AAAM Vol. 1, Iss. 1

METER AS ALGORITHMIC TRANSFORMATION: COMPARATIVE HISTORIES OF AFRICAN MUSIC1

(In memory of Akin Euba and Andrew Tracey)

Martin Scherzinger

One should not be too hopeful of a ship sailing from Europe.

(West African Aphorism)

In 1959, as resistance forces in Nyasaland were battling military operations unleashed by a Declaration of a State of Emergency by the colonial governor, a Viennese ethnologist, 25-year-old Gerhard Kubik, gathered research notes on the phenomenon of inherent patterns in the instrumental music of, among others, the Tumbuka people. Tumbuka-speaking areas of Nyasaland had received scant investment for developing infrastructure or commercial crop-farming from the colonial government; and, by the early sixties, these areas were afflicted by environmental degradation, poverty, and systematic underdevelopment—disparagingly referred to as the “Dead North.”2 In the subsequent publication of his findings in the African Music Society Journal (founded five years earlier in South Africa, where the passing of the Group Areas Act had recently consolidated the basis for apartheid), Kubik does not mention the representative circumstances of the region, save for a single reference to another place, Bondo, in Province Orientale, Congo. “July 8, 1960,” we read, in an oblique and understated way, “was not the best time for a peaceful research trip.”3 While conditions under the State of Emergency in Nyasaland deteriorated—a dramatic escalation of force by the police and troops, including the burning of houses, destruction of property, and the beating of dissidents—Kubik’s aim was to lay forth the precise structural conditions and attendant audile conundra inherent to the drumming patterns of the Vimbuza dance, and to consolidate the reach of his observations across a broader cultural terrain.

[2] Perhaps Kubik’s schweigen could be regarded as kind of defiance; a way of reminding his readers that there is a perspective in which the struggle for independence, achieved two years later, was on the order of fait accompli; that colonial occupation was a historical disruption, however massive, of the inevitable forces leading toward the autonomous self-governance of nations. Kubik’s homeland, Austria, had itself restored independence only four years earlier, following a decade-long allied occupation; and the establishment of a Nyasaland free from colonial occupation could perhaps be considered similarly self-evident, a historical inevitability. In the post-war period, the prospective character of national self-determination had, it seems, become a dominant ethos. For Kubik, the music of Nyasaland, from the vantage of similarly anti-colonial circumstances, ought possibly to be considered with a comparable degree of analytic integrity—as an autochthonous cultural practice, regulated by its own internal musical features and deserving of serious disciplinary attention. It comes as no surprise then that the language of the article partook liberally of the latest trends in the Austro-German intellectual milieu, including the school of psychology known as “Gestalt” theory.4 Kubik took note, for example, of the way African music exploits cognitive archetypes—“psycho-acoustic facts”— whereby musical phenomena are produced not by the mere linking of units, but units cohering in higher order objects. Of particular interest was one such higher order object, an elusive rhythmic activity that Kubik called an “inherent rhythm.”5 In later writings, Kubik termed these inherent patterns “subjective patterns.”6 The phenomenon of inherent rythms presents a challenge to the temporal conceptions guiding music in the European tradition of the late 18th and 19th Centuries. Reaching back further in time, Kubik briefly laid out the formal conditions for their emergence in African music with comparisons to J.S. Bach and other Baroque and Renaissance composers, who, while resembling the crucial patterning required for inherent rythms, do not quite achieve the sophistication of the African type.7 Inherent rythms in European music, for example, are regular, while those in African music are “additive”, by which Kubik likely meant that these are patterns of regular irregularity—rhythmic groupings that cut against the grain of the regular motor patterns by which they are formed. It is this “fifth condition” for the production of inherent rhythms—the irregularity of rhythm—Kubik argued, that surpassed the intellectual achievement of its European counterpart.8

[3] How does this African musical idea—this additive conceptual schema—weigh upon the content of musical meter? Consider the motional patterns deployed to render simple rhythms recorded and notated by Kubik in Thethe village near Rumi in Central Nyasaland in 1962, two years before independence. The rhythms notated in Figure 1 emerge in one particular section of the Vimbuza dance, performed on conical drums with single membranes made of cowhide, called Mohambu. The dance, practiced among the Tumbuka people, partakes of a biomedical tradition that involves possession by the Vimbuza spirits. Forbidden by missionaries because of its association with local resistance to colonialism, the Vimbuza dance intersects the natural and the supernatural world in an effort to overcome social grievances as well as physical and mental illness. Drummers produce a high (“small”) and a low (“big”) tone depending on how closely to the middle of the membrane they tap the drum.9 These simple motional patterns are striking for the varied ways in which they render their respective rhythmic groupings. Noting particularly the morphology of the second pattern, Mohambu II, Kubik makes a distinction between the “musician’s hands (motor image)” and the “pattern actually coming out (acoustic image).”10

[4] Kubik notates three patterns of the Vimbuza dance, marking the sticking patterns of left and right hands with “l” and “r” above and below the notated rhythms of Mohambu I and II respectively (see Figure 1). He concerns himself with the misalignment between motor and acoustic images in the music. In other words, motor images, which are clearly grasped by the performing musician, cannot be instantly registered by the listening dancers. This principle of audiovisual misalignment is a fairly widespread African musical idea. In the Mahume (sometimes spelled Mheme) drumming of Wagogo women from the Dodoma region of central Tanzania (then Tanganyika), for example, the motor movements of left and right hands are different, frequently marking tacet pulses—“beating into the air”—along with sounding pulses, thereby demarcating embodied temporal patterns that are asynchronous with the concomitant sounding ones associated with them.11 Feathered extensions attached to the upper arms of the women, along with exaggerated shoulder movements, further create visual illusions of temporal patterning at odds with the sounding rhythms of this ritually restricted sacred music.12 The lead drummer, along with the backing drummers, frequently beat into the air with their left hands.

Figure 1: Gerhard Kubik’s (1962, 39) representation of sticking patterns for the rhythms of the Vimbuza dance. This music and dance is a healing ritual of the Tumbuka people living in the village of Thethe near Rumi in Central Malawi (then Nyasaland).13

