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.


Storms in Chang-an: On the Music Debate of Kai-huang Period

Rujing Huang

In its pursuit of national rejuvenation, the Xi administration in China has shown a renewed interest for traditional culture. Integral to this latest wave of cultural renaissance is a movement to reconstruct—based on treatises of various historical times—the nation’s classical music theory, an effort driven by a group of musicians, scholars and self-acclaimed literati (wenren) in Beijing. Accompanying the intensifying anxiety over the absence of a unified, national music theory is a re-emerged faith in the ancient Chinese association between the quality of music and the quality of rule. Throughout early and imperial Chinese history, the consolidation of political powers had almost always entailed a large-scale reform of the state sacrificial music system, a crucial step in harmonizing the new regime with larger cosmic patterns and in turn confirming its legitimacy. Such efforts ranged from the adjustment of musical temperament to that of modal scales, and from the re-sizing of ritual bell chimes to the renewal of the official repertory. Marking one of these pivotal moments is the Music Debate of Kai-huang Period (thereafter “the Debate”), a series of conferences summoned by the emperor between 582 and 594 CE to reform music theory. The Debate is best documented in the “Treatise on Music” of the Book of Sui, official dynastic history and a major source for the music of the Sui court (Figure 1).


Figure 1: Wei Zheng (580-643), Book of Sui (Suishu), Harvard-Yenching Rare Book T 2605 2124

A Backdrop

Historians today have pointed to the period between 500 and 800 CE as the “first great divergence” between China and Europe: while in Eastern Eurasia the Sui dynasty reunited China following the long-time fragmentation of the Northern and Southern Dynasties, in Western Eurasia the decline of the Roman Empire had brought about a tumultuous era of political disintegration. In the musical realm, when Boethius was working to safeguard Greek musical knowledge, the Sui court was confronted with the sudden influx of foreign music—an after-effect of a newly unified China—that had started to threaten the ritually proper sacrificial music known as yayue. Taking place in the capital city of Chang’an, the Debate eventually evolved into a fierce political battlefield.  Its account captures the unique role that music theory played in shaping early Chinese conceptions of the government, the empire, and the universe.

Largely absent in the English-language literature, the Debate nevertheless remains a topic of interest among music historians and theorists in China today. Explicit mention of this milestone event is rare among avid revivalists of classical music theory in Beijing, but many theoretical contributions of the Debate continue to underlie the ongoing revivalist campaign. A re-examination of the Debate is timely, not only for its relevance to current discussions about the recovery of classical Chinese music theory, but also for the gateway it provides to understanding the intersection between music and politics during a historical period of frequent inter-ethnic exchange, the effects of which are still felt today.

The Debate

In 582 CE, an imperial decree was issued by Emperor Wen of Sui (Figure 2), founder of China’s Sui Dynasty, to “check the Bureau of Music and to alter the tones in a modal scale along with the standard tuning pitch-pipes,” initiating a twelve-year reform aimed at rectifying the official system of imperial ritual music. According to the Book of Sui, the Emperor, then dissatisfied with the initial slow progress of the reform, lamented, “The Mandate of Heaven has been bestowed upon me for seven years. How can the Music Bureau still praise the virtues of the bygone era!” Underlying the Emperor’s anxiety is a century-old belief in the necessity of “correcting” the official music of the former court immediately after the rise of a new imperial regime.


Figure 2: Emperor Wen of Sui, from the Portraits of Former Emperors (Lidai diwang tu) by Yan Liben (600–673); Museum of Fine Arts, Boston

The first watershed moment of the Debate occurred in 587 CE, when the Emperor summoned a cohort of scholar-official elites to resume the discussion over yayue. Zheng Yi, the Duke of Pei, famously proposed the theory of Wu-dan Qi-diao (“five note-sets, seven mode-keys,” or WDQD) based on the teachings of Sujivha, a Kuchean musician who arrived at the Chinese court with Princess Ashina of the Göktürk empire—a nomadic confederation of Turkic peoples in medieval Inner Asia (Figure 3). It was recorded that Sujivha, using five heptatonic “note-sets” (dan in Kuchean terms, or yun in Chinese modal terminology) built on distinct home pitches (gong), demonstrated the possibility of initiating mode-keys (diao) on all seven scale degrees of a note-set (Figure 4 a and b).[1] In citing Sujivha, Zheng hoped to revive a classical tradition of modulation known as “rotating the gong and shifting the diao (xuangong zhuandiao), which calls for the monthly modulation of the basic, heptatonic note-set—by altering its gong pitch—in order to achieve seamless alignment between musical structure and the changing cosmic elements.

tang china

Figure 3: A Map of Tang China at 700 CE, with Kucha and Chang’an highlighted in red (image credit: Ian Mladjov). A Central Asian oasis kingdom, Kucha became the seat of a Chinese protectorate (officially “the Protectorate General to Pacify the West”) during the Tang Dynasty.

An erudite of the National Academy (guozi boshi) and a key opponent of Zheng, He Tuo insisted that the “yellow bell” or huangzhong pitch (C), a musical signifier for “the virtue of the ruler,” be established as the only gong sound of the Sui dynasty. To gain imperial favor, officers who sided with He argued that only by securing the majestic “yellow bell” pitch as the gong would the newly founded empire rightfully prevail over the previous Northern Zhou rule, a regime believed to be overtaken for falsely selecting a musical symbol for “the ruled”—the linzhong or “forest bell” pitch (G)—as its official gong. He’s narrative is rooted in an ancient method of pitch generation known as “A-Third-Removing-Extending” (ATRE, see an earlier post by Guangming Li), which specifies that the “yellow bell” pitch pipe “generates downward”—and hence governs—the “forest bell” pitch pipe. He’s argument soon won over the heart of the Emperor, who overturned Zheng’s proposal and concluded that the gong of Sui be none other but the pitch of “yellow bell.”


Figure 4a. Illustration of the Wu-Dan Qi-Diao (WDQD) theory:
the basic heptatonic note-set (dan) built on C



Figure 4b.  Illustration of the Wu-Dan Qi-Diao (WDQD) theory: the seven mode-keys (diao) of a dan

The second turning point of the Debate came in 589 CE, immediately after the Sui conquered its neighboring Chen state. In fear of having the popular music (suyue) and foreign music (huyue) from the newly annexed lands “pollute” the orthodox, elegant ritual music (yayue) of the Sui court, the Emperor ordered for another grand meeting to revise yayue. In the limelight was Niu Hong, Minister of the Court of Imperial Sacrifices, who advised that the Emperor re-consider Zheng Yi’s earlier proposal for restoring the classical method of “rotating the gong” (xuangong) throughout the year. The emperor, having fully subscribed to He’s argument on the potentially disastrous, symbolic power of music, rejected Niu’s recommendation and once again affirmed the status of “yellow bell” as the sole, legitimate gong of his rule.

Five years later, a renewed system of imperial ritual music was in place, with its “yellow bell” ringing, sounding the most auspicious gong of an august emperor. This historic debate on music theory and its symbolic power, its full details beyond the scope of this post, eventually developed into ruthless political warfare, with hundreds of scholar-officers found guilty for fueling political factionalism. This included Su Kui, an overall supporter of Zheng, and his father Su Wei, Duke of Pi and the grand chancellor of the Sui dynasty.

