutations, Chapter 5 studies the modes that were actually used in different periods. While musicians at the Sui (518-618) and early Tang (618-907) courts first formulated the eighty-four-mode system and labeled the twenty-eight they used, only seventeen modes remained in northern opera and thirteen in southern opera by the sixteenth century. Finally, after first introducing the gongche 工尺 notational syllables used by opera performers, Chapter 6 catalogues nearly a thousand modally ordered fixed tunes and concludes by discussing the contours, cadences, permissible mutations, and characters of different modes.


Image 2: Two slide charts (also known as volvelles), the first one rotating counterclockwise and the second one clockwise, in f. 62r, illustrating all possible combinations of the twelve-tone octave and the seven-note diatonic scale. Rotating charts are particularly fitting here, since the word for transposition in Chinese, xuangong 旋宮, literally means “rotating the tonic,” and the word for mutation, zhuandiao 轉調, literally means “turning around the mode.” © 中国国家图书馆 National Library of China

“Western” Music a Millennium Apart

It was precisely to rectify the use of modes in opera and better train opera performers that the Qing found Guido’s hexachords helpful. Though they governed China as a minority conquest regime, the Qing’s Manchu rulers surpassed all preceding Chinese regimes in opera patronage. Not only did opera’s penchant for historical themes help the Manchus appropriate the Chinese historical imagination, but sponsoring these immensely popular theatrics also brought the court closer to the Chinese landed gentry and merchant guilds, the Empire’s bread basket and tax base who were also opera aficionados themselves. Thus, even though they remained on the su 俗 “vulgar” end of the echelon of genres in contrast to the yayue 雅樂 “elegant music” that Confucianism stipulated for a virtuous ruler, opera dominated all court rituals of the Qing, from monthly feasts to diplomatic receptions and military triumphs. And while theater troops from across China performed for special occasions and imperial tours, the court also established an academy of music, nanfu 南府, to oversee regular opera performances and train eunuch performers, who had been responsible for all types of music at the court since the 17th century.

Faced with this increasing demand of performers, the Qing turned to musica practica. Though European missionaries bragged only about their musical tutelage of the Kangxi Emperor and his sons, an edict on August 2, 1714 indicates that most of their pupils were actually eunuchs. Addressing the chief eunuch, the Emperor asked that eunuch musicians learn their “u le ming fa shuo la (i.e. ut re mi fa sol la)” from the Italian Lazarist Teodorico Pedrini (1671-1746) in order to master not only their voice and instruments but also lülü “music theory.”[8] And the type of music theory they should master by studying ut re mi fa sol la was exactly the use of modes in opera. Published less than five months after the edict, Orthodox prefaces its last volume on musica practica with a history of how Suzup (fl. 568),[9] a pipa player from Kucha, helped Chinese musicians of the Sui Dynasty develop the eighty-four-mode system. Drawing a parallel between the 6th-century Suzup from xiyu 西域 “Western Region” or Central Asia and the 18th-century European missionaries from xiyang “Western Ocean” or Europe, the preface portrays the latter’s Guidonian hexachords as an effective pedagogy for the former’s modal system. And just as Suzup’s Central Asian music helped rescue Chinese music from centuries of disrepair at his time, the preface argues, Guido’s hexachords promised to rescue Suzup’s system of eighty-four modes from a millennium of misuse in Chinese opera after the Sui and Tang eras, to the point that it had become “empty words in history books.”

As it turns out, this idea of using Guido’s transposing and mutating hexachords to learn the eighty-four modes and rectify their use in opera was in In-c’i’s mind as he was reading Putong Guji 15251. While he left no comment on staff notation or rhythmic notation in the copy of Elements in ff. 1-51, In-c’i concentrated his remarks on chapters concerning the Guidonian gamut, annotating every ut re mi in the main text with its corresponding gongche syllable, the notation system opera performers used (Image 3).


