Chris Stover

[Please stay tuned for a revised version of this talk with example slides embedded——coming soon!]

Eight Axioms for a Theory of Timeline Spaces

(invited talk, University of Oslo, 16 October, 2017)

Most of the existing literature on timelines focuses on one of two guiding themes. First is their morphological characteristics—for example, the asymmetrical ways that their interonset sequences traverse the metric grids with which they coexist. [Slide 2] Here are four well-known timelines, with labels that reflect common usage by practitioners and/or scholars; note, among other shared attributes, a binary relation with the four-count metric cycle in which the cycle beginning is articulated by a timeline onset, while the mid-point is not. Second is their formal properties, such as, in the case of the so-called “standard pattern” [Slide 3] to which the most scholarly attention has turned and shown here in a mostly metrically unmarked notation, [Slide 4] its rotational possibilities; that is, its potential to begin on any onset; [Slide 5] the isomorphic potential of its interonset sequence with diatonic collections in ordered pitch-class space, [Slide 6] and its maximal evenness and the maximal individuation of its beat-class members in terms of the singular constellation of intervallic relations each holds with the other six members. A small percentage of this literature begins to point to a more fundamental question: what is the role of a timeline in its musical context? What are timelines for? Or, better, why are there timelines?

Here is sampling of definitions that respond to the “why” question. [Slide 7] Kwabena Nketia defines a timeline as [Slide 8] “a constant point of reference by which the phrase structure of a song as well as the linear metrical organization of phrases are guided.” [Slide 9] For Gerhard Kubik, a timeline is “a regulative element in many kinds of African music” and one of the basic “principles of timing.” One page later Kubik locates the timeline as [Slide 10] “the structural core of a musical piece, something like a condensed and extremely concentrated expression of the motional possibilities open to the participants.” [Slide 11] Kofi Agawu defines timelines as “short, distinct, and often memorable rhythmic figure(s) of modest duration…[that serve as] point(s) of temporal reference.” Three years later he modifies this slightly to read [Slide 12]: “a distinctly shaped and often memorable rhythmic figure of modest duration that is played as an ostinato throughout a given dance composition.” [Slide 13] Meki Nzewi, lastly, asserts that a timeline is “a phrasing referent, not a structural referent in African musical thought and ensemble composition.” There have been many more definitions, of course; for example Ruth Stone’s bare-bones “a rhythmic pattern that continually repeats throughout a composition.” But in these six statements a number of compelling themes emerge—[Slide 14] phrase structure, regulative element, structural core, motional possibilities, temporal reference, ostinato, dance, phrasing referent. These will linger in the background as I theorize what I am going to call timeline spaces.

What do all of these rich, evocative descriptions have in common? They all foreground the distinctive character of a timeline—the importance of its recognizability, engendered in large part through repetition. More important, I think, is how they assert an expressive organizational role for timelines: what kinds of movement- or phrasing possibilities are articulable within a timeline space; perhaps even foreclosing certain kinds of music-temporal expressions. This latter is what Chris Washburne refers to when he examines salsa improvisations that are or are not “en clave,” following the discourse of a micro-community of salsa musicians in New York City. I’ll push against this characterization shortly.

Let me go straight into a musical example in order to provide some context and clarify some terminology. But first: timeline music is participatory music, so can I invite you all to join me in a demonstration?

[at this point the participants stepped a four-count metric cycle in a LRRL sequence and clapped the pattern 2-3-2-2-3 over the footwork’s articulated 3-3-3-3. When this felt comfortable we added a six-count (2-2-2-2-2-2 clapping layer.]

[Slide 15] Here’s one way of notating what we just did. The top staff shows the timeline you were clapping, known in the Cuban music community as a version of rumba clave [Click], which I’ll call 12-cycle rumba clave since its onsets scan to an underlying 12-cycle isochronous grid. The second staff shows what I’m going to provisionally call the main beat [Click], a four-count metric framework coextensive with rumba clave, and aligning with on what’s shown here as beat 1. I’ll sometimes refer to this as the 3-cycle pulse, since it parses the 12-cycle [Slide 16] into isochronous groupings of three. The third staff shows an additional isochronous layer, a six-count metric grid superimposed on the four-count one [Slide 17]. A hemiola relation obtains between the four-count and six-count layers; David Locke refers to this as the “cross-beat relation” [Slide 18].