Throughout, both the minister’s and the emperor’s concerns over the absolute gong pitch foreground the rich symbolic systems in which classical Chinese music theory participated. Pitch names, in particular, have long been regarded as forces capable of reflecting and altering social order. In convincing the emperor, He tactically fell back on the ancient and semi-legendary association between the “yellow bell”—often regarded as the imperial bell—and the ruler as well as the divine will. For He, only when the gong pitch is fixed on this most auspicious tone, with no monthly or seasonal adjustments, would the new dynasty be brought into ultimate harmony with the cosmos.

A Closer Look  

The large-scale, inter-ethnic migration that marked the turbulent Southern and Northern dynasties (420–589 CE) induced much anxiety in the succeeding Sui court, whose ruling class feared the infiltration of foreign and popular melodies into the sacred realm of imperial ritual music. It was against this backdrop that the Debate unfolded. Standing at the center of the storm was Zheng Yi, the musical mastermind whose theoretical contributions remain the focal point of existing scholarly writings (Figure 5). Via the work of Zheng, I hope to bring to light the subtle dynamics between popular music (suyue), foreign music (huyue), and imperial ritual music (yayue), three fluid constructs central to early Chinese perceptions of music that continue to inform discussions over the state of classical music theory in China today.


Figure 5: Tomb inscriptions (muzhi) of Zheng Yi (540-591), Tang West Market Museum,
excavated in 2014

Music historians today have turned to Zheng’s connection with Sujivha when dissecting the inflow of foreign music into the Central Plain during the Sui, a period often dismissed for the “barbarianization” (huhua) of Han Chinese culture. Over the decades, scholars have probed the possible Karnatik (Hayashi 1936), Hindustani (Xiang 1937), Central Asian (Guan 1980), and Persian (Wang 1931; Shen 1993) roots of Sujivha’s WDQD theory. At issue is Zheng’s bold proposal that mode-keys be initiated on all seven scale degrees—including the two altered (bian) tones—of the standard heptatonic note-set. Traditionally, the bian tones, highlighted in Figure 4a, were treated as auxiliary notes in the Chinese modal system, which was commonly regarded to comprise five rather than seven diatonic modes. Aside from importing foreign practices, Zheng is also remembered—sometimes criticized—for disseminating knowledge that had been previously kept secret. This includes his decision to illustrate the classical art of modulation on the Kuchean pipa, a secular instrument, leading to the propagation of Sujivha’s theory outside the palace during the ensuing Tang dynasty.

Musicologists who work against a simplistic portrait of Zheng as an enthusiast for “the foreign” (hu) and “the popular” (su) have called for a re-evaluation of his contribution. Shen Tung (1993), for one, argues that Zheng is as much a classicist as he is a syncreticist, and that underneath Zheng’s efforts to “apply” (yong) imported knowledge is his unwavering faith in classical Chinese learning, the “essence” (ti) of his campaign.[2] Scholars sharing this viewpoint re-associate Zheng’s proposal for “rotating the gong” with similar teachings in the Book of Rites, a Confucian classic deemed unequivocally Chinese. Attention also lands on Zheng’s adherence to the exclusive usage of the yayue scale in imperial ceremonial contexts. The yayue scale, so named for its historical association with the yayue tradition, is a modal scale that features a prominent augmented fourth (A4) between the gong and bianzhi scale degrees (Figure 6a). It is recorded that Zheng once condemned the Imperial Music Office for interfusing the yayue and xinyue (new music) scales, deriding the bianzhi scale degree of the latter—which forms a perfect fourth (P4) with the gong—as “betraying the Way of pitch generation” (Figure 6b). Shen, in her textual analysis of the Book of Sui, traces Zheng’s criticism of the xinyue scale to the classical method of ATRE, and proceeds to argue that behind Zheng’s acceptance of WDQD is his realization that the imported theory from Sujivha “fits like matching tallies” the scales generated under ATRE. A connection is thus established between Zheng’s proposals and sources of classical authority. In this narrative, Zheng is no longer the radical who “impurified” yayue and sidetracked the “authentic” history of Chinese music theory. Instead, he has become a guardian of traditional knowledge who—in bridging it with the foreign—develops and empowers it.


Figure 6: Yayue Scale and Xinyue Scale

The Debate, which took place during one of the most short-lived regimes in Chinese history, extends far beyond the musical realm. In his analysis of early Sui politics, Wang Li-zeng seeks to uncover the hidden agenda behind the Emperor’s response to each core participant of the Debate, arguing that in rejecting or accepting a given proposal, the Son of Heaven was in essence gesturing to the nuanced political network behind the screen.[3] The Debate thus brings to the surface a classical association between music, cosmology, and government theory that was in effect then and is gradually re-entering the Chinese consciousness today. Finally, with its participants ranging from historians, music theorists and musicians to Confucian scholar-officials and political strategists, the Debate meaningfully contrasts intellectual exchanges in China today, often limited by disciplinary boundaries and scholar-practitioner divides. The wind keeps blowing, as I write, leaving many of us wonder whether yet another storm is brewing.

[1] A standard heptatonic “note-set” within the context of Chinese yayue refers to a collection of seven pitches within an octave—in ascending, stepwise arrangement—that in its basic formation resembles a Lydian modal scale and that functions as the basis of scalar and modal construction. Laurence Picken and Ernest Rowland. Music from the Tang Court: Some ancient connections explored. Cambridge: Cambridge University Press, 2000.  By “mode-key,” I adopt Picken and Rowland’s translation of diao, an ambiguous Chinese character that falls in between Western notions of “key” and “mode.”

[2] Shen Tung, “‘On the Music Conference of Kai-huang Period.’ 隋代開皇樂議研究,” New History 4, no.1 (1993): 1-42.

[3] Wang Li-zeng, “‘The Kai-huang Music Debate and Early Sui Politics.’ 開皇樂議與隋初政治,” Journal of Tianjin Conservatory of Music, no.4 (2003): 33-36.

Huang 2


Rujing Huang is a Ph.D. Candidate in Ethnomusicology at Harvard University’s Department of Music. This year, she also serves as a Graduate Student Associate at the Fairbank Center for Chinese Studies.

Researching the Transfer of Central-European Music Theory and Composition Treatises to China

Gesine Schröder

My interest in the circulation of European music theory in China was initially piqued through the doctoral work of Wang Ying, for whom I served as an external advisor. Now a lecturer of music theory in Guangzhou, Dr. Wang was a doctoral candidate at the Central Conservatory of Beijing at the time and spent two years of her studies in Leipzig. Like her, many students from China have been coming to Central Europe seeking to learn some “original European theory” at a university or conservatory. But at the same time these students have been bringing with them to Europe an idiosyncratic understanding of Western music theory that seems to synthesize recent, partially North American theories with outgrowths of theories from late nineteenth-century Leipzig and with relics of the British theoretical tradition. Trying to understand the roots of this amalgamation was the beginning of a research project that I initiated, along with collaborators Cheong Wai-Ling from the Chinese University of Hong Kong and Zhang Wei of the Shanghai Conservatory of Music.

Together with these colleagues we organized workshops that involved primarily young scholars and students. The aim was to trace the paths through which Central European theories arrived in China, focusing on theories and composition treatises from the late nineteenth century to the present day and particularly those dating from the decade following the founding of the People’s Republic of China in 1949. How did these theories change along the way and how were they adapted? When and in which form did they develop an independent existence in the new location? How could one describe the current status of Chinese music theories used in pedagogy?