Image 3: A full ladder presentation of the Guidonian gamut spanning three octaves, in ff. 12v-13r. Unfortunately, I have not been able to acquire colored scans of the source, yet all the tiny Chinese characters written in the diagram were annotations in vermilion by In-c’i, translating ut re mi fa sol la into their equivalent gongche syllables. © 中国国家图书馆 National Library of China

And while he pasted additional papers onto Chapters 5 and 6 of the commonplace book just to accommodate his comments on the fixed tunes and the modes, he read through its first three chapters on musical tuning too cursorily to even punctuate them. What’s more, at the same time he incorporated musica practica from Elements into Orthodox (1714), In-c’i oversaw two modally ordered compendia of qupai or fixed tunes that built on the last three chapters of the commonplace book: The Emperor’s Library of Lyrical Tunes (1715) featuring tunes from before the 13th century, and The Emperor’s Library of Opera Tunes (1715) featuring tunes from after the 13th century.[10] After In-c’i’s political demise in the 1720s, his brother and former apprentice In-lu (1695-1767) assumed Inc’i’s library and continued his research agenda. This culminated in Grand Compendium of Northern and Southern Opera Tunes in All Modes in 1746, which features more than two thousand tunes, four thousand musical scores, and extensive discussions on the history, usage, and characters of the modes (see Image 4).[11] It was around the same time that the Qing court codified the ritual use of opera, assigning different ceremonial functions to some two-hundred newly composed chengyingxi 承應戲 “on-demand intermezzi.”


Image 4: The first qupai 曲牌 or “fixed tune” presented in the gao dashidiao 高大石調 “High Arabic Mode”—comparable to D#-Mixolydian in the Western system—in Grand Compendium of Northern and Southern Opera Tuens in All Modes (1746), Vol. 44. The title of this fixed tune, man’er wu yunqi 蠻兒舞雲旗 “the barbarian kid waves the cloud banner,” has nothing to do with the lyrics of the actual opera aria presented here as an example of the tune; the aria is drawn from a seasonal ritual opera performed on the 7th day of the 7th month in the Chinese calendar, also known as the Chinese Valentine’s Day. © 中国国家图书馆 National Library of China


Thus, even though no cross reference exists between the ostensibly “Western” treatise in ff. 1-51 and the digests on “Chinese” music theory and opera in ff. 52-83, these two parts of Putong Guji 15251 were nonetheless compiled and consumed with a common objective: to perfect the use of modes in opera, so as to better harness their political power in service of the Qing Empire. And though I am yet to chance upon an 18th-century Chinese score of opera tunes notated with Guido’s syllables or staff notation, the entanglements both inside and outside the manuscript between the Qing court’s operatic ambitions and its interest in Western musica practica show that the latter played a far more significant role than mere foreign exotica.



[1] Lülü zuanyao 律呂纂要. The earliest record of this text is the entry on May 1, 1691 in the diary the French Jesuits Jean-François Gerbillon and Joachim Bouvet kept on their lectures to the Kangxi Emperor between January 1690 and November 1691. See Bibliothèque nationale de France, MS 17240, f. 277r.

[2] See Archivum Romanum Societatis Iesu, Jap. Sin. 145, f. 82v, in Verbiest’s letter to Charles de Noyelle in Rome on August 1, 1685.

[3] See Jonathan Hay, “Foreword” to Petra Ten-Doesschate Chu et al ed., Qing Encounters: Artistic Exchanges between China and the West (Los Angeles: The Getty Research Institute, 2015), vii-xix.

[4] The Kangxi Emperor effectively disinherited his two eldest sons in 1708 and 1712 respectively.

[5] Lüli yuanyuan 律曆淵源, literally “Origins of Pitch Pipes and Calendar”; I have translated “pitch pipes and calendar” as “cosmological sciences” due to the cosmological resonances inscribed onto pitch pipes and calendar in the Confucian tradition to which the treatise appeals.

[6] See No. 2307, 2310, 2321, 2324, 2326, 2328, 2329 in Kangxichao manwen zhupi zouzhe quanyi 康熙朝滿文硃批奏摺全譯 (“Complete Translation of Vermilion-Annotated Memorials in Manchu from the Kangxi Era”), 1996.

[7] Lülü Zhengyi 律呂正義.

[8] I have not seen this source myself, only as cited in various credible secondary sources. Purportedly it is located at the Propaganda Fide archives in Rome.

[9] His name, here in Tocharian, is also known as Sujiva in Sanskrit and Suzhipo 蘇祗婆 in Chinese.