Let’s examine these relationships more closely. None of this is new information: I describe all of this in detail elsewhere, as do David Peñalosa and others. But I’d like to draw your attention to a few key features. First, to repeat, the metric cycle or cycles are coextensive with one iteration of the timeline [Slide 19]. The concerted event of two metric pulses plus a timeline onset might, in fact, be a clue that guides our hearing of that event as a beat one [Slide 20]. [ASIDE: this is not quite true, but it can serve as a way in] Second, timeline onsets align with the four-count pulse at two locations, beat 1 and beat 4 [Slide 21]. I describe this elsewhere using language borrowed from Victor Zuckerkandl, as a wave that moves away from and then back to alignment. Third, if we think of a conventional (Western) definition of a four-count metric cycle as a series of beats of greater or lesser metric “weight” (that is, stronger or weaker beats), with beat 1 carrying the “strongest” metric implications, then beat 3 is usually considered the next strongest. I cannot overstate how careful I want to be with these words, strong and weak—since crippling misunderstandings about what each term means has driven practically every critique of Western notation and its putative inappropriateness for rendering non-Western musics in visual form. We can talk about this if you like, but for now please understand that, first and foremost, “stronger” does not mean “louder,” for me or for any reputable theorist of rhythm and meter (with the possible exception of Moritz Hauptmann). What is important to notice now is that, if beats 1 and 3 of the four-count cycle are taken to have a structurally significant relation to one another—a reading reinforced by a stratified account alongside the six-count cycle, à la Yeston or Lerdahl and Jackendoff,  [Slide 22]—then a binary relation with the timeline obtains: beat 1 aligns with a timeline onset, while beat 3 does not [Slide 23]. I mentioned this at the outset in regard to timelines’ morphological characteristics, but now I’m starting to push toward their expressive or organizing potential.

[Slide 24] Let me lightly paraphrase these three points, since they undergird everything I’m going to talk about today. First [Slide 25], timelines are coextensive with one or more metric cycles. Second [Slide 26], a wave of coincidence to non-coincidence and back animates the relationship between a timeline and what we’re provisionally calling the “main” metric cycle. And third [Slide 27], a binary relation exists between the timeline and the metric beginning point v. halfway point, or between beats 1 and 3 of the 3-cycle; in almost all cases this is again a relationship of coincidence v. non-coincidence. We’ll see that it matters little whether it is beat 1 or beat 3 that a timeline onset aligns with; what matters is that a timeline onset coincides with one or the other but not both.

Here’s one example that tests the rule [Slide 28], the fourth rotation of the standard pattern. However, because I conceptualize the first of each pair of short interonset intervals [Slide 29] as syntactically connected to, or even diminutions of, the longs they follow—and therefore conceive of the standard pattern as an elaboration of this underlying five-onset sequence [Slide 30], Jones’s “African signature tune”; Lehmann’s “syntactic background”; what I will call the 12-cycle son clave timeline; we can still speak of a binary relation where beat 1 is articulated by a structurally significant timeline onset whereas beat 3 is only marked by an embellished onset that links syntactically to the structurally prior onset that follows.

[Slide 31] Let’s modify our earlier example slightly. Will you please step and clap with me again? What we’re going to do is slightly different than what you see here.

[At this point participants again stepped a four-count metric cycle as LRRL, to which we added a 3-4-3-2-4 clapping pattern. When this felt comfortable we added an addition 3-3-2 / 3-3-2 layer.]