Through financial support from the Austrian Eurasia Pacific Uninet foundation, our team could hold four small conferences dedicated to these questions. These conferences took place in Hong Kong (January 2014), Shanghai (April 2015), Vienna (January 2016), and again in Hong Kong (April 2017). Contributions from the first three meetings were published in Spring 2017 in a special issue of the online journal Zeitschrift ästhetische Bildung ZÄB. The following discussion makes reference to some of these articles. But first, I start with some notes about the last meeting (2017) that is not yet documented in the publication.

lou - blick von oben 1

Figure 1: In the apartment of Luo Zhongrong 罗忠鎔 (seated), with his wife Li Yamo 李雅莫, singer; Gesine Schröder (Wien/Leipzig); Kang Xiao 康啸 (China Conservatory Beijing); Odila Schröder (Heidelberg/Nottingham); Jonathan Stark (Vienna); Tobias Tschiedl (Vienna/now Montréal); Hong Ding (Soochow/Hong Kong); N.N. (China Conservatory Beijing)

This meeting concluded with a week-long research stay in Beijing during which we documented materials in the libraries of the Central Conservatory and the China Conservatory. During this stay, our team of six researchers from China, Austria, and Germany also visited Luo Zhongrong (see Figure 1). A pioneer of new music and music theory in his country, Luo is an important eyewitness to their development. Almost 94 years old, he lives with his family in the outskirts of Beijing. Luo’s work as composer and as translator of music theory treatises illustrates one way in which European music theories arrived in China, how they were transformed and sinicized.

In the mid-1940s, Luo Zhongrong studied in Shanghai with Tan Xiaolin who himself had been a student of Paul Hindemith’s at Yale. Through Tan, Luo became familiar with Hindemith’s harmonic concepts. When he was incarcerated in the cultural revolution, Luo translated Hindemith’s The Craft of Musical Composition (originally Unterweisung im Tonsatz [Mainz 1937; English ed. London 1942]) into Chinese (using the English version as his model), and later the first volume of Hindemith’s A Concentrated Course in Traditional Harmony (London 1944). Both translations were printed only in 1984 and 1980 respectively. Luo also translated Schoenberg’s Harmonielehre (again from the English edition), as well as monographs by George Perle and Allen Forte. Luo is regarded as the first composer in China to have used Schoenberg’s “method of composing with twelve tones which are related only to one another,” such as in his song with piano accompaniment Picking Flowers in the Lotus Garden from 1979. But how did he become acquainted with Schoenberg’s method if he had not studied outside his home country?

In the People’s Republic of the 1950s to 70s snippets of information about the Second Viennese school circulated furtively, having been imported by Jewish émigrés during the Nazi period. Wolfgang Fraenkel, exiled from Berlin, taught at the Shanghai Conservatory and from 1947 onwards his successor Julius Schloß from Vienna (see here). Besides the regular music-theoretical subject matter, both of them also introduced their students to the atonal compositional methods and the style of the Second Viennese School. Such teachings are evident in the atonal piano piece Night Scenery (1947) by Sang Tong, which was composed already before the founding of the People’s Republic. As colleagues at the Shanghai Conservatory, Sang and Luo exchanged ideas about their new compositional experiments long before Sang’s piece was published in 1981.

Another way to learn about atonal composition—and one that, according to Luo’s account, was paradoxically more efficient—was through music theory. He referred specifically to the Chinese translation of Czech author Ctirad Kohoutek’s book that contained two chapters on the twelve-tone method. He remembers that the discussion of the composition method was presented with the same harsh critique that had to be applied to any so-called formalistic method during the regime of socialist realism. However, Luo was able to appreciate the descriptions of the compositional methods between these layers of criticism. In an idiosyncratic approach, Luo sinicized the method by composing the twelve-tone row from pentatonic segments, as multiple Chinese music theorists have analyzed (see here and here). In later dodecaphonic compositions, Luo also applied the ordering principles of the row to rhythmic formulas that he derived from traditional Chinese genres of music.

In the first half of the twentieth century, modern Chinese music theory was influenced the most through the knowledge brought back from students who had spent time in Central and Western European countries and the United States. Up until the 1950s, some impact can be traced as well to newer Japanese theories, which often had their roots in theories from the Paris Conservatory, in North American treatises or in the work of Hugo Riemann. Among Chinese scholars who had studied abroad, individual personalities such as Xiao Shuxian were important in particular because of their teaching service besides their treatises. Xiao, for example, taught counterpoint at the Central Conservatory in Beijing for decades. She had studied in Brussels, following an adapted French curriculum. Having lived in Switzerland for over a decade as the wife of conductor Hermann Scherchen, she was well acquainted with new music composed in Europe.

Her uncle Xiao Youmei was even more influential for the early history of Chinese music theory. He had studied in Leipzig in the years preceding World War I. While Xiao received his doctorate under Riemann from the University of Leipzig, his introductory music and harmony treatise shows more similarity with ideas that he picked up in his parallel studies at the Conservatory of the same city. The theories of Salomon Jadassohn were still widespread at the Conservatory even years after his death—despite Riemann’s vehement criticism of his former teacher. Similarly, Jadassohn’s thought persisted at the Conservatory of Shanghai over decades through Xiao’s students. Sang Tong’s harmony treatise, for instance, uses the analytical ciphers that disseminated far beyond Leipzig in the late nineteenth century, as can be seen in an excerpt from that treatise in Figure 2.


Figure 2: Excerpt from Sang Tong’s harmony treatise (Shanghai 2001, 121)

Riemann’s ideas, meanwhile, still played an influential role in conceptualizing a Chinese music pedagogy and musicology—a “plan of musical knowledge”—in the years preceding the foundation of the People’s Republic. Also Chinese harmonic theory bears traces of Riemann’s ideas. This, however, only occurred later—after the death of Xiao Youmei and the foundation of the People’s Republic—and via a different route: the Soviet Union. The so-called Brigade-treatise, written by four theory professors of the Moscow Conservatory and first published in Russian in 1937 and in a Chinese translation in 1957 and 1958, had a far-reaching impact (as is discussed here and here).


Figure 3: From the Chinese translation of the harmony treatise of the so-called theory-brigadiers, p. 201. The annotations come from Sun Yongdan, the owner of this copy.

However, what was transmitted to China as “function theory” through the Soviet pedagogy treatises bears little resemblance to Riemann’s ideas. (See Figure 3: In this passage from Mozart’s The Marriage of Figaro, act II, scene 1, the parallelism is shown to lie in the analogy of the functional symbols, showing the progression S->T [in the first instance this is hidden in the second letter of the functional symbols respectively]).

Moreover, China absorbed music-theoretical ideas and concepts from scholars who visited the country. Among the first was Boris Aleksandrovič Arapov (1905–92) of the Leningrad Conservatory. He taught at the Central Conservatory between 1955–57. Further guest lecturers were invited to China from other countries of the Warsaw Pact as well, such as Paul Schenk from Leipzig, who taught at the conservatories of Beijing, Shanghai und Guangzhou in 1959 for seven weeks (see here, esp. 11–13). A translation of his lectures is preserved in the library of the Central Conservatory. Schenk advocated for a pragmatic music theory, which built on the work of his teacher Sigfrid Karg-Elert and initially opposed Riemann’s dualistic thinking through a more radical polaristic concept. Its pedagogically simplified version was the music-theoretical approach of reference to anyone wanting to become a musician in the GDR and remained so up until 1989 and beyond.