[10] Yuding qupu 御定詞譜and Yuding qupu 御定曲譜. See No. 3115 in Kangxichao hanwen zhupi zouzhe 康熙朝漢文硃批奏摺(“Complete Translation of Vermilion-Annotated Memorials in Manchu from the Kangxi Era”), 1985.

[11] Jiugong dacheng nanbeici gongpu 九宮大成南北詞宮譜.

IMG_7192Lester Hu is a PhD Candidate in Music History and Theory at the University of Chicago. His dissertation, Music and Qing Imperial Formations, c. 1680-1820: Negotiating Historiography and Ethnography in the Global Music History, examines the shifting epistemologies of music and sound in early modern China in a transregional context. When not working on academic stuff, he enjoys cooking—particularly braising, which allows him to do work at the same time.


Rhythm, Number, and Heraclitus’ River

David E. Cohen

The following short passage was recorded by an anonymous student of Aristotle or his school in the section on music in the pseudo-Aristotelian work known as the Problems:

We enjoy rhythm because it possesses number both familiar and ordered, and moves us in an orderly way. For ordered movement is by nature more akin [to us] than disordered, as indeed [it is itself] more natural.[1]

We enjoy rhythm, this tells us, in part because it “possesses number,” and in part because it moves us in a particular, “natural” way. The following is a reading of this passage as an invitation to situate some fundamental issues concerning rhythm in a historical context.

Let us begin with “rhythm” (rhythmos) itself. The word has a number of meanings in ancient Greek, including “measure, proportion or symmetry of parts.” In the present (musical) context, its use would have implied the application of those concepts to the proportional relationships of time durations specifically, manifested first in the long and short syllables of the Greek language, especially as these were exploited in verse, and then in the analogous temporal relations evidenced in the tones of melody and the bodily movements of dance.

Thus for an educated Greek of antiquity, to say that rhythm is a phenomenon of quantity, measure, and proportion would have been to state the obvious. But the word “number” (arithmos) implies a more specific claim, namely, that rhythm consists in “discrete” quantity, that is, a kind of quantity that is intrinsically “quantized,” coming in distinct, segregated units, or discrete batches of a common unit, this being precisely what was meant in ancient Greek by “number” in the specific and proper sense of the word: a countable collection of units.[2] And this idea of units of measure, whether conglomerated into a set or concatenated in a (temporal or spatial) series, reminds us that the durational relations of measure, proportion, and symmetry mentioned above all entail regularity or periodicity, the recurrence of a single, consistent time span. As we are told in an earlier section of the Problems, “every rhythm is measured by a determinate motion, and equal motion is of this kind.”[3]

All this implies that, for the writer and readers of our quoted passage, rhythm might have had a quality akin to that which we now call meter, characterized by periodicity, regularity, symmetry, the repetitive and predictable recurrence of identically “spaced” modules of identical “length.”[4]

On the other hand, it is also possible—though unlikely—that in the phrase “rhythm possesses number,” “number” is being used generically or metaphorically to denote “quantity” in general (to posón). If so, then it could be read so as to include continuous quantity, the kind of quantity found in the lines, planes, and solids of geometry, which the Greeks also called “magnitude” (megethos). Such a conception of the quantitative nature of rhythm might be taken to emphasize its continuity, its fluidity, its flow (rheuma). It might remind us that the word itself, rhythmos, is possibly related to the verb rhein, “to flow.” One might even go so far as to recall Heraclitus’s doctrine of universal flux: that “all things flow” (panta rhei) so that they are ever new, as in his saying that “upon those who step into the same rivers new waters are ever flowing.”[5] Such a reading would stress precisely those features that we now tend to associate specifically with rhythm: variety, creativity, unpredictability, what Christopher Hasty has characterized as the “spontaneous creation of the ever new,” in opposition to meter as the “constant repetition of the same.”[6]

That possibility is tempting. Nonetheless, the passage seems clearly to support the normal and proper meaning of “number” as “discrete quantity,” distinct groups of discrete units. For the passage goes on to characterize this “number” possessed by rhythm as both “familiar” and “ordered.” The word translated as “familiar” (gnôrimon) is from gnôrizein, “to come to know,” with its resonance of gnôsis, “knowledge,” suggesting that the “number” of rhythm is “familiar” in the sense of being “known” to the intellect, which recognizes something in it, meaning that the number in question is an entity of such a kind as to be, by nature, capable of being cognized (or “re-cognized”) by the mind; indeed its adverbial form gnôrimos means “intelligible,” “capable of being understood.” But for a Greek of the fourth century BC, that characterization would inevitably have implied the properties of regularity and finitude, as opposed to the chaotic and unknowable infinite.