[5] A closer analysis of the details of the Mohambu II of the Tumbuka Vimbuza dance in 1962 is instructive. First, the hands alternate—effectively tapping the membrane in a simple interlock of binary time—as they simultaneously track back-and-forth between small and big tones—effectively tracing a simple ternary time. Although this is a simple two-note monophonic line, the pattern thereby nests two alternate temporalities in its motional vectors. Which of these embodied perceptions of the beat is dominant? Both the standard modern theories of meter and the projective analyses of meter in the West associated with the phenomenological turn (including their sixteenth-century forebears) are of limited use here. At first glance, one might argue, the way this drumming technique inflects a kind of binary feel into a ternary situation resonates with the more embodied, elastic conception of duration found in phenomenologically-inflected theories of rhythm and meter in the West. Christopher Hasty, for example, writes: “Nothing that is actual—that is, nothing becoming or having become—is without duration.”14 Hasty reconceptualizes meter in processual, or projective, terms. In triple meter, for example, beats are not subdivided into three isochronous units, but projected in a duple manner. He writes: “Since projection is essentially binary and requires that the two terms be immediately successive, and since projection results in equality, a projective account of triple, unequal meter is problematic”15 As a result, Hasty regards the second and third beats of triple meter as a kind of prolongation (and “deferral”) of the first beat. Deferral in triple time is therefore intensified in the context of the binary beating of time. The music theorist hereby returns to the analytic scene a kind of motional praxis for guiding metric perception that recalls pre-modern treatises on meter. In his Musica Figuralis Deudsch (1532) Martin Agricola, for example, defines “the beat (tact)” as a “motion of the hand,” likewise construed as an oscillating down-up gesture.16 In short, by reaching across the disembodied “Newtonian” moment in the 18th Century (about which more below), Hasty brings the German construal of embodied “Paarigkeit”, or pairedness, to bear on a contemporary interpretation of musical perception.17 Meter as Rhythm— a modern reinscription of the premodern tactile tact.18

[6] But, for all the recent intellectual effort placed into returning to musical meter an aspect of temporal embodiment, the African case discussed by Kubik gives embodiments of this sort the slip. In the Vimbuza dance, the simple temporal inflection of dual grouping is produced in a quite different way. First, the “projection” (in Hasty’s terms) of binary qualities in the context of 3/4 meter in the Vimbuza dance is of unequal duration. In other words, the projective deferral in Hasty’s phenomenological account—no less than the lowering and raising of the hand in Agricola’s premodern account—is a motional praxis that discloses binary and ternary qualities within the elastic bounds of equal duration, while the African case disrupts the equality of that duration. Second, the peculiar temporality of the Vimbuza rhythm additionally encapsulates an inverse projective capacity that the European case does not—a “projection” of ternary qualities in the context of a binary weave. In other words, where triple meter was “problematic” for Hasty on account of its “essentially binary” character—no less than it was for premodern European theories, where triple meter often garnered special treatment—In the African case, the pattern is delicately suspended between the projection of binary qualities in triple time and the projection of ternary qualities in duple time.19

[7] No amount of plasticity wrought by motional praxes of meter formation can quite capture the particular African inflection of this basic temporality. To give a sense of this, let us speculatively cast the Mohambu II Vimbuza pattern of 1962 in terms of Agricola’s Musica Figuralis Deudsch of 1532. What is at stake is the separation of conducting hands, whereby one hand is simultaneously lowered and raised in the tempo of a (triple time) “proporcien tact,” while the other is lowered and raised in between the arcs of the first hand. The latter could be in the same tempo as the former, or, as is frequently found in southern African music, in a contrasting tempo, such as a (duple time) “half tact”. In the case of the relatively simple amadinda xylophone piece “Olutalo olw’e Nsinsi” (“The Battle of Nsinsi”) transcribed in Buganda in 1962, for example, the rhythmic groupings of the two parts okunaga and okwawula clearly articulate, on the one hand, a binary metric scheme and, on the other, a ternary one. In western terms, these simultaneous meters could be construed either as 2/4 alongside 3/4 or, if two notes measure the quarter, as 3/4 alongside 6/8. Kubik’s transcription of the beginning of this piece in Figure 2 demonstrates the different rhythmic grouping of okunaga and and okwawula parts. The two interlocking parts are best grasped as melodic-rhythmic entities than purely metric ones, or what Meki Nzewi calls “melo-rhythm.”20 Bode Omojola describes such entities thus: “Melo-rhythm designates and empowers rhythm as an active determinant of melodic identity.”21 Strikingly, both melo-rhythms proffer metric schemes that are nested within a sustained interlocking structure; each pattern situated in the offbeat gaps of the other.

Figure 2: Kubik’s Short Transcription of “Olutalwo Olwe Nsinsi” (1962:16)

[8] The inflection of the western musical tact with malleable internal temporalities cannot in fact reckon with the dual inflections of the simultaneous interlocking tact we find here; nor can a simple appeal to perception or embodiment suffice. In fact, it would not be an exaggeration to say that the more embodied accounts of metric entrainment reinforce the very unit of measured beats that they flexibly inflect.22 This point applies to both pre- and (ostensibly) post-Enlightenment European accounts of meter as much as it applies to many other ethnographic and popular accounts of non-European meter. Stated differently, the phenomenological resistance to the Euclidean geometries of absolute time by way of micro-temporal variation (processual elasticity, non-isochronous beat subdivision, swing-like groove, and so on) is a kind of constrained resistance—one that preserves and intensifies its object of critique by paradoxically endowing its disembodied quantities with embodied qualities. (Colloquially speaking, the feel of African music’s swing, one might say, is the true topos of the neo-Newtonian legacy!)

[9] Kubik draws attention to a different aspect of the short drumming fragment recorded in Thethe village in 1962, namely the sound of the drummer’s total rhythmic configuration. This is a third perception—which Kubik calls the “acoustic image”—and it is heard apart from both the ternary and binary embodiments of the drummer’s motional arcs. The pattern adds an additional cross-metric relation to the rhythmic line— the “dotted note” pattern that proffers a four-beat measure in what is colloquially called “compound time” in the West. This abstract inherent rhythm, issued forth by the woven temporalities of two distinct embodied rhythms, carries the unmistakable micro-temporal inflections of an irregular thrown voice—the swing (if swing must be the word) not of perception, but of material musical ventriloquism. The production of disembodied but inherently salient, melo-rhythms such as this are a central aspect of African musical practices. Inherent patterns are particularly manifest on material instrumentaria that pre-date the ravages of colonial conquest. Their nano-chronemics cannot be collapsed into the culturally variable embodiments of some kind of biologic knowledge, but emerge instead at the technical crossroads between the human perceptual faculty, a precise division of impersonal, unedited motor processes, and the technical interfaces of music’s material media.

[10] Inherent patterns in these examples, furthermore, instantiate a third meter, set adrift of both the alternating metric groupings that constitute them. In “Olutalo olw’e Nsinsi” (The Battle of Nsinsi), for example, the inherent pattern (known as the “okukonera” in Luganda), forged at the intersection of 2/4 in the spaces of 3/4, elaborates a kind of compound time at double tempo. This melo-rhythmic line (in Nzewi and Omojola’s terms), doubled by the okukonera part (in Figure 2) is the melodic fallout between the two lowest notes that is doubled as a vocal line, thereby recapitulating the tonally inflected words of the accompanying song. Specialists in the field recognize the affinity between these kinds of phantom patterns and local cosmological practices. Kubik, for example, appears to credit Joseph Kyagambiddwa, who described the hallucinatory appearance of okukonera as a “voice” that “mysteriously looms up” in his book African Music from the Source of the Nile in 1955, and then elaborates the idea, by describing them as chimeric polyphonic lines—“as if the spirit [mizimu] were talking.”23

[11] While it is this indirect, emergent meter that may constitute the primary metric entrainment for observers, listeners, and dancers, it is demonstrably realized by neither performer in the ensemble. In fact, once again, the motional iterations of the performers’ patterns (each with their own micro-temporal inflection) directly cut against the grain of the illusionistic emergent meter in two distinct ways. As a result, the resultant meter found in both of these pre-colonial fragments from 1962—the Lugandan amadinda xylophone and the Tumbuka Vimbuza dance—does not function as a kind of prioristic grid for the elaboration of rhythmic ideas (as it is imagined in Western theory, about which more below) as much as it is indirectly revealed in the interlocking interaction of two distinct meter-like rhythmic processes. The place of meter in this African music, in other words, might count as a reversal of the Western case—meter less grounded in a basic series of isochronous beats framing rhythmic activity and more as an algorithmic derivation wrought by weaving multiply metric melo-rhythmic patterns. It is in the sparse conditions of this ephemeral archive that a global theory of musical meter—genuinely dislocated from the vexing antinomies of the modern measure of time—might begin.