Figure 4

See for example Figure 4, which explains Richard Strauss’ “Eulenspiegel-chord” (the so-called “Australian Sixth”) through Karg-Elert’s system as a mixture of two polar second-degree third-related chords: SM and Dm.

Other guest lecturers came from the UK, such as composer Alexander Goehr (born in Berlin in 1932). He lectured on new compositional techniques in Beijing in 1980—the year of publication of Luo’s dodecaphonic song—and again a little later. But also permanent teachers at the Central Conservatory such as Yao Heng-lu elevated the relevance of England for Chinese music theory. The Central Conservatory had been closed during the Cultural Revolution and Yao was among the first students to matriculate there in 1976 upon its reopening after the reign of the Gang of Four. In the 1980s, Yao completed his PhD in England. Afterwards he taught many generations of Chinese students at the Central Conservatory in music analysis.


Figure 5: from Yao Heng-lu’s Essential Training for Composition, Beijing 2011, p. 125.

He is the author of many pedagogical treatises. His harmony treatise demonstrates how chord symbols that Yao imported from England merged with Soviet symbols and more traditional Roman numerals. An example is the symbol for the augmented sixth-chord in Figure 5, an excerpt from Wagner’s The Valkyrie, act II, scene 4.

Examples like those quoted above made it clear to our research team that it would be too narrow to consider only relics of German-language music theory in China. Instead, it would require a team of scholars who are trained in the theory traditions of other countries as well to adequately trace and differentiate the influence of Roman numerals of presumably German origin, Soviet chord symbols, which adopt a Riemannian system, and British signs which root in ideas from the early nineteenth century. The British tradition of music theory has played a strong role also for political reasons: Since Hong Kong was integrated into the People’s Republic, it has constituted a pole of attraction for students from mainland China, who come to study within a music theory system that still bears clear traces of its past as a former British colony.

Today there are numerous contacts between China and music theorists from around the world. Among the many guest lecturers—including Steven Laitz from the Juilliard School respectively the Eastman School of Music, Reinhard Bahr from Hamburg, or Ariane Jeßulat from Berlin who was invited to participate in the Forum Music Analysis (2016)—Allen Forte was particularly influential. He visited Shanghai in 2009 during the conference of the Chinese Society for Music Analysis. Still more influential are, as mentioned, Chinese teachers who had studied abroad and who return to their home country with the acquired knowledge, which is usually adapted in manifold ways. These adaptations of imported theories pertain most commonly to the quality of the respective theories (however that might be assessed), rather than to the historical or local contexts from which they arose and that they originally referred to. This is particularly evident in the discipline of counterpoint.


Figure 6: A page from the first index to Chinese Art Education Encyclopaedia. Music Volume (Shanghai 2010, 3 vols.).

The theories are dissolved from their original contexts and it seems as if they thus become particularly adept for contemporary composition in China—at a time, notably, when these theories increasingly lose relevance for compositional pedagogy across Western and Central Europe. It is common in China that pedagogical treatises partake in the decontextualization and sinicization of materials and theories by demonstrating compositional principles through examples of recent Chinese music. Figure 6, a page from the first index to Chinese Art Education Encyclopaedia. Music Volume, for example, shows such a mixture of compositions from diverse historical and geographical origins, including many works by Chinese composers.
shanghai 2017 mai


Gesine Schröder is Professor of Music Theory at the University of Music and Performing Arts in Vienna and at the Hochschule für Musik und Theater in Leipzig.



Blog post translated by Stephanie Probst




Conference Report


On November 8th and 9th, the SMT  History of Theory Interest Group / AMS Study Group co-sponsored a conference on “Instruments of Music Theory,” at the Eastman School of Music in Rochester, NY co-organized by Prof. Andrew Hicks (Cornell University), Prof. Nathan Martin (University of Michigan), Dr. Caleb Mutch (Indiana University), Dr. Carmel Raz (Columbia University), and Prof. Anna Zayaruznaya (Yale University).  The event, which attracted nearly eighty registrants, featured three keynote speakers, eleven papers by scholars in all career stages (from graduate students to full professors), and a hammered clavisimbalum concert with music from the Faenza Codex and other recently discovered manuscript fragments.

The conference sought both to build upon and to reinforce the increasingly eclectic and interdisciplinary set of questions now being asked within music studies and the humanities more broadly, to which the history of theory, as an inherently interdisciplinary field of study, has already begun to make significant contributions. Its theme, “Instruments of Music Theory,” explored Prof. Alexander Rehding’s recent call to “reconsider the relationship between music-theoretical instruments and the music theory they occasion” (MTO 22.6 [2016]), while also foregrounding the broader global context in which theories and instruments of music are situated.

The first day began with a session examining the use of “instruments in theory” as well as the uses of “theories as instruments,” chaired by Prof. Stefano Mengozzi (University of Michigan). The first speaker, Etha Williams (Harvard University), discussed the gendered meanings of sensibility in Denis Diderot and Anton Bemetzrieder’s Leçons de clavecin (1771), an instructional treatise detailing the keyboard lessons given to Diderot’s daughter, Marie-Angélique. The next paper, given by Lester Hu (University of Chicago), attempted to reconstruct the engagement between the Jesuit writer on Chinese music, Jean-Joseph Marie Amiot, and the contemporaneous Qing treatise that proposed a fourteen-tone temperament. The third speaker, Prof. Karl Braunschweig (Wayne State University), examined the changing role of music-theoretical reductions between the eighteenth and early twentieth centuries, focusing on the ways they provided material forms for elusive language conditions in music. Dr. Scott Gleason (Oxford University Press), the final speaker on the panel, used the writings of contemporary music theorist David Lewin to investigate how historical music theories can be leveraged as instruments for music-theoretical exploration.

After lunch, the conference reconvened for the first keynote address by Dr. David Catalunya, an accomplished medieval musicologist and early music performer currently based in Würzburg, Germany. Dr. Catalunya’s presentation featured organological demonstrations about the sensorial perception of music-theoretical precepts, focusing on the organ but also including Pythagorean bells, which he had reconstructed in collaboration with bell makers, as well as his own newly reconstructed hammered clavisimbalum, a medieval precursor to the fortepiano and built following the description of Henri Arnaut de Zwolle in a mid-fifteenth-century manuscript. Although de Zwolle does not describe the hammer action in detail, Dr. Catalunya and his collaborators were able to reconstruct it using the plans for the hammer of a clock also detailed in the manuscript.