At the same time, this “number” is ordered (tetagmenos); this word comes from a verb (tattein) that means to organize things or people into a definite arrangement, for example, to draw troops up into a battle formation; it is the same word that gives us “syntax” and its derivatives. It calls to mind the fact that another attested meaning of rhythmos is “form,” “shape.” And it is this property of being “ordered” that is responsible for rhythm’s powerful effect on us, which is to “move us in an orderly way” —something that rhythm can only do, of course, because it is already “ordered” itself, in a way that must somehow be related to the order that subsists in ourselves.

This is why our passage’s second sentence tells us that “ordered movement is by nature more akin [to us] than disordered, as indeed [it is itself] more natural”:  the word translated as “more akin,” oikeotera, characterizes things and people that are domestic, or related by blood, or friendly, or proper, or appropriate. Again, there is no fully satisfactory English equivalent, but all of those resonances amount to an implicit assertion of a profound and intimate relationship between us as living beings and the general phenomenon of “order.” We respond with pleasure to this phenomenon of order which, we are told, is also “more natural in itself,” because order is already within us as a principle of nature. And it is why the answer to the broader question of the passage as a whole, which asks why we take pleasure in melody and consonance as well as rhythm, is that we naturally enjoy those things that embody a proportion or ratio (logos), because “a ratio is [itself] an order (taxis); and order is, by nature, pleasant.”[7]

Measure, proportion, regularity, knowability, order: these properties would surely have seemed, to the early readers of this passage, far more characteristic of discrete quantities than of continuous ones, which—as had already long been known—are rife with the possibility of irrationality. (Think, for instance, of the value of pi.)

It is the fragmentary Elementa rhythmica of Aristoxenus, himself a student of Aristotle, that provides the earliest detailed theoretical account of rhythm in terms of proportional durational relations[8] —more specifically, of ratios between the durations of the upward motion (anô, arsis) and the downward motion (katô, basis) of the metrical verse foot, measured in multiples of a temporal unit he calls the prôtos chronos (the “primary duration”).[9] Rhythm, Aristoxenus says, occurs when the rhythmized material (speech in poetry, tones in music, movements in dance) is divided into parts that are, again, “knowable” (gnorimois), and so produce the special kind of determinate arrangement (taxis, from tattein) of temporal durations that qualifies as “rhythmic” (enrhythmon).[10] This requires (among other things) that the upward and downward motions of each foot be mutually commensurable: that each of them be an integral multiple of the protos chronos, which is therefore their “common measure” (metron koinon). Such a foot is “rational” (rhêtos). “Irrationality” (alogia) occurs when that is not the case. But this is not “irrationality” in the proper mathematical sense: it is not that the lengths of the arsis and basis have no possible “common unit.” Rather, Aristoxenus calls a foot “irrational” when the common measure of its up and down motions is a duration shorter than the perceptually indivisible protos chronos. Such a duration is, simply for that reason, “arrhythmic,” and all such irrational feet are “not proper (oikeiai) to the nature of rhythm” itself.[11] Consequently, only the very small set of verse feet with ratios that are, in this sense, “rational,” and are moreover “familiar” or “knowable” (gnôrimon) because their proportional relation of arsis to basis is easily perceptible, are “proper (oikeiai) to, and capable of being ordered in accord with, the nature of rhythm.”[12] All this, of course, rules out a fortiori any proportional relations that are truly irrational (in the proper mathematical sense of having no common unit at all), and so restricts the domain of rhythm to that of arithmos, “number” in the Greek sense explained earlier.

A few moments ago I mentioned Heraclitus’s maxim, “Upon those stepping into the same rivers ever new waters flow.” Unlike his other, more famous saying, “It is not possible to step into the same river twice,” or the catch phrase used to summarize his thought, “everything flows,” this one, which seems to acknowledge the sense in which people actually can “step into the same river” more than once, permits us to think that perhaps Heraclitus did recognize some sort of stability along with the flux. It prompts us to recognize that, while everything may be continually changing, it is precisely in and through that ceaseless alteration that the seemingly stable entities of our empirical world apparently persist and subsist. The parallels, and contrasts, with rhythm and meter are intriguing.