II

[12] For an essay invested in unpacking the distinct structural processes of African musical practices across a broad terrain (from then-Rhodesia to the Congo by way of Buganda), Kubik’s article takes a sudden developmentalist turn, almost tacked on as an afterthought. Having repeatedly demonstrated the delicate precision of the permutational framework underlying various African compositions, Kubik, with wistful certainty, nonetheless adds: “This he [the African composer] certainly does not achieve by intellectual or even mathematical experiments.”24 Conscious analytic thought, Kubik explains, is simply foreign to the creation of this music, adding finally: “We should keep this in mind when trying to understand the structure of African music.”25 Here the ethnologist encourages the reader to keep in mind an imagined African frame of mind—a mind characterized by a lack. African creation, it seems, is ensnared in the embodiments of creation—mere patterns stored in muscle memory; a state of mind at odds with conscious analytic abstraction. On the one hand, then, we find in Kubik’s detailed understanding of African music a kind of rhythmic complexity that heretofore “has been considered beyond comprehension,” and, on the other, a kind of non-mathematical mindset, unconsciously creative, “belonging to another sphere of human experience,” and so on.26 Beyond whose comprehension, one might be tempted to ask, is this mathematical dimension of African music? Does a framework that lies beyond European comprehension entail lying beyond an African one? Against odds, then, Kubik at once offers compelling evidence of precisely calibrated permutational frameworks and technical practices in Africa, and then withdraws that evidence by abbreviating the dimension of abstract mathematical thought. 

[13] That said, the hauntingly brief ethnographic turn of mind—which detects in African practices an arena of embodied creation, devoid of abstract experiment—is resolutely abandoned in Kubik’s extensive subsequent work on pre-colonial African culture. By the 1980s, he describes African practices of this sort in abstract, mathematical terms: Angolan Tusona ideography, for example, is framed as a geometric system reflecting “abstract logical thinking,” while some of the Kigandan court compositions of Uganda and the timbrh lamellophone music of Cameroon are described in terms of mathematical discoveries.27 In Kubik’s later view, then, these ideographs and compositions are autonomous constructions, reflecting a rigor whereby “each constituent part of a whole is designated to be an essential, irreplaceable entity with multi-lateral relationships.”28 Against Kubik’s revised idea of African practices regarded in rigorously autonomous terms—abstract, self-contained constructions marked by permutational logic—the earlier ethnographic validation of embodied creation proved to be symptomatic of a general condition that held widespread institutional currency well into the 21st century. Music scholars from Africa, ranging from Kofi Agawu, Daniel Avorgbedor, and Akin Euba to Jean Kidula, Willie Anku, and Kwasi Ampene, have long insisted on granting African music a dimension of autonomy, as against the various swerves toward grasping African music under the rubric of culture, affect, embodiment, collectivity, and other tools of “ethnographic segregation.”29 The turn in the Humanities toward affect and vitalism—away from discretization, formalism, and abstraction—in the early 21st century exacerbated the condition. When it comes to music, it is paradoxical that this kind of swerve toward performance, movement, expression, experience, entrainment, and other forms of embodied knowledge is strikingly at odds with the contemporary consumer-electronics model that underwrites most actual musical production and distribution across the globe today. Although engagements with meter and rhythm today deploy insights from a wide range of disciplines (from acoustics to neuroscience; from anthropology to cognitive science), they tend nonetheless to coalesce around a dynamic thesis for a philosophy of rhythm that is curiously delinked from the primary modus of their own technological age—the material infrastructures of digitality, characterized by discretization, abstraction, statistical and symbolic orders, and the computational modeling of formal systems directly. It could be argued, of course, that the means of rhythmic production today is not the central subject of music scholarship, and that the perception and experience of rhythm should be delinked from such concerns. Furthermore, it could be shown that much work on rhythm, in fact, offers more nuanced, dialectical accounts of rhythm that productively oscillate between an experiential dynamic thesis, on one hand, and an abstract static one, on the other. Nonetheless, barring the occasional acknowledgment of the part played by technology in rhythmic experience, discussions of technical objects or encoded rhythm in scores or graphic equalizers remain limited.30 The absence of precise engagement with the technical mediation of rhythm today, as well as its mode of production, tends to escape the attention of this otherwise wide-ranging work.

[14] Perhaps a deeper methodological symptom lies in the nature of the dialectical rubric itself— between abstract and experiential (or what Peter Cheyne calls “encoded” and “embodied”) theories of rhythm—which can be genealogically traced to a relatively localized historical and geographical locus.31 Indeed, the very idea of meter construed as a sequence of isochronous beats with a specific cardinality (number of beats per measure)—in turn hierarchized both internally and at higher levels of organization—reflected an abbreviated mode of engaging and experiencing music that is of extremely recent vintage. In Beating Time & Measuring Music in the Early Modern Era (2014), Roger Grant demonstrates how modern conceptions of meter actually fragmented earlier understandings of meter, which were generally more imbricated in multidimensional facets of embodied motion, note duration, musical character, and tempo.32 In Cartographies of Time: A History of the Timeline (2010), historians Daniel Rosenberg and Anthony Grafton furthermore point out that the framework of absolute time— adumbrated most forcefully by Isaac Newton in the late seventeenth century—only fully emerged as a dominant scheme for time-reckoning in the age of Enlightenment.33 Contending with the fact that the rate of the earth’s rotations was continually subject to variation—external “forces” that retarded or accelerated its motion—the consolidation of absolute time—regarded as pure duration or the persistence of objective existence—was achieved against considerable empirical, religious, and conceptual odds. Matrices for framing musical time were theorized along similar lines; and, along with pitch spaces, instruments, electrical currents, political nation-states, devices, legal policies— instrumentality itself—were gradually standardized and scaled. Key technological artefacts for keeping time in music emerged throughout this period—including D.N. Winkel’s musical chronometer of 1814 or J.N. Maelzel’s newly-patented metronome of 1816. Likewise, music’s metric division in the mid-18th century—following, among others, J.P. Kirnberger’s landmark Die Kunst des reinen Satzes in der Musik (1771-79)—was re-theorized as a kind of undifferentiated flow of absolute time [Zeit], intercalated by an isochronous circular time that was subdivided by accentuations grouped in two, three, and four beats [Tackte]. This reconceptualization of time in music became normative in the late 18th century. Johann Forkel’s theory of musical time, for example, was ensnared in the exactitude of then-modern timekeeping devices, effectively inverting the temporal priority of time in music. Music did not shape time as much as pass through a kind of conduit of abstract clockwork: “In this regard, the partitioning of rhythm into measures and phrases with bar lines—the marks that express them—is to be seen in like manner to the hour hand, indicating to us the musical relationship of time that every musical phrase must pass through.”34 The archetypal and epistemological composition of these theories and technologies guided and augmented modes of action and thought, which came to enjoy a world monopoly.