Dr. David Catalunya’s Keynote Address

The afternoon session featured three papers that engaged with “theories of instruments.” Chaired by Prof. Alice Clark (Loyola University New Orleans), the panel began with Dr. Leon Chisholm (Deutsches Museum), who discussed the embodied ecology of music-making that underpins the shift from a voice-centered to keyboard-centered paradigm of musical thought in the sixteenth century. The next speaker, Prof. Rebecca Cypess (Rutgers University), considered the dependence of seventeenth-century basso continuo practice on the embodied knowledge of skilled instrumentalists, as well as on compositional ingenuity, uncovering traces of an experimental process among harpsichordists in Luzzaschi’s Madrigali (1601) and Frescobaldi’s Toccate (1615). The last paper on the panel, jointly delivered by Prof. Bryan Parkhurst (Oberlin College) and Prof. Stephan Hammel (University of California-Irvine), considered how the study of musical instruments within the broader anthropological and sociological study of human practices and institutions might open the door for a Marxist telling of the history of music.

The conference then reconvened for the second keynote address by Prof. Rehding (Harvard University) entitled “Global Thoughts on Music-Theoretical Instruments.” Occasioned by the coincidental calculation of equal temperament in both Western Europe and Ming Dynasty China in the late sixteenth century, Prof. Rehding asked us to consider the divergences that underlie such apparent similarities. He then ventured an alternative methodology for a transcultural history of music theory, drawing from Shigehisa Kuriyama’s analysis of cultural comparisons from the realm of medical history.



Prof. Alexander Rehding’s Keynote Address

Following the dinner reception, co-sponsored by the Heyman Center for the Humanities at Columbia University and the Journal of Music Theory, the evening concluded with a spellbinding concert performed by Dr. Catalunya on his hammered clavisimbalum featuring music from the fifteenth-century Faenza Codex and other recently discovered manuscript fragments.

The second day of the conference began with a panel on “instruments of theory” chaired by Prof. Thomas Christensen (University of Chicago). The first speaker, Elizabeth Lyon (Cornell University), examined Jean Gerson’s Tractatus de Canticis (fifteenth century) as an illustration of the ways in which allegorical uses of music theory contribute to knowledge of extra-musical domains and may also feed back into the aesthetics of sounding music. The second speaker, Prof. Joon Park (University of Arkansas), considered ways in which the terms “murky” and “transparent,” commonly used in Chinese and Korean calligraphy and music, both do and do not map onto ideas about high and low pitches in Western music theory. The third speaker, Prof. Abby Shupe (Colorado State University), surveyed the way in which the eighteenth-century French music theorist Jean-Philippe Rameau used both real and imaginary experiments with household implements in order to ground his claims in the empirical science of his day. The last speaker on the panel, Prof. David Cohen (Columbia University), analyzed discrepancies in accounts of the Pythagoras myth and argued that the role played by quantified string tensions in the Nicomachus of Gerasa’s version demonstrates a Neopythagorean desire to appropriate the Aristoxenian conceptualization of pitch as “tension” (tasis).

Thursday’s activities ended with a final keynote by Prof. Gabriela Currie (University of Minnesota) on “Instrumental Globalities: Object, Thought, Practice in Pre-modern Eurasia.”


Prof. Gabriela Currie’s Keynote Address

Prof. Currie’s lecture outlined the linguistic, archeological, and iconographic   evidence for the early dissemination of musical instruments in the intellectual and artistic exchanges along Eurasian trade routes. It was along these routes that the early outlines of global modernity first emerged, as people from diverse cultures facilitated a Eurasian transcultural commerce. The music-iconographical legacies of these complex Eurasian networks allow us to map a new model of instrumental and music-theoretical “globalities.”

The last session of the conference took place on Friday evening during the AMS Study Group’s dedicated session, chaired by Prof. Andrew Hicks (Cornell University). The first speaker on the panel, Lars Christensen (University of Minnesota), discussed cosmological diagrams produced during the Northern Song dynasty (960-1127), showing how similar kinds of discrepancies had to be silently normalized in calculating a calendar of twelve months and deriving a gamut of twelve pitches. The next speaker on the panel, Steffi Probst (Harvard University), was unfortunately unable to attend in person but contributed nonetheless a lively paper on musical visualization strategies in the “AudioGraphic” player piano rolls created by Percy A. Scholes between 1925 and 1930. Prof. Jennifer Iverson (University of Chicago) followed with a paper describing Oskar Sala’s Mixtur-Trautonium, an early synthesizer that reflected contemporary vowel experiments of Carl Stumpf, as a boundary object that drew various several unexpected cultural and scientific strands together. The fourth speaker on the panel, Siavash Sabetrohani (University of Chicago), discussed the role of the oud in the reception of Greek music theory during the Islamic Caliphate in ninth-century Baghdad. The panel concluded with a paper by Prof. Alexander Bonus (Bard College) on the influence of the metronome on various aspects of modern musical and scientific engagements with the notion of time.


Historians of Music Theory Admiring the Clavisimbalum

Additional photos and videos can be found on the conference website

With thanks to:

The Central New York Humanities Corridor (from an award by the Andrew W. Mellon Foundation), The Westfield Center for Historical Keyboard Studies, The Heyman Center for the Humanities at Columbia University, The Society of Fellows in the Humanities at Columbia University, The Journal of Music Theory, The Eastman School of Music, Cornell University, and the Society for Music Theory

Teaching Solfège in Socialist East Germany

Anicia Timberlake 

What can music pedagogies for children tell us about how grown-ups think about music? My recent work examines the politics of classical music in socialist East Germany (GDR) through the lens of children’s music education. Officially, the doctrine of socialist realism demanded that citizens of the new socialist society should perform and listen to new music that emphasized socialist ideals. Stylistically, this music was meant to both draw from the German classical tradition and also move beyond it. But there’s a difference between theory and practice: East German policymakers dictated what was to be sung, not how it was to be learned. They rarely stepped into the nation’s classrooms, where teachers often operated by a different set of assumptions. I look at children’s pedagogies to see how music was taught to children in practice, and, by extension, to learn about how the people in charge of bringing music to the state’s youngest citizens conceived of it.

As the Soviet occupied zone (and later the GDR) sought to restructure its educational system, many teachers—not policymakers—looked to the progressive pedagogies (Reformpädagogik) of the late nineteenth century and the Weimar era for inspiration, believing that the utopian aims of the earlier time could be taken up again and, perhaps, fulfilled. Many of these pre-war German progressives used aspects of English tonic sol-fa, a singing pedagogy developed in the mid nineteenth century that used syllables, hand gestures (cheironomy), and a simplified notation system to teach adult amateurs to read music. In adapting tonic sol-fa for children, they jettisoned the simplified notation to focus solely on the syllables and the cheironomy.


Fig. 1: Wilfried Friedrich, Heinrich Martens, Richard Münnich, and Karl Rehberg, Tonika-Do, Eitz, Jale (Berlin: Volk und Wissen, 1949), 8.

Tonic sol-fa (likely already familiar to many Western readers of this blog) is a movable-do system that sets a major scale to adapted Guidonian syllables: do re mi fa so la ti do. East Germans used the following cheironomy (Fig. 1).

Many teachers used the system as outlined here.[1] Others developed their own solmization (or solfège) systems meant to teach everything from music literacy to absolute pitch.