I’d like to close with the suggestion that this dialectic of flux and stasis, sameness and difference, unfolds not only over short spans but in and through the course of history as well. It is evident in our feeling that the anonymous Greek passage we’ve been examining says things that seem both archaic and alien, as in the emphasis on an otherwise undefined “number,” and on the other hand, things that seem quite current, such as its author’s unmistakable sense that within the phenomenon called “rhythm” and its powerful effect on us there is something that is both extraordinary and significant and yet deeply, intimately familiar. Rhythm does indeed move us, cause us pleasure, and, at least in many cases, seem profoundly natural while doing so. And we are still seeking to understand why, although the kinds of answers we prefer now are usually very different from those of over two thousand years ago. But what I especially like about the passage I’ve just been discussing is its quiet sense of wonder at the marvelous phenomenon of rhythm, and its calm confidence that this can be explained by being defined and situated within a world of nature that has not, as yet, been disenchanted. It is by studying the history of such questions and answers that one can sometimes come to a new understanding, not only of who we are, how we got here, and what we have gained, but also of what we have lost along the way.[13]


[1] ῥυθμῷ δὲ χαίρομεν διὰ τὸ γνώριμον καὶ τεταγμένον ἀριθμὸν ἔχειν, καὶ κινεῖν ἡμᾶς τεταγμένως· οἰκειοτέρα γὰρ ἡ τεταγμένη κίνησις φύσει τῆς ἀτάκτου, ὥστε καὶ κατὰ φύσιν μᾶλλον (Problemata, Book 19, Chap. 38; 920b33-36).

[2] The study that established the meaning of arithmos in ancient Greek mathematics is Jacob Klein, Greek Mathematical Thought and the Origin of Algebra, trans. Eva Brann (Cambridge, Mass.: M.I.T. Press, 1968), originally published in German in 1934.

[3] πᾶς ῥυθμὸς ὡρισμένῃ μετρεῖται κινήσει, τοιαύτη δ’ ἐστὶν ἡ δι’ ἴσου οὖσα (Problemata, Book 5, Chap. 16; 882b2-3).

[4] I draw here on the brilliant critique of this conception of meter by Christopher Hasty, Meter as Rhythm (Oxford & New York: Oxford University Press, 1997), esp. pp. 1-21.

[5] ποταμοῖσι τοῖσιν αὐτοῖσιν ἐμβαίνουσιν, ἕτερα καὶ ἕτερα ὕδατα ἐπιρρεῖ. (H. Diels and W. Kranz, eds., Die Fragmente der Vorsokratiker, 6th ed. (Berlin: Weidmann, 1951), Vol. 1, Chap. 22, Pt. B, Frag. 12).

[6] Hasty, op. cit., pp. 4-5.

[7] ὁ μὲν οὖν λόγος τάξις, ὃ ἦν φύσει ἡδύ (Problemata, Bk. 19, Chap. 38; 921a3-4).

[8] Aristoxenus’ Elementa rhythmica is well translated in Andrew Barker, Greek Musical Writings, Vol. 2 (Cambridge University Press, 1989). I quote here from the edition by G. B. Pighi (Bologna: Patron, 1959), as reproduced in Thesaurus Linguae Graecae (URL: stephanus.tlg.uci.edu).

[9] Aristoxenus defines the prôtos chronos as “that [duration] which cannot be divided” by the least division of any rhythmized material (speech sounds, musical tones, bodily motions): Καλείσθω δὲ πρῶτος μὲν τῶν χρόνων ὁ ὑπὸ μηδενὸς τῶν ῥυθμιζομένων δυνατὸς ὢν διαιρεθῆναι (ed. cit. p. 19:21-22). It is not, of course, indivisible in the absolute sense, since time itself for Aristotle and his followers is continuous.

[10] Ἀναγκαῖον οὖν ἂν εἴη μεριστὸν εἶναι τὸ ῥυθμιζόμενον γνωρίμοις μέρεσιν, οἷς διαιρήσει τὸν χρόνον. … Ἀκόλουθον δέ ἐστι … τὸ λέγειν, τὸν ῥυθμὸν γίνεσθαι, ὅταν ἡ τῶν χρόνων διαίρεσις τάξιν τινὰ λάβῃ ἀφωρισμένην, οὐ γὰρ πᾶσα χρόνων τάξις ἔνρυθμος (Elementa rhythmica, ed. cit., p. 18: 15-16, 18-20).