[Slide 32] Here’s one way to notate what we just did. The timeline in the top staff is also called rumba clave, this time stretched from the first version to conform to a 16-cycle grid [Slide 33]. The second staff again shows the four-count metric cycle [Slide 34]. The third staff shows two iterations of an onset pattern called tresillo [Slide 35], a Spanish word that has made it into common parlance in meter and rhythm theory. Tresillo outlines the pattern L-L-S, where each long is half again as long as the short; in the additive terminology of early African music studies, we could call this a 3+3+2 configuration [Click].

If we compare the 12-cycle and 16-cycle examples, several features jump out. First, the four-count “main beat” remains constant. Second, the first and fourth timeline onsets remain attached to their main beat correlates. And third, the second, third, and fourth timeline onsets, and the second and third of each trio of six-count onsets, are delayed; we might say they are pulled back in time by the gravitational attraction of the 16-cycle. [Slide 36] Here is an animation that shows one way to think of this: as a transformation of the 12-cycle version into the 16-cycle version: notice how those timeline and six-count onsets slide slightly to the right so that their onsets align with 16-cycle timepoints. [Click (animation)] Arrows highlight the three timeline onsets that shift. The 12-cycle and 16-cycle examples are isomorphic with one another in the sense that each should be conceived as a stretched version of the other. Each defines a context of which an interpretation of the five-onset rumba clave timeline is a key element. And, we’ll soon see, all three layers need one another in order to define that context.

Four more vocabulary words [Slide 37]. If we draw a dotted line through the halfway point of the cycle [Slide 38], again suggesting, through empirical experience, that that is a salient location to which we should be paying attention, then we end up with two syntactic halves that some sources describe as existing in a sort of call/response relation to one another. These are very commonly refered to as the “3-side” and “2-side” [Slide 39], since the timeline has three onsets in the first half and two in the second: “1–2–3; 1–2”. Some insiders, especially in Cuba, reverse these terms, “1–2; 1–2–3” [Slide 40], aligning the third onset cognitively with the second half, as an anacrusis. We should always be careful, of course, not to impose a discursive framework that forecloses potentially viable perspectives, especially when those possibilities arise within the very discourses of the practitioners whose music we are studying. Still, I will occasionally refer to these as the 3- and 2-side in turn [Slide 41], because there are other Cuban and diasporic practitioners that do use them regularly, and because they have currency across several discursive communities. Each second and third tresillo onset also has a name: bomba and ponche respectively [Slide 42]. David Peñalosa has refined these terms to describe a primary bomba and ponche corresponding to the 3-side of clave, and a secondary bomba and ponche that align with the 2-side [Slide 43]. I adopt Peñalosa’s terms–their significance will become increasingly clear as I refine my thesis today.

[At this point we sang and/or clapped the various layers of the rumba guaguancó basic strata.]

Here is one way to notate what we just sang [Slide 44]. The segundo and tumbador strata, which interlock to form a single hocketed melody, are shown in the top two staves, followed by catá, clave, and the less overtly articulated metric and tresillo strata. Accents are shown on the first and third metric stratum staff: one or both of these onsets are often articulated by a chekere. Note that the catá part is an embellished form of the rumba clave timeline, that also articulates the four-count metric stratum. Tumbador, the lowest drum, articulates primary and secondary ponche with an open tone, and primary bomba with a muted bass tone. Segundo articulates beat 3 of the metric cycle and secondary bomba. These tumbador and segundo parts, which are intended to be heard together as a single ongoing hocketed melody, serve as prototypes for improvisatory activity. The degree of improvisational elaboration changes from player to player, ensemble to ensemble, and subgenre to subgenre; there has been something of a gradual move to increased improvisational complexity as rumba has transformed over the last eight or nine decades. Some of that elaboration, importantly, is by subtraction, so for example in the contemporary rumba subgenre guarapachangeo, the tumbador–segundo basic melody is sometimes nearly impossible to discern. I’ll come back to this important point from another perspective shortly.