Heinrich Werlé’s Solfège Method

In 1949, as part of this trend, the teacher and choral conductor Heinrich Werlé published a guide to his own solfège method, developed on the basis of 40 years of working with children.[2] The method was meant to let what he considered to be the child’s natural musical tendencies unfold by themselves, according to their own pace. Werlé claimed that all newborn infants cried at, or close to, the pitch a’, which formed the tonal center of their childish lives. Children up through the age of 10 continued to produce the pitch spontaneously and intuitively: “not through thinking,” he emphasized, but rather “out of their bodily and spiritual instinct [Antrieb]” (3). From the infant’s scream, Werlé also deduced the inborn nature of diatonic harmony. Careful observation of the infant revealed that she hit a second note when breathing in between cries. This note was always higher by an octave, a fifth, or two octaves, proving the primacy of the intervallic relationships that structured tonal music. The relationship between tonic and dominant was fundamental to the infant’s body, as it was to the world of physics. Thus, Werlé concluded, “the foundation for harmony is already there […] the choice of so-called primary overtones indicates […] that a process is unfolding subconsciously which points back nearly to the hour of music’s birth within acoustics” (5). From these earliest tones, toddlers would go on to improvise songs that used the notes of the tonic triad; children of five or six would “naturally” add the sixth scale degree.

Werlé’s solmization system was meant to cultivate from within those musical tendencies that he believed to be present within the child since birth. Like tonic sol-fa, his used syllables and cheironomy to activate children’s muscle memories: Werlé, like other solfège practitioners, believed that children’s physical abilities developed faster than their rational brains. And like tonic sol-fa—at least as it was practiced in East Germany—his system focused exclusively on diatonic repertoire.

But there, the similarities between the methods ended. Teachers of tonic sol-fa generally started the youngest learners with the pitches so and mi, a falling third common to children’s songs. Children first learned the hand signals for just those notes; they then added (in order) do, la, re, ti, and fa. Thus the method encouraged children to learn songs based on a major triad and a pentatonic scale before moving to repertoire that used a full diatonic scale.

In contrast, Werlé started with the single primal pitch, a’, to which he assigned the syllable fe and a helpful mnemonic gesture (Fig. 2).


Fig. 2: Heinrich Werlé, Musik im Leben des Kindes, 54.

This gesture and syllable, which worked together as a “unity,” formed the basis for children’s tonal education. Werlé recommended that the learning child train himself by suddenly singing fe with his arm outstretched and checking the pitch with a tuning fork. The child would slowly rediscover, or simply refine, the natural tonal center with which he had been born.

What about that fe gesture?

When I’ve given talks on this method, the fe gesture—right arm outstretched parallel to the ground—always prompts uncomfortable giggles. The resemblance to the Hitlergruß is striking for North American audiences. Curiously, none of Werlé’s contemporaries commented on it. Perhaps, to their eyes, the gestures weren’t actually that similar, as the Hitlergruß is angled upward. (Indeed, Werlé’s system studiously skips over the danger zone: when the arm is held at ca. 30 degrees above level, the palm is rotated so that it’s held perpendicular to the ground.) Or the similarity may have simply been irrelevant: East German music educators were willing to overlook the fact that many of the Weimar-era pedagogies they were intent on using had been used throughout the Nazi era as well. From the teachers’ perspective, the gestures and syllables might have seemed apolitical, being devised to teach skills and not any propositional content; perhaps the gesture’s context was too innocent to raise eyebrows.


Fig. 3: Heinrich Werlé, Musik im Leben des Kindes, 54.

The next steps were a guided exploration of the tonal space around a’, already familiar to children from their spontaneous musical improvisations. First, children were to proceed to f#’ (a minor third below a’), given the syllable Wa and a gesture similar to fe but at an angle (Fig. 3).


Fig. 4: Heinrich Werlé, Musik im Leben des Kindes, 55.

This step was nearly automatic, Werlé wrote, as f#’ was a “primary tone of sympathetic resonance […] already present in the subconscious as we sang fe, as it is one of the notes whose sum produces fe.” (The falling minor third, of course, was central to tonic sol-fa education as well, but not for psychoacoustic reasons: teachers simply noted that most children’s songs and speech tended to feature the interval.) Next the teacher was to drop his hand to his side, producing d’, or Mu (Fig. 4).

The rest of the pitches of the D major scale would follow slowly over time. Werlé assigned them the syllables Mu Ro Wa la fe bü zu mu,[3] and the following gestures, in which the arm traces a 180 degree arc from bottom to top, and for each step upwards, the palm alternates between being held parallel and perpendicular to the ground:


Fig. 5: Heinrich Werlé, Musik im Leben des Kindes, 63.

The whole-arm gestures were meant to indicate to children the relative height of pitches in a way that a purely hand-based cheironomy of other solfège systems could not. The choice of D major as the center was determined by the child’s natural inclinations; children were never to sing in another key.


Why does this matter?

All children’s pedagogies negotiate between the material to be learned and the abilities of the growing child. Thus the design of pedagogies can tell us what aspects of music are believed essential (the material to be learned), who the child is when she begins learning (what mental and physical skills are believed abilities that are natural and easy), and the processes by which the gap between the two can be closed.

Tonic sol-fa seems to assume that children are born musically unformed, and can be trained to be proficient in a complex musical system through methods tailored to their developmental stage. Werlé, by contrast, believed that children were born with all the elements of tonal harmony already in place—complete with mostly reliable absolute pitch—and that the teacher’s job was to help them strengthen their natural abilities. The hand gestures and syllables were mnemonic devices; they did not represent skills to be learned from scratch. Werlé also cautioned the teacher against “learned, theoretical concepts of music” (“[das] lernhaft musikalisch Begriffliche”) (34), which would overtax the children’s memories, destroying their organic relationship to music. Thus the teacher’s job was less to teach children than to preserve for them a space in which they could remember that which they already knew.

Werlé moreover seemed to believe that the nature of music itself ought to be identical to the capabilities with which the child was born: no more, no less. For every childish ability he described, he pointed to a corresponding acoustic phenomenon. The intervals between children’s cries and breaths were reproduced in the overtone series, and the ease with which children sang a falling minor third (a’-f#’) was due to that same overtone relationship. Thus he implied that tonal music was the only natural form of music, not only because of its acoustic resonances (and here he ignored all the questions of tuning and temperament that such statements often prompt) but also because of its organic presence within the childish body.

This firm conviction that children can produce tonal music organically—not only because they hear it from a young age, but because they are essentially born singing it—is a funny fit for a socialist nation. Marxist theories of history emphasize that societies, and the people who live in them, change qualitatively over time; most East German theories of education therefore shied away from anything that ascribed too much importance to an “inborn” and unchanging human nature. The idea that there could be a music that was biologically best for children was therefore distinctly un-Marxist. The method, however, still enjoyed moderate success: it was taught at several pedagogical institutes, and reportedly had a good reputation in East Germany (and in West Germany as well).[4]

To say that Werlé’s method was un-Marxist is not to say that it was explicitly anti-socialist, or even merely apolitical. The idea that tonal music is baked into human nature has obvious political ramifications. To close, I’ll draw out just one. The method was clearly designed to be German: instead of relying on internationally well-known tonic sol-fa syllables, Werlé chose a new set—Mu Ro Wa la fe bü zu mu. (In this, he might have been following the lead of the teacher Richard Münnich, whose 1930 solmization system Jale used phonemes particularly useful for singing in German.) Werlé had explained that the vowels he chose were physiologically determined: the progression u – o – a – e – i moved the child’s larynx upwards, easing the singing of the ascending scale (56). While it is possible that physiology also prompted Werlé to substitute “i” for “ü” – the mouth could be held in the same position for and zu, and “ü” might be easier to sing than “i” – the vowel “ü” at the same time reinforced a distinctly German sound world. Similarly, the syllables wa and zu relied on a knowledge of German orthography. These markers made the method essentially unexportable. But even more, there are hints of a familiar chauvinism in a method that uses specifically, even exclusively, German-language sounds as an expression of an innate, universal human musicality. Policymakers may have called for new music for a new society, but teachers delivered pedagogy that traded on older notions: tonality is universal, and universality is German.