[15] The Enlightenment connection between absolute time and musical time deserves closer attention. For example, Kirnberger’s student, Johann Georg Sulzer, like Heinrich C. Koch, offer a generative approach to rhythm—from its putatively simple origins in a Schlagfolge (Sulzer) or Schlagreihe (Koch) to its sentence-like higher-order elaborations. Here we find an important perspective on the nature of what Michael Spitzer calls a “space-time” dialectic in the rhythmic imagination of late 18th-century music.35 Spitzer describes passages from Mozart’s String Quartet in C major, K.465 in the metaphors of linguistic punctuation; an analysis that hinges upon a kind of “material-formal” distinction marking the music’s unique punctuation.36 What is interesting about Koch’s language-like description of music’s temporality is its teleological underpinnings. Spitzer alludes to the ideas of Étienne Condillac, Jean-Jacques Rousseau, Johann Forkel, and J.G. Herder to contextualize a framework for examining the evolutionary relationship between music and language—highlighting a historical shift from a “language of feeling” (Empfindungsgespräch) to a “language of ideas” (Ideensprache). Notably, this shift from feeling to idea was often accompanied by racialized undertones. For instance, in his Allgemeine Geschichte der Musik (1788), Forkel argues that “primitive nations are only capable of rhythmic music,” describing it as a percussive style resembling the child-like, affective outbursts of language—limited to “interjections and simple words” (137).37 On the subject of Forkel’s imagined music history, Spitzer writes: “Just as primitive races are slow to achieve this kind of linguistic sophistication, it takes them a long time to ‘arrange a series of tones in such a way … that a melody, akin to a spoken sentence, can arise’.”38 Spitzer does not remark on the degrading evolutionary references to “Man, in his earliest state,” “primitive man,” or the “primitive nations” capable only of “rhythmic music” that function as a placeholder for the evolutionary underpinnings of Koch’s theory. For Forkel, the “regular repetition of simple things,” which he associates with early man, is called “meter” in music; a simple state that has not yet risen to the language of “ideas.”39 Could it be that the very concept of meter—the regular affective repetitions of primitive men—reflects a racialized framework for conceptualizing rhythmic practice?

III

[16] It is essential to contextualize this conception of musical time-keeping in relation to an era characterized, on one hand, by industrial development, evolutionary thought, and colonial expansion, but also, on the other, by idealism, rational reform, and social progress. Perhaps the development of a rhythmic theory that emphasized the properly punctuated musical sentence as an emancipatory idea reflected the aspirations of what Herder understood as the “newly found orderliness of social and political organization.”40 As Kirnberger was honing Die Kunst des Reinen Satzes, Frederick II (r. 1740- 86) declared war on the Habsburg monarchy, following the annexation of Bavaria, no more than a hundred miles south of his home town of Saalfeld. Frederick’s ability to unite seventeen rulers of a decentralized Germany to the cause, known as the Fürstenbund (League of Princes), was co-terminus with a new idealism that attempted politically to navigate an alliance beyond barriers of state, class and religion. Various principalities merged under the Holy Roman Empire, albeit in a defensive act of self-preservation, ushering in a period marked by educational reform, articulated civil codes and social norms, religious tolerance, and the gradual emancipation of peasants. Kirnberger’s modern theory was grounded in the scientific turn of this era, including a general model of time-reckoning that formalized the abstract continuity of time that transcended and downplayed the diverse proliferation of everyday, organic, and quotidian temporalities. The equable flow of time was regarded as a universal given, indifferent to both the older cosmological orders of motional timekeeping and the fragmented multiplicity of localized concepts and frameworks of time. Kirnberger concluded his work in 1779, on the eve of the revolution in France.

[17] Not surprisingly, the reign of absolute time in music, increasingly buttressed by technological artefacts for time-keeping (from chronometers and metronomes to drum machines and beat-trackers), simultaneously generated its own culture of critique. The very stylistics of music in the heyday of the industrial era—the turn to romantic ideas of transformation, variation and becoming writ large—was largely resistant to the reign of mechanical clock time. Theorists, from Jean-Jacques Rousseau to François-Joseph Fétis, by way of Pietro Lichtenthal and others, construed the metronome as a threat to true musical sensibility—guided by “impulses of affect” and “sentiment” (Lichtenthal), “expression” and “eloquence” (Fétis), and so on.41 Historical musical performance practices, such as tempo rubato, for example—a term that indicated the non-observance of notated metric time in service of an expressive ideal—emerged as key conceptual coordinates for musical performance in the industrial age. Indeed, with frequent references to music, entire intellectual currents of critique emerged in the nineteenth century—ranging from Wilhelm Heinrich Wackenroder and Arthur Schopenhauer to Søren Kierkegaard and Friedrich Nietzsche—at variance with the dominant mode of time-reckoning during this period.

[18] Contemporary scholarship on meter and rhythm owes a genealogical allegiance to this culture of critique. With frequent references to the work of Henri Bergson, Edmund Husserl, Gaston Bachelard, and others, an extensive recent collection of essays titled The Philosophy of Rhythm: Aesthetics, Music, Poetics, for example, consciously invokes this tradition; but the allegiance inflects chapters with a contemporary focus as well. In “Rhythm and Popular Music,” for example, Alison Stone argues that popular music “challenges rather than reinforces the domination of clock time over the lived, bodily world.”42 Stone draws attention to the music’s rhythmic “pulls-and-pushes”—bodily experiences solicited by way of dynamic backbeats and syncopations— as the site of this resistance, and even argues that, by subverting “the hierarchy bound up with meter,” such music can be said to “reject the broader set of power relations bound up with measured time.”43 Stone historicizes the dominant conception of meter—a “Western form as it … crystallized from 1600 onwards”—which, she argues, is critically subverted in her examples from popular music.44

[19] However, this characteristic critical gambit (ubiquitous in recent cultural studies of music) needs itself to be historicized within an Enlightenment culture of critique. The history of the western beat, one might say, has always been a history of its discontents; the transformational agogics of romanticism pitted against industrialized clock time. Even Forkel’s oddly evolutionary description of a “civilized” musical sentence entailed moving beyond the mere beating of a clock toward the flexible temporalities of a rhythmacized musical sentence. Whatever the ethics of this latent idealism and romanticism, I here wish only to draw attention to the curious resonance between the “dynamic,” “processual,” “critical” arguments against the purported reign of the abstract conception of meter today and a set of provincial Enlightenment ideals. In other words, the dynamic thesis actually marches in sync with the ideological demands of the Enlightenment conception of time in music.