Here is an example that illustrates how these performance strata come together, as well as how they might be varied and transformed in the early moments of a performance [Slide 45]. In this transcription of Los Muñequitos de Matanzas’s 1987 recording of “Parece mentira,” the vocal melody, beginning with a wordless diana before progressing to the composed melodic cuerpo or head, is shown in the top staff. The second staff shows quinto, the lead drum, which plays a mostly improvisatory role within constraints imposed by the clave–tresillo–meter assemblage. The third and fourth staves show segundo and tumbador respectively, and the fifth staff shows catá in upward-stemmed notes and clave, the timeline, in downward-stemmed notes. The metric cycle and tresillo are not notated, since they are not explicitly articulated, but please notice their implied presence. I have notated everything using two bar measures for what we might call each timeline measure, first of all because this follows common notational practice, and second because there are aspects of the musical surface that “flip” the perceived downbeat such that the 2-side of clave is heard as the first half of the cycle. The vocal entrance just before what I’ve indicated as measure 12 is one such example. As we listen, I suggest that you focus on the segundo–tumbador dialogue; especially how they vary their basic melodic shapes. As we’ll zero in on shortly, this version of rumba, rumba guaguancó matanceros—rumba from Matanzas, Cuba—has a simpler basic melody: instead of <sing Havana melody>, <sing Matanzas melody>. Let’s listen. [Play; advance slides 46, 47, 48, 49, 50]

There is a lot that we could talk about in this example, in terms of improvisatory dialogue, microtiming variation, tempo, inter-ensemble relationality, the role of dance, social context, and much more. Perhaps we can touch on some of these in the discussion that follows. For now, though, I’d like to use this example to summarize a few key points that I’ve either stated explicitly thus far, or that have lurked under the surface of my narrative. Here, then are some general observations about timelines and their contexts that follow from my exposition so far.

1)    [Slide 51] Both the cycle beginning and halfway point are highly salient locations around which music and dance are organized.

2)    [Slide 52] Timelines exist in asymmetrical relations of alignment and non-alignment with a “main” metric level.

3)    [Slide 53] Silent or “ghosted” pulses can be as structurally important as articulated ones.

4)    [Slide 54] Meter in timeline music is multidimensional or polymorphic, conceived either as a main beat and at least one secondary beat  or as the co-occurrence of multiple, hierarchically equal metric strata.

Multidimensionality is David Locke’s term; polymorphism is the word I prefer to use to conceive of a cycle’s multistable temporal gestalt.

5)    [Slide 55] This plural metric background is interwoven or braided with the surface features of the performed music.

I borrow the term “braiding” from James Burns.

6)    [Slide 56] Multiple layers, not just timelines, come together to contribute to the identity of the cycle that grounds timeline musics; there may be three or four layers, or there may be many more than that.

Meki Nzewi describes this as the ensemble thematic cycle; Nzewi, I believe, has come closest to accurately describing, without overdetermining, the role that the timeline plays in its ecological environment.

7)    [Slide 57] 12-cycle and 16-cycle timeline spaces can be thought of as beat-span transformations of one another, with a primary locus of that transformation in the relationship between the isochronous six-count 2-cycle in 12-cycle music and tresillo in 16-cycle music.

We saw one example of this with the comparison between 12-cycle and 16-cycle rumba clave. This is the focus of some other work of mine; its relation to today’s talk lies most pointedly in how it might affect microtiming fluctuations that arise within the context of a performance.

8)    [Slide 58] Different strata in timeline music are articulated not just by event onsets, but by differently-directed motions that we might describe as “away-from–back-to” (following Zuckerkandl) or “up–down.”

This is, again, to reiterate emphatically that event onsets do not necessarily signal beginnings, and that gestures comprised, say, of several off-beat onsets can be considered as a single gestalt that conveys a sense of directional impetus. We’ll see and hear this clearly when we turn to a samba example shortly.

These eight observations form the empirical underpinning of my current research. I would now like to take a more speculative turn for the remainder of my time today, and present a series of axioms for a theory of timeline music [Slide 59]. The following axioms appear in a slightly different form in a forthcoming article—such is the nature of academic publishing that ideas have often evolved, sometimes radically, by the time a theory makes it to publication!

1) [Slide 60] Timelines occupy and define a liminal space between meter and rhythm.