[1] East German pedagogues traced their solfège lineage directly back to the efforts of Sarah Glover (1845) and John Curwen (1885) in Victorian England. They made no mention of French solfège or of the Kodaly method. For more information on English tonic sol-fa, see Charles McGuire, Music and Victorian Philanthropy: The Tonic Sol-fa Movement (New York: Cambridge University Press, 2009). Other German solmization systems common in East Germany included Carl Eitz’s Tonwort (1911), a fixed-do system designed to teach absolute pitch and account for chromatic tones and modulations, and Richard Münnich’s Jale (1930), a movable-do system that operated much like tonic sol-fa but with different cheironomy and syllables designed to fit the German language.

[2] Heinrich Werlé, Musik im Leben des Kindes (Dresden: Ehlermann, 1949).

[3] Werlé indicated that only the first three syllables were to be capitalized, but gave no explanation.

[4] Letter from H. Becker to E. Zaisser, 19 May 1950, DPZI 14.

Anicia Timberlake researches the politics of music education in postwar East and West Germany. She is on the musicology faculty at the Peabody Institute of the Johns Hopkins University.

Chromatic Scale Construction in Ancient China

Guangming Li

Music theory is an essential part of Chinese music and culture. Its centrality to understandings of cosmology and social order may seem familiar to music theorists trained in the European tradition. Yet because the distinct characteristics of Chinese music theory have only rarely been incorporated into discussions within the history of Western music theory, exploring corresponding approaches to questions such as tuning and scale generation remains a meaningful point of departure. This was demonstrated in the Global Perspectives in Histories of Music Theory conference at Columbia University in February, which showcased the impressive scale and potential of engaging across cultural contexts.

In addition to learning from a range of excellent scholars, I had the opportunity to present my own research on the construction of the chromatic gamut as first recorded in an exchange between the King Jing (d. 520 BCE) of the Eastern Zhou period (770 BCE – 256 BCE) and the court music official Ling Zhoujiu. In this exchange, Zhoujiu mentions the creator of the chromatic scale, describes the purpose for creating pitch reference, the name of the tuning apparatus, and the twelve disyllabic names, among other things. Yet, Zhoujiu offers no direct explanation for how the pitch reference was created and how the twelve disyllabic names were selected.

At present, Chinese music theorists rely on a method called the “a-third-removing-extending” (or ATRE) technique to explain the process of generating a chromatic gamut. Akin to the cycle of fifths, the ATRE method was first recorded in Guanzi, an encyclopedic treatise that dates from the seventh century BCE.  While the ATRE can generate a complete chromatic scale, it cannot account for the scale introduced in Zhoujiu’s exchange with the King.  For instance, it fails to address the two major-third-chains in the pitch name pattern of the chromatic gamut in the process of generating fourths and fifths.

For the purpose of this post, I will explain the basics of the ATRE method and briefly discuss the motivation for my own research on a new understanding of the ancient method for generating the twelve standard pitches.

“A-Third-Removing-Extending” Method

The details of this classical method for generating the pentatonic scale are found in Guanzi, which is named after the philosopher Guan Zhong (719-645 BCE).  Famed for his achievements as the Prime Minister of the Qi State, Guang Zhong came to be called the “Pioneer of the Legalists,” “The Teacher of the Saints,” “First Prime Minister of the Land,” and worshipped as a divine figure by Daoists.  The mathematical process of constructing a pentatonic scale, known as “三分损益,” which I translate as the “a-third-removing-extending” (ATRE) method, is found in a chapter entitled “Diyuan” of Guanzi. The five monosyllabic names for each of the scale degrees in the pentatonic scale are gong, zhi, shang, yu, and jiao.

This process can be illustrated as the following: begin by stopping on a horizontal string at any point no less than 1/4 of the original string length from the left, and pluck the string on the right. Take the tone from the vibrating portion of the string as the initial length, gong (arbitrarily assigned to C4). To obtain the next pitch, zhi (G3), extend ⅓ of the string length of gong (C4) leftward; to obtain the next pitch shang (D4), stop at ⅓ of the string length of the zhi (G4) from the left; to obtain the next pitch yu (A3), extend ⅓ of the string length of the shang (D4) leftward; to obtain the last pitch jiao (E4), stop at the point ⅓ of the length of the yu (A3) from the left.

Arranging these sequences of ratios in an ascending order based on the ratio results in an ascending pentatonic scale: zhi(G3)-yu(A3)-gong(C4)-shang(D4)-jiao(E4).

Figure 1. “A-Third-Removing-Extending” Method (designed by G. Li)

Limitations of “A-Third-Removing-Extending” Method

Meanwhile, in his reply to King Jing, Zhoujiu introduces the names of twelve standard disyllabic pitch names in a complete chromatic scale.  They are: huangzhong, dalü, taicu, jiazhong, guxian, zhonglü, ruibin, linzhong, yize, nanlü, quyi, yingzhong. The relationship among these pitch names in the Western chromatic scale are C, C#/Db, D, D#/Eb, E, F, F#/Gb, G, G#/Ab, A, A#/Bb, and B. The process of applying the ATRE method to construct a complete chromatic scale commonly found in Chinese classics (e.g. in Lüshi Chunqiu, 241BCE) is as follows:

huangzhong (C ) – linzhong (G) – taicu (D) – nanlü (A) – guxian (E) – yingzhong (B) – ruibin (F#/Gb) – dalü (C#/Db) – yize (G#/Ab) – jiazhong (D#/Eb) – wuyi (A#/Bb) – zhonglü (F).

Figure 2. Chromatic scale based on the ATRE method and the issue of major third chains (designed by G. Li)

However, this method cannot account for the appearance of major third chains in the pitch-naming pattern of the gamut. Two major third chains are found among pitches at even number positions in the gamut, and assigned a common suffix-like morpheme in their names. The pitches jiazhong (4), lingzhong (8), and yingzhong (12) share a common morpheme zhong and they form a chain of major thirds Eb4-G4-B4; the pitches da (2), zhong (6), and nan (10) share a common morpheme and they form a chain of major third Db4-F4-A4.

Given that the series of pitch names appeared in an official record that captured dialogue between the King and his music director, the particular sequence of major thirds is not likely to be the music director’s spontaneous creation, but rather a reflection of a well-established convention.  Further, the fact that the music director did not refer to the ATRE method suggests the possibility that a different method of directly building on major thirds and the chromatic gamut could have been applied.