[20] In a curious act of assimilation, the abundant references to non-western music in an extensive collection like The Philosophy of Rhythm often function similarly, as placeholders for a critique of (or as resistance to) the simplified isochronous model of musical meter apparently upheld by the European Enlightenment. Here we find non-western music prominently upholding the dynamic model of time implicit to Enlightenment ideals. African music, for example, is variously identified in this collection with embodiment and dance (pp. 21 and 45), polyrhythm (pp. 66, 73, and 190), the affective and social binding of bodies (p. 120), and the displacement of normative accents (p. 149).45 Encapsulating this tendency in a sentence, Tenzer even distinguishes “African musical rhythms” from the “European rhythmic experience” with reference to the work of John Miller Chernoff and Anne Danielsen (on West African music and African American funk respectively): “repetition in a highly pulsating and rhythmically layered groove engenders participation, entrainment, embodiment and pleasure.”46 Tenzer’s chapter, by far the most genuinely global and diverse in its description of rhythmic practices of the world in Philosophy of Rhythm, is nonetheless caught in the grip of this European industrial dichotomy.

[21] The remark deserves closer explanation. While attentive to the entangled nuances of a differentiation between rhythmic experiences, Tenzer distinguishes two types of “rhythm object,” namely A- and B-series rhythm. “A-series rhythm is continuous,” he writes, “B-series rhythm has discrete parts” and is captured in fixed forms (notations, recordings, etc.).47 Drawing on the work of Rainer Polak and Justin London, Tenzer asserts the uneven division of beats in an African music genre (Mande drumming), which is then rhetorically affined with an African-American one (jazz improvisation)—which are described as the “micro rhythmic idiolects” of black musical genres.48 Tenzer does in fact warn against the racialism implicit in granting “blackness an exclusive purchase” on such embodied qualities (or “A-series” rhythmic activity) and suggests that “Afro-Euro-American entanglement” produces similar (A-series) experiences among diverse actors. Importantly, however, of the discrete “B-series” rhythmic patterning implicit in African music, the author offers no account.

[22] This is a curious omission given both the relative rhythmic precision required for the performance of, no less than the abundance of discrete triggering instruments (comparatively speaking) found in, the African music under discussion. In fact, Polak has noted that the precision of beat-production in the live context of African drum ensembles easily eclipses that of the average European professional orchestra, a point echoed by Kofi Agawu in his various critiques of ethnographic prohibitions on thinking African temporalities as either unit-based or embodied.49 In sum, the “micro rhythmic idiolect” of the European orchestra is further set adrift from, rather than proximate to, the very “B-series” rhythms that African musical drumming (and rhetorically attendant “grooves of ‘blackness’”) are recruited to elude.50

[23] The striking omission in the current Euro-American literature of considerations of African music’s formal dimensions, encoded “B-series rhythms,” and so on, is symptomatic of what Agawu would call ethno-theory. On Simha Arom’s assertion that traditional African music “does not require any formal theory on the part of those who inherit it and perform it,” Ampene writes: “It is a serious intellectual misrepresentation for Arom to assert that Africans are oblivious to the underlying conventions and theory of the music traditions they create and pass on from generation to generation. It is also a travesty for Arom to assert that Central Africans can only compose and perform music but that the key to unpacking the cognitive processes that govern the musical expressions of his interlocuters lies squarely with him.”51 As it was for Agawu and Kidula, Ampene detects in this disavowal of theory a segregationist impulse that cannot be sustained in the face of the evidence proffered by genuine analytic engagement.

IV

[24] Again, this kind of lopsided dialectical encounter with global rhythmic practices needs to be historically situated within the Euro-industrial order of things, to which the turn to embodiment, experience, and performance is symptomatically tethered. Consequently, this casts some doubt on the question concerning universals (biological, cognitive, psychological and neurological affordances and constraints) across the cultural terrain of rhythmic reckoning. Interestingly, the work directed most prominently toward the cognitive, tactile, vestibular, and neuro-psychological systems of human perception are often the most generously attentive to the variability of its specific cultural realization. Jenny Judge’s notion of cross-modal “perceptual completion” (which occurs with partially, or incompletely, seen or heard objects), for example, usefully captures the open-endedness of the sensory system, and thus also the constitutive role played by culture, form, and history.52 Likewise, Udo Will’s neuro-psychological approach to the processing and experience of rhythm considers internal bodily periodicities in conjunction with external event sequences. Although Will outlines a series of irreducible perceptual thresholds (or constraints), temporal processing also involves an aspect of variability and volition.53 On the subject of meter, for example, Will demonstrates that its 17th-Century Western conceptualization does not seem to apply in the context of either poetry or non-Western music. This is because the detection of repeating patterns involves training, memory, attention, and different sensory-motor networks. (Vocal and instrumental rhythms, for example, engage different neural pathways, which explains why instrumental rhythms are more regular than speech rhythms). Using aboriginal Pitjantjatjara speech and song as a central referent, Will thereby leverages a neuro-psychological account to demonstrate the cultural specificity of rhythmic experience.

[25] Justin London’s model too steers between universal features of rhythmic perception—the perceptual threshold for intervals between event onsets, for example—and the cultural variability when it comes to music’s actual rhythmic structuring (via entrainment).54 London shows that it is precisely because of the active contribution of the auditory system—to “fill in” incomplete data (absent beats, and the like)—that perceptual variability and volition becomes possible. The role of learned conventions and techniques for encultured listeners, including the presence of timekeepers, choreographic supplements, performance variables, tempo and dynamics, ensemble interactions, and other ethnographic evidence further mediate the scene of beat-making in context. London advances the idea of “subjective rhythmization” (partly innate, partly culturally conditioned), demonstrating how beat formation in the context of metrically malleable music differs vis-à-vis differently enculturated listeners (p. 178).55 To illustrate the point, London uses the “standard” gankogui pattern to contrast African with non-African ways of hearing metric beats within it, and notes that variable modes of entrainment alter the identity of the pattern considerably. It is precisely the perceptual capacity to supplement incomplete information—infer “missing” beats and interpolate syncopations—that opens the possibility for divergent entrainment.