Are timelines rhythmic or metric phenomena? This is a question that has impinged on a great deal of recent research, from Ruth Stone defining a timeline as “a rhythmic pattern that continually repeats throughout a composition” (thereby locating them fully on the rhythm side of the equation, even if only by invoking that word) to Justin London’s speculations as to whether a timeline might articulate a non-isochronous meter. I begin, then, by considering timelines as neither rhythm nor meter, but also at the same time both rhythm and meter, opening both to a transitional space, “between and betwixt” in Victor Turner’s words, neither/nor and both/and and something else entirely. This is probably my most radical claim today, but it grounds all the rest of my claims, since we’ll soon see that every aspect of temporal process in timeline music has an essential liminality to it that pushes against structuralizing claims or binary co-determinations.

In other words, in timeline music there are rhythmic phenomena, there are metric phenomena, and there are timeline phenomena. But more important, all of these flow into one another, disrupting any sense of categorical distinction. For example, a timeline becomes-rhythm, in a Deleuzian sense, as a tending toward, or as a becoming-actual of a virtual that was inherent in the original condition or conception. It becomes-rhythm when it is articulated on a bell: a pattern gets embellished, timbral and loudness variations are enacted, microtiming variations unfold. Like the orishas of the Yoruba tradition, a timeline is both real and irreal (in Nelson Goodman’s sense): made manifest through real actions in music–dance–ritual space, at once metaphysical and here-and-now, transcendental and immanent.

When I say that a conception of timelines–as–liminal grounds the rest of my claims, I mean exactly that: thinking in terms of timelines is a heuristic for deconstructing binary structures in such a way that the original terms still hold, but the borders that determine where one ends and the other begins are made essentially and productively porous. This is the definition of liminality. In her study of the congada tradition in Minas Gerais, Glaura Lucas describes a process practitioners call repica (or repique), “transforming reality in 2 into reality in 3,” which manifests as a constant process of stretching a nominally duple rhythm almost, but not quite into its triple equivalent. The performed in-between utterances are intended to express, at one and the same time, both realities. In other words, the transformation of duple into triple is not the replacement of the former with the latter, it is the staking out of a new in-between space that is both duple and triple and something altogether different. And this is essential to congado practice:

[Slide 61] “In this way, certain repeated rhythmic figures, arising within similar musical contexts, present an internal articulation that vary between duration that now tend toward a binary division, now toward a ternary division, but executed, most of the time, as realizations in an intermediate duration” (Lucas 2002, 126; translation mine; emph. added).

[Slide 62] Axioms 2 and 3 go together:

2) [Slide 63] Timelines are dependent on meter for their identity and function.

3) [Slide 64] BUT they also partially constitute meter; that is, they partially determine how meter behaves in the music and music-dance contexts in which they occur.

Axiom 2 is basically an argument against timelines as somehow being metric—as articulating a non-isochronous metric layer with or without one or more coextensive isochronous layers. Claims of NI-meter status, from A.M. Jones and Rose Brandel to Justin London and beyond, all hinge on an assumption that event onsets determine metric salience, a claim that doesn’t hold up even in the more metrically incontrovertible common-practice musics that most theories of meter and rhythm were designed to help explain. Christopher Hasty makes this point with great force and subtlety, even if not exactly in these words.

If timelines are partially dependent on the organizing force of a metric grid (or multiple strata of metric grids), they also help determine how we experience and organize those grids. This is what Axiom 3 suggests. Note that I am conflating two epistemological perspectives here, which is also intentional. The axiom itself points toward the empirically-there, how meter behaves, as impinged upon by a timeline, a complex of properties to be revealed through experience. But my verbal lemma turns to the experiential, to our embodied perception of sonic phenomena. The space of Husserlian phenomenology is also a liminal space: at once a turn to the analysis of experience and an acknowledgement that all experience is an experience-of that proceeds right from the middle of the action of experiencing that which is experienced. So to develop an understanding of how timelines help determine how we can or should experience or organize meter is to extend Steven Friedson’s Heideggerian notion of “being in meter” to go something like “being in the lively now-unfolding meter–rhythm–timeline assemblage” as a place of action, events, and affect.