By drawing together a linguistic analysis of ancient texts with acoustic principles and string performance techniques, I offer a different method for constructing the chromatic gamut called the “harmonic-and-stopped-tone-mutual-converting” (or HSTMC) method. Based on ancient Chinese zither design and existing performance techniques, the HSTMC method could have easily been obtained on a monochord. This is based on the fact that a single-stringed zither or yixianqin (akin to the monochord) was commonly known in antiquity.  And given that the historical text describes the creator of the chromatic pitch system as a “divinely blind” music master, the HSTMC method requires only understanding the physical behavior of the string through haptic touch, without the aid of mathematical calculation or the ATRE method.  When applied, HSTMC better corresponds with Zhoujiu’s numerological description of pitch system construction and better aligns with the tenets of Chinese natural philosophy.

Details of the HSTMC method for generating the chromatic gamut remain forthcoming in a paper titled, “New Findings on Non-mathematical Methods of Constructing the 12-tone Chromatic Scale with the Monochord in Ancient China.” This study not only enriches understandings of music theory and music culture in ancient China, but also affords a new perspective on understanding Western music theory beyond exercises in abstract mathematical contemplation.

 Special thanks to Leon Chisholm, Carmel Raz, Nori Jacoby, and Lan Li for their assistance and feedback in writing this post. Special thanks also to the Heyman Center for the Humanities at Columbia University and the Global Perspectives in Histories of Music Theory Conference.

Li picture blog

Guangming Li, PhD (UCLA), is a scholar, performing artist, and educator.  His areas of research include issues in music archaeology and history, music theory, music cognition, and music aesthetics.

Some Reflections about Thoroughbass Pedagogy Today

by Peter van Tour

In recent times, quite a few European conservatories have been reintroducing thoroughbass methods in their music theory curricula.[1] The surge of interest in these methods, commonly known as “partimento,” has led to a reevaluation of theory pedagogy more broadly. While this development testifies to a generally healthy trend towards integrating the various subdisciplines in music theory, it also raises questions.

Thoroughbass methods may have a lot to offer in terms of stimulating fluency in playing and of integrating practical playing and singing with theoretical reflection. However, the modern student may ask whether it is really necessary to learn yet another “harmonic” system, not to speak of the practical thresholds of reading C-clefs, which appear unavoidable in the teaching of partimento.

Since I, from time to time, am invited to talk about such issues at conservatories and since I regularly experience both the benefits of the method and the hesitations of students and teachers, I would like to take this opportunity to describe some of my practical experiences. How might some aspects of partimento practice be successfully integrated into modern music theory curricula?

A first and obvious benefit of partimento pedagogy is that its exercises enable the student to work practically with simple textures. Most Neapolitan partimenti are optimally realized in three voices, especially in partimenti with a rather contrapuntal texture. Today it seems that we have limited our attention to four-part harmony to such extent that we have forgotten quite a lot about the appeal of three-part style.

Secondly, the Neapolitans were very much aware of the necessity of practical singing and playing in their education in music theory: they talk about “contrappunto prattico,” and written two-part contrapuntal exercises are termed “solfeggi.” In other words: anything you write should be sung and played.

Thirdly, the Neapolitans were never afraid of clichés. Harmonic and contrapuntal clichés were, in fact, the cornerstones of their teaching in composition.[2] But—and this is essential—these clichés were always varied in numerous ways, in both performance and written exercises.

In my own teaching in aural training at the Royal College of Music in Stockholm, I developed in the past years a common practice at the start of my lessons, of having students play back short three-voice phrases that I then modified in various ways. In my classroom, I had two pianos placed with their backs against each other and played phrases that my students could play back directly afterwards. I offered these exercises to students who studied Western classical music in a Master’s program in choral conducting and / or orchestral conducting. As part of their curriculum they received individual lessons in aural training of ca. 45 minutes a week.

Such a lesson could start something like this:

  • I inform the student that the exercise will be in the key of A minor.
  • I play a bass line of just a few bars and ask the student to play it back as accurately as possible.[3]


[bass line]

  • As soon as the student has managed to imitate the bass line, I add a second voice over this bass and ask the student to imitate the two voices, playing the bass line in the left hand and the upper voice in the right hand.

[bass line + voice 1]

  • I repeat this procedure, now with a different upper voice. Like this:

[bass line + voice 2]

  • As soon as the student has managed to play this second voice over the bass line, I play them once again together and ask the student to play all three voices. Like this:

[bass line + voice 1 + 2, variant 1]

  • After the student has managed to put this together, I play a new and modified version and ask the student to play it back.

[bass line + voice 1 + 2, variant 2]

  • I repeat this procedure once again, now with a few other changes.

[bass line + voice 1 + 2, variant 3]

Now, this is of course just a very short example of how playful attitudes in historic milieus can be reused to make music theory lessons more attractive. Such exercises can, of course, be varied in many different ways, both musically and pedagogically. An exercise like the one above allows the student not only to get acquainted with a typical harmonic formula (or cliché), it also teaches how such formulas can be modified and elaborated on. In other words, it teaches not only the standard clichés in eighteenth-century music, but also the playful attitudes through which such music was improvised or composed.

Especially when I use fragments that are taken from larger exercises, I find it useful not only to show the exercise in its entirety, but also to give the student examples from famous classical works in which these models appear. After having worked practically with this kind of models, the student will inevitably recognize the models in real music.

Finally, I would like to emphasize that in many cases it is not really necessary to put too much emphasis on thoroughbass figures. They can be used as an aid in the student’s aural orientation and I thus use figures for memorizing fragments that are later varied in many ways. What at first may appear to be a burden of yet another system may in fact become a playful way of engaging in musical clichés, enabling students to recognize commonly used patterns in the music that they play and listen to.

Together with other authors who have started to develop new teaching materials in this vein, I believe in the pedagogical value of these trends, of integrating aural skills training into the training of counterpoint and harmony.[4]

[1] See, for example, David Lodewckx and Pieter Bergé, ”Partimento, Waer Bestu Bleven? Partimento in the European Classroom: Pedagogical Considerations and Perspectives.” Music Theory and Analysis 1/1-2 (2014): 146–69.

[2] See: Robert Gjerdingen, Music in the Galant Style (Oxford: Oxford University Press, 2007), Giorgio Sanguinetti, The Art of Partimento: History, Theory and Practice (New York: Oxford University Press, 2012).

[3] This particular example is adapted from Carlo Cotumacci’s first disposition and was taken from: [24] Disposizioni a tre, e quattro parti, ossiano Partimenti Del Sig.r Carlo Cotumacci.” MS: I Mc Noseda E 66-16, olim 7759. Cotumacci was teacher of counterpoint and composition at the Conservatorio di Santa Maria di Onofrio in Naples between 1755 and 1785.

[4] A few examples that may be mentioned here are Lieven Strobbe’s Tonal Tools for Keyboard Players (Garant Publication, 2014) and Job Ijzerman’s forthcoming book on harmony (Oxford University Press) in the field of harmony, and Peter Schubert’s Modal Counterpoint: Renaissance Style (Oxford: Oxford University Press, 2008) and Barnabé Janin’s Chanter sur le livre: Manuel pratique d’improvisation polyphonique de la Renaissance (Lyon: Symétrie, 2014) in the field of counterpoint.


Peter van Tour is visiting professor at the University of Leuven in Belgium. His PhD dissertation Counterpoint and Partimento: Methods in Teaching Composition in Late Eighteenth-Century Naples was recently awarded the 2016 Hilding Rosenberg scholarship in musicology by the Royal Swedish Academy of Music. In his current research, Peter is investigating fugal improvisation in Italy and Germany between 1680 and 1720.