[26] London’s account of the unique challenge presented by African rhythms of this sort invites a reconsideration of the dualism between embodied and encoded rhythms that guides much scholarship on meter and rhythm. Culturally speaking, the standard pattern under discussion, for example, is entrained differently not only in African and non-African cultural settings, but also in different cultural settings within Africa. In accordance with conventions of Ghanaian drumming ensembles, for instance, London locates the African starting point for the ternary interpretation of the rhythm on the first of the two quarter notes in the pattern.56 In the context of a makwa (hand clapping) pattern to accompany a Zimbabwean mbira dza vadzimu ensemble, in contrast, the ternary inflection would begin on the second quarter note (according to London’s figure), while the analogous pattern in the Mohambu III of the Zambian Vimbuza dance would begin on the ninth. Even if we discount the question of an unequivocal downbeat in either context, the entrainment schema in one setting occurs within the spaces of the schema in another, while the ostensibly non-African schema occurs elsewhere still. The striking point about asymmetric African timelines such as this concerns what music theorists might call “grouping dissonance,” or what one might accurately call metric multiplicity by design.  Of the three possible ternary groupings associated with the formation of ternary (“compound”) time in a 12-pulse sequence, the gankogui/makwa pattern has the formal capacity to entrain each with almost equal validity. Likewise, of the two (or four) possible binary groupings associated with the formation of binary (“simple”) time in a twelve-pulse sequence, the pattern can do likewise. In other words, the seven-note timeline successfully embeds these metric schemes, whether composed of two-, three-, or four-beat units—or whether beginning on any of the twelve time-points—with near-equivalent plausibility. The way the timeline nests ternary and binary components across the entirety of a single pattern explains the uniquely differential distribution of binary and ternary qualities, and hence its maximal metric variety within a bounded mathematical set.


Figure 2: Nested Metric Ambiguity in the Gankogui/Makwa Rhythmic Pattern57

[27] Of course, in practice, the pattern is heard in one or other metric scheme at any given time, but it is “encoded” in such a way as to admit of multiple possibilities. This crucial feature of African music, structurally resonant with the Mohambu II of the Tumbuka Vimbuza dance no less than the interlocking lines of the Lugandan amadinda music discussed above, extends beyond the signature timeline pattern under discussion here, and suggests a modality of listening that is not legible to a narrowly “dynamic” account of rhythmic phenomena in music. Due attention to the discrete “B-series” rhythmic structure of African music may be a key element to grasp the full extent of its permutational logic. The late African music theorist Willie Anku regards this kind of timeline pattern precisely as a kind of encoded “B-series” structure—a “controlling structural concept”—and further explores the role of rotational interpolations in the context of circular musical forms from West and Central Africa.58

[28] How might this kind of example challenge the dominant music theory of the Atlantic north-west? How might one, following the example of Mozambiquan philosopher Elísio Macamo, craft a “study of Africa aimed to understand the world.”59 In the European Enlightenment ideal, music is raised from its primitive state (marked by “regular repetition”) to a civilized one (marked by “ideas”) by way of melodic punctuation—“arranging tones in such a way … that a melody, akin to a spoken sentence, can arise” (138). Therein lies its “linguistic sophistication,” its “newly found orderliness,” its “civilized syntax,” and so on (p. 138). Against a repetitious metric ground, one might say, this evolutionary theory of music locates figures of rhythmic-melodic flexibility that raise the music to the status of an “idea”. But what kind of idea is raised when music’s rhythmic-melodic entities, through an intricately woven lattice work, actually produce the illusory stability of metric schema? Less the beating of a primitive German Schalfolge or Schlagreie, and more the resultant effect of an algorithm, African meter of this sort exhibits a kind of chimeric genealogy, woven by inaudible polyphonic threads. It is as if the figure and ground (or the perceived cause and effect) of music’s rhythm-meter relation (and perhaps thereby too its affect-idea relation) are reversible.

[29] Transposed into 18th-Century European terms, we find the following formulation: If metric beating, in the European Enlightenment, is associated with affective outbursts (Empfindungsgesprach), then in precolonial Africa it is instead elicited—composed (in both senses of the term)—from interacting melo-rhythmic ideas (Ideensprache).  This is a musical meter not built on a grid (as a starting block for rhythmic invention), but one that culminates (from rhythmic interaction) in elusive entrainment. Here we find a metric system construed as the sonification of a kind of algebraic expression; a pattern-generator for threading sound. It is the relationship between human play and coded keys that generates uncanny metric patterns, both present and absent, which are woven into audible appearance. Of spirit, one might say —vimbuza, mizimu, mudzimu— this meter—though wrought by weaving mallets and hands—is not under the direct supervision of a single performer. Meter, in these precolonial African examples, emerges as an algorithmic fallout of a transformational situation. The inherent metric schema are sonic figures of xenogenesis, a snippet of mathematical space rendered audible—the beating not of the child, but of the spirit.

[30] Could it be that the entire European conceptual edifice is thrown into question by a properly formal analysis of meter (and rhythm) in these African examples? In contrast to the Euro-industrial conception of it, rhythmic-melodic entities in the precolonial Vimbuza dance music of the Tumbuka people in modern-day Malawi, no less than the amadinda (and akadinda) music from the Kampala region in southern Uganda from the era before the destruction of the Lubiri Court, often remain stable, while metric schemes are meticulously permutable. As shown above, the systems that undergird the performance practices of these musical types involve the interlocking of parts, polymetric grouping, and inherent pattern formation that elicit beat entrainment set adrift of the embodied motor patterns of performers. In the case of amadinda music, procedures for pitch transposition further rotate distinct metric schemes, effectively recouping a kind of rhythmic-melodic identity under transformational metric conditions.60 A great variety of African musical types—ranging from pre-colonial matepe music from the Korekore region in northern Zimbabwe to timbila music from the Nyungwe region of Mozambique—similarly upend the Newtonian-inflected relation of rhythm and meter in important ways.61 How might one craft a study in reimagining a theory of meter, beyond refusal or redemption, to decenter the legacies of musical Newtonianism? The article has shown that the reign of absolute time is not the only target of critique here, but that the equally ubiquitous affective, embodied resistance to it can serve as its uncanny alibi. In both cases, meter is forged at the intersection of technical encoding and embodied technique, but legacies of their Western representation, against evidentiary odds, betray an uneven handling of this intersection in the African case. When Africa is regarded from the perspective of its immediately apparent cultural shell (instead of its rational kernel), its systems of mathematical self-organization appear as mere epiphenomena of observed human behavior. How, then, might one relativize Euro-industrial practices of meter (and its attendant rhythm-concept) as well as Africanize those metric practices that go as universal? Can we assume, in the spirit of the central protagonists of this story, that the Eurogenetic inflection of musical time is but an historical outburst, however massive, of the inevitable forces leading toward an insurgent set of universal ideas? Can the establishment of a global music theory free from racializing tropes perhaps be considered similarly self-evident, a historical inevitability? What perspective can be gained from methodological considerations that are asymmetric to the current state of affairs? Resisting both the affective dimensions of performance alone and the positivism of the Digital Humanities (already ensnared in a networked ideological infrastructure), this article hopes to widen the historical and conceptual window of what counts in musical meter.