[11] Δεῖ δὲ μηδ’ ἐνταῦθα διαμαρτεῖν, ἀγνοηθέντος τοῦ τε ῥητοῦ καὶ τοῦ ἀλόγου, τίνα τρόπον ἐν τοῖς περὶ τοὺς ῥυθμοὺς λαμβάνεται. … Τὸ μὲν γὰρ κατὰ τὴν τοῦ ῥυθμοῦ φύσιν λαμβάνεται ῥητόν, τὸ δὲ κατὰ τοὺς τῶν ἀριθμῶν μόνον λόγους. Τὸ μὲν οὖν ἐν ῥυθμῷ λαμβανόμενον ῥητὸν χρόνου μέγεθος πρῶτον μὲν δεῖ τῶν πιπτόντων εἰς τὴν ῥυθμοποιίαν εἶναι, ἔπειτα τοῦ ποδὸς ἐν ᾧ τέτακται μέρος εἶναι ῥητόν· τὸ δὲ κατὰ τοὺς τῶν ἀριθμῶν λόγους λαμβανόμενον ῥητὸν τοιοῦτόν τι δεῖ νοεῖν οἷον ἐν τοῖς διαστηματικοῖς τὸ δωδεκατημόριον τοῦ τόνου καὶ εἴ τι τοιοῦτον ἄλλο ἐν ταῖς τῶν διαστημάτων παραλλαγαῖς λαμβάνεται. Φανερὸν δὲ διὰ τῶν εἰρημένων, ὅτι ἡ μέση ληφθεῖσα τῶν ἄρσεων οὐκ ἔσται σύμμετρος τῇ βάσει· οὐδὲν γὰρ αὐτῶν μέτρον ἐστὶ κοινὸν ἔνρυθμον (Elementa rhythmica, ed. cit., p. 23:1-3, 9-19).

[12] Διὰ ταύτην γὰρ τὴν αἰτίαν τὸ μὲν ἡρμοσμένον εἰς πολὺ ἐλάττους ἰδέας τίθεται, τὸ δὲ ἀνάρμοστον εἰς πολὺ πλείους. Οὕτω δὲ καὶ τὰ περὶ τοὺς χρόνους ἔχοντα φανήσεται· πολλαὶ μὲν γὰρ αὐτῶν συμμετρίαι τε καὶ τάξεις ἀλλότριαι φαίνονται τῆς αἰσθήσεως οὖσαι, ὀλίγαι δέ τινες οἰκεῖαί τε καὶ δυναταὶ ταχθῆναι εἰς τὴν τοῦ ῥυθμοῦ φύσιν (Elementa rhythmica, ed. cit., p. 19:3-8).

[13] The foregoing is an expanded and much revised version of remarks read at the opening plenary session of the fifth annual Mannes Institute for Advanced Studies in Music Theory, held at the Mannes School of Music in New York City in the summer of 2005. The Institute’s topic that year was rhythm, and I was to lead a workshop that would examine theories of rhythm historically. Each workshop leader delivered a brief preliminary address. I wish to thank Carmel Raz for suggesting that I share mine with the readers of the AMS / SMT HoT blog.


David E. Cohen is Senior Research Scientist with the research group, “Histories of Music, Mind, and Body” at the Max Planck Institute for Empirical Aesthetics in Frankfurt, Germany. His research focuses on the history of music theory from Greek antiquity through the nineteenth century. A PhD graduate of Brandeis University (1993), he has held professorships at Columbia, Harvard, and Tufts Universities, and visiting professorships at Yale and McGill Universities. His article, “‘The Imperfect Seeks its Perfection’: Harmonic Progression, Directed Motion, and Aristotelian Physics” received the 2001 Best Publication award of the Society for Music Theory. Among his current projects are an essay about the musical note as the “element” of music, a study on Rameau’s harmonic theory, and a book, The End of Pythagoreanism: Music Theory, Philosophy, and Science from the Middle Ages to the Enlightenment.