  1. This article draws on and supplements aspects of work previously published in my “Afro-Electric Counterpoint,” Rethinking Steve Reich  (eds. Sumanth Gopinath and Pwyll ap Sion). Oxford: Oxford University Press, 2019, 259-302; “Temporalities,” The Oxford Handbook of Critical Concepts in Music Theory (Eds.Alexander Rehding and Steven Rings), Oxford: Oxford University Press, 2018; and “The Rhythm of Philosophy: Inertia of an Enlightenment Ideal?” Revue de Musicologie, 107/2, 2021, 449-457. I would like to thank Kofi Agawu, Kwasi Ampene, Daniel Avorgbedor, Seth Brodsky, Rick Cohn, James Currie, George Worlasi Kwasi Dor, Roger Grant, Jean Kidula, Elísio Macamo, Kelli Moore, Alexander Rehding, Steven Rings and Viktoria Tkaczyk for their input at various stages of this work. ↩︎
  2. Vail, Leroy, “The making of the ‘Dead North’: a study of the Ngoni rule in northern Malawi, c. 1855-1907,”in J. Peires, ed., Before and After Shaka: Papers in Nguni History. Grahamstown, 1981, 143-145; Baker, Colin. 1974. “Depression and Development in Nyasaland: 1929-1939,” The Society of Malawi Journal,
    Vol. 27, No. 1 (January, 1974), 7-26. ↩︎
  3. Kubik, Gerhard, “The Phenomenon of Inherent Rhythms in East and Central African Instrumental Music,”African Music 3/1, 1962, 33–42, 37. ↩︎
  4. Gestalt psychology was a theory of visual experience that emerged in Austria and Germany in the early twentieth century. Kubik detected a notable feature of gestalt psychology—namely, the contiguity of experience in the context of a succession of static images or stationary objects—in his discussion of illusory sonic motion in the music of Africa. ↩︎
  5. Kubik, “The Phenomenon of Inherent Rhythms,” 33. ↩︎
  6. See, for example, “Subjective Patterns in African Music,” Cross Rhythms, Vol. 3. Papers in African Folklore,  129-154 (Ed. Daniel Susan Domowitz, Maureen Eke and Enoch Mvula). Bloomington: African Folklore Publications, Trickster Press, 1989. Kubik writes: “They hide in the inner structure of the aural image of a musical composition, hence: inherent patterns. As they are subjectively in perception and cannot be traced by any objective means of measurement, we may also call them subjective patterns.” The new terminology is also self-referenced in “The Emics of African Musical Rhythm,” Cross Rhythms, Vol. 2. Occasional Papers in African Folklore/Music, 27-66. (Ed. Daniel Avorgbedor and Kwesi Yankah). Bloomington: African Folklore Publications, Trickster Press, 1985. I would like to thank Daniel Avorgbedor for pointing out this subtle shift in terminology. ↩︎
  7. Kubik, The Phenomenon of Inherent Rhythms, 36. ↩︎
  8. Ibid., 36. ↩︎
  9. Ibid., 38. ↩︎
  10. Ibid., 39. A short documentary produced by the Malawi National Commission for UNESCO in 2008 (Vimbuza Candidature Video) offers a recent glimpse into the world of the Vimbuza healing dance, even though the drumming of Mohambu II on this film no longer strictly follows the principles adumbrated by Kubik (UNESCO 2008).  ↩︎
  11. Ibid., 40. ↩︎
  12. A recent—staged—example of Wagogo women’s “Muheme” drumming, performed by the Nyati Group from Nzali Village in Dodoma, Tanzania, is available online (Malago Shiro, 2012). ↩︎
  13. The analysis of the Vimbuza dance appears also in my “Temporalities,” 2018. ↩︎
  14. Hasty, Christopher. 1997. Meter as Rhythm. Oxford: Oxford University Press, 69. ↩︎
  15. Hasty, Meter as Rhythm, 67. ↩︎
  16. Agricola in Roger Grant. 2014. Beating Time & Measuring Music in the Early Modern Era. New York: Oxford University Press, 68. ↩︎
  17. Hasty, Meter as Rhythm, 135) ↩︎
  18. Grant points out that, while fundamentally allied with them, Hasty’s theory actually reverses the basic duple construction of triple meter found in historical treatises. We find in Hasty a temporal projection from short to long, while the “proporcien tact”— Agricola’s triple meter—consists of two motions, the first a lowering of the hand twice as long as the subsequent raising” (2014, 69). ↩︎
  19. In his book, Dancing Prophets, Steven Friedson considers the experiential valences of Vimbuza. (See Dancing Prophets: Musical Experience in Tumbuka Healing. Chicago: University of Chicago Press, 1996). Additionally, Friedson addresses the experience of ternary and binary combinations in terms of a “complementary relationship” or “structured ambiguity” (instead of “cross-rhythm”): “It is an ontological structure, a musical way of being-in-the-world in multiple ways, a fundamental feature of a widespread and historically deep African rhythmic praxis.”(Remains of Ritual: Northern Gods in a Southern Land. Chicago: University of Chicago Press, 2009). Daniel Avorgbedor has critiqued Friedson’s analysis of metric ambiguity in relation to a projection of mythical cultural difference. Avorgbedor writes: “Perhaps in search of an elevated play with/on ‘ambiguities’, [Friedson] sustains his arguments in support of polymeter and cross-rhythm, not so much in clarification of any phenomenological analytical approach, but to reify the ‘other’, albeit in subtle ways” (Avorgbedor, Daniel. Ethnomusicology Forum. Dec 2013, Vol. 22 Issue 3, 379-382, 381). The analysis to follow reckons with Avorgbedor’s critique without relinquishing the case for metric specificity/complexity in African music. ↩︎
  20. Nzewi, Meki. “Melo-Rhythmic Essence and Hot Rhythm in Nigerian Folk Music.” The Black Perspective in Music, vol. 2, no. 1, 1974, pp. 23–28. ↩︎
  21. Omojola, Bode. “Composing and Contemplating African Melo-rhythmic Polyphony,” Ghana Studies Journal, Special Issue on The African Imagination in Music (edited by Kwasi Ampene), Vol. 24, 2022, 69-75. ↩︎
  22. For a key text in the scholarship on musical entrainment, see In Time with Music: The Concept of Entrainment and its Significance for Ethnomusicology (eds. Martin Clayton, Rebecca Sager and Udo Will), to appear in ESEM CounterPoint, Vol. 1, 2004, 1-83. ↩︎
  23. Gerhard Kubik, Theory of African Music: Vol. 2. 2010. University of Chicago Press, 108. See also Joseph Kyagambiddwa’s African Music from the Source of the Nile. Indiana University Press, 1955. ↩︎
  24. Kubik, “The Phenomenon of Inherent Rhythms,” 42. ↩︎
  25. Ibid. ↩︎
  26. Ibid, 32, 42. ↩︎
  27. In “The Emics of African Musical Rhythm,” Kubik underscores the importance of numerical abstraction in musical contexts such as those described in the analysis: “How is movement behavior acquired in the process of learning? There are many indications that in Africa what is actually learned by the music/dance apprentice is not the audible or visible manifestations of a pattern alone, but its abstract structural content which is defined by a numerical relationship or configuration. The transformation of motion patters from one phenotype into another in their practical application to music and dance is accomplished with strict retention of their numerical configuration. Since it is the latter which the learner internalizes, it follows that motion patterns in African cultures are convertible. At any given moment they can be recast and transformed from spatial/temporal into aural/temporal phenotypes and the reverse” (See “The Emics of African Musical Rhythm,” in Cross Rhythms, Vol. 2. Occasional Papers in African Folklore/Music, pp. 27-66. Ed. Daniel Avorgbedor and Kwesi Yankah. Bloomington: African Folklore Publications, Trickster Press, 1985). I would like to thank Daniel Avorgbedor for pointing out this important reference. ↩︎
  28. Kubik, Gerhard. 2010. “African Space/Time Concepts and the Tusona Ideographs in Luchazi Culture,” in Theory of African Music: Vol. 2. University of Chicago Press, 283, 321. ↩︎
  29. Agawu, Kofi. 2006. The African Imagination in Music. New York: Oxford University Press, 22. See also Jean Kidula’s emphasis on the musical sound (against cultural product) in Kidula, Jean. 2006. “Ethnomusicology, the Music Canon, and African Music: Positions, Tensions, and Resolutions in the African Academy. Africa Today 52 (3): 99-113, 104. ↩︎
  30. See, for example, Michael Tenzer’s comprehensive anthropological account with reference to rhythm and technology in “How Many Kinds of Rhythm are There?” in The Philosophy of Rhythm: Aesthetics, Music, Poetics. Oxford: Oxford University Press, 2019, 199-215, 209-13; the reference to beat trackers in Justin London’s cross-modal account of entrainment in “Metric Entrainment and the Problem(s) of Perception, in The Philosophy of Rhythm: Aesthetics, Music, Poetics. Oxford: Oxford University Press, 2019, 171-182, 178; or Peter Cheyne’s reference to technical media in relation to his investigation of encoded rhythm in “Encoded and Embodied Rhythm: An Unprioritized Ontology,” in The Philosophy of Rhythm: Aesthetics, Music, Poetics. Oxford: Oxford University Press, 2019, 255-271, 255, 265. ↩︎
  31. Peter Cheyne. “Encoded and Embodied Rhythm: An Unprioritized Ontology,” in The Philosophy of Rhythm: Aesthetics, Music, Poetics. Oxford: Oxford University Press, 2019, 255-271, 255. ↩︎
  32. Roger Matthew Grant. 2014. Beating Time & Measuring Music in the Early Modern Era. New York: Oxford University Press. ↩︎
  33. Daniel Rosenberg and Anthony Grafton. 2010. Cartographies of Time. New York: Princeton Architectural Press. ↩︎
  34. Forkel, in Grant, Beating Time, 126. ↩︎
  35. Michael Spitzer, “Mozart’s ‘Dissonance’ and the Dialectic of Language and Thought in Classical Theories of Rhythm,” in The Philosophy of Rhythm: Aesthetics, Music, Poetics. Oxford: Oxford University Press, 2019, 125-140, 130. ↩︎
  36. Ibid., 134. ↩︎
  37. Ibid., 137. ↩︎
  38. Ibid., 138. ↩︎
  39. Ibid., 138. ↩︎
  40. Aldo D. Scaglione. 1981. The Theory of German Word Order from the Renaissance to the Present. Minneapolis: University of Minnesota Press, 74. ↩︎
  41. Lichtenthal and Fétis quoted in Grant, Beating Time, 206. ↩︎
  42. Alison Stone, “Rhythm and Popular Music,” in The Philosophy of Rhythm: Aesthetics, Music, Poetics. Oxford: Oxford University Press, 2019, 141-155, 145. ↩︎
  43. Ibid., 145. ↩︎
  44. Ibid., 148. ↩︎
  45. Peter Cheyne, Andy Hamilton, and Max Paddison (eds). 2019. The Philosophy of Rhythm: Aesthetics, Music, Poetics. Oxford: Oxford University Press. ↩︎
  46. Tenzer. “How Many Kinds of Rhythm,” 205. ↩︎
  47. Ibid., 206. ↩︎
  48. Ibid., 211. ↩︎
  49. In The African Imagination in Music, Agawu rhetorically asks, “what is one to make of the contrast between an ostensibly casual African sense of time—dictated by sun and shadows (‘African time,’ which euphemistically denotes perpetual lateness)—and the amazing precision in timing that one finds, say, in xylophone ensemble performances?” (The African Imagination in Music, Oxford: Oxford University Press, 2016, 158). ↩︎
  50. Ibid., 205. ↩︎
  51. Kwasi Ampene, “Introduction and Provocative Musings,” Ghana Studies Journal, Special Issue on The African Imagination in Music (edited by Kwasi Ampene), Vol. 24, 2022, 69-75. On “ethno-theory,” see Kofi Agawu, On African Music: Techniques, Influences, Scholarship, Oxford: Oxford University Press, 2023, 129-147. ↩︎
  52. Jenny Judge, “‘Feeling the Beat’: Multimodal Perception and the Experience of Musical Movement,” in The Philosophy of Rhythm: Aesthetics, Music, Poetics Oxford: Oxford University Press, 2019, 76-90, 86. ↩︎
  53. Udo Will, “Temporal Processing and the Experience of Rhythm: A Neuro-Psychological Approach,” in The Philosophy of Rhythm: Aesthetics,Music, Poetics. Oxford: Oxford University Press, 2019, 216-230. ↩︎
  54. London, “Metric Entrainment”. ↩︎
  55. Ibid., 178. ↩︎
  56. Ibid., 178 ↩︎
  57. For a full analysis of the metric permutations in the gankogui/makwa pattern, see my “Afro-Electric Counterpoint,” in Rethinking Steve Reich (eds. Sumanth Gopinath and Pwyll ap Sion). Oxford: Oxford University Press, 259-302; 287-293. I am grateful to Rick Cohn for his helpful discussion of the pattern with me. For a properly detailed account of the structural set analysis of West African music, see Willie Anku, Structural Set Analysis of African Music. Vols. 1 and 2, Adowa. Legon, Ghana: Soundstage Production, 1992. ↩︎
  58. Willie Anku, “Circles and Time: A Theory of Structural Organization of Rhythm in African Music,” Music Theory Online, Vol. 6 No.1, 20 ↩︎
  59. Macamo, Elísio. 2017. “Urbane Scholarship: Studying Africa, Understanding the World,” Zentrum für Afrikastudien Basel, June 2017, https://www.youtube.com/watch?v=tooclmpUmnc ↩︎
  60. For an analysis of the miko transposition system in amadinda music, see my “African Music and the History of Time,” History of the Humanities, Special Issue on “Decentralizing the History of the Humanities,” Vol. 6, No. 2, 2021, 397-426. For a detailed account of the music of Uganda, see Sylvia A. Nannyonga-Tamusuza’s Baaksimba: Gender in the Music and Dance of the Baganda People of Uganda. New York: Routledge, 2005. See also Joseph Kyagambiddwa’s early account of this music in African Music from the Source of the Nile. Indiana University Press, 1955. ↩︎
  61. For an analysis of the metric implications of interlocking structures in the music of the mbira dza vadzimu, see my “Temporal Geometries of an African Music,” Music Theory Online, Vol. 16, No. 4, December 2010. ↩︎