This leads, then, to my fourth axiom:

4) [Slide 65] Timelines only exist within their ecological contexts; when extracted or relocated into other contexts they lose their identity as timelines.

This is partially self-evident from the two preceding claims. But it is also a controversial position to take, since it suggests that some musics that have rhythmic patterns that unfold as the same series of interonset intervals as common timelines; say, “Hey Bo Diddley,” do not get to count as timeline music. So we may hear a “timeline rhythm”—son clave in Miami Sound Machine or Santana’s “Oye Como Va”, but that does not make these timeline musics. Please note that this is not a value judgment: I make no claim as to the aesthetic appeal of timeline v. non-timeline music. I only claim here that timeline music is a specific category of musical expression, and one of its defining features is a multi-stable array of temporal strata in a complex ecological relation with one another. More on this in a moment.

So the presence of an interonset pattern that scans to that of a timeline does not, in itself, qualify a musical expression to be timeline music. A roughly corrollary claim emerges in my fifth axiom:

5) [Slide 66] A timeline does not have to be audibly articulated for something to be timeline music.

There does not have to be someone literally articulating the timeline on a bell or claves or a hoe blade or shaker for the timeline to operate on other performance layers in its music or music–dance ecology. A great deal of the batá repertoire from Cuba is structured such that the asymmetrical impulses of a virtual timeline constrain or enable certain performance criteria, including but not limited to how the individual drum parts interact and where in the metric cycle certain kinds of improvised expressions are allowable. But in most cases no single instrument is sounding the timeline. [Slide 67; play] There are also many examples in West and Central African vocal music. Bertram Lehmann provides several illustrations of the latter; his way of characterizing this phenomenon is that what he calls the syntactic background is “not externalized in the music.” We’ll consider an interesting middle-ground example momentarily, when we turn to samba.

To reiterate my fourth and fifth claims, then: the presence of a timeline does not guarantee timeline music, and timeline music need not have an overtly articulated timeline onset pattern. To fold these claims into the trio of axioms that I began with, these are also expressions of liminality: of opening liminal spaces that call categorical thinking into question, to acknowledge and embrace the ontological premise that contexts are complex, are always in processes of coming into being, are products of conscious and unconscious, culturally-embedded actions and relations. Take, for example, this well-known performance from Central African Republic. [Slide 68; play] There is no timeline overtly articulated, but the durational profile of the monophonic melody is impinged upon by a virtual timeline; we might say that it is constrained by the rhythmic impetus of the virtual timeline. I have included a six-count metric stratum in the lower staff, which reflects how I tend to hear the metric organization of the melody. Notice how the three sung melody notes in each cycle reflect an on-beat v. off-beat double relation: down in the first half, up in the second [Slide 69]. If we reorient to a four-count metric cyle these relations hold [Slide 70; play again].

Here is another small example, a prayer song, also from Central African Republic [Slide 71]. The repetitive melody sung by the lower voices outlines the five-count 12-cycle timeline that, as I described earlier, has been dubbed the “syntactic background” or the “African signature tune.” This unfolds alongside an isochronous four-count handclap. The first three melody onsets, which are reinforced by the upper voices, also align with a virtual (but also real, as Deleuze would say) six-count metric cycle. [Slide 72; play] So these are very much examples of timeline music; they express, in their own minimalist ways, all of the axioms I’ve suggested so far: they articulate a space that is both rhythm and metric and somewhere in between the two, they depend on a multi-metric substrate but also help guide our way through those strata, they are fully embedded in the ecological conditions within which they are operating, and they operate even if not overtly articulated. Furthermore, these songs, and many more like them, are created within a cultural ecology that also includes a great deal of overtly articulated timeline music; for instance, drum–song–dance music one layer of which is a bell playing a timeline. These various community music-making practices have, over many generations, bled into one another in mutually emboldening ways. If I may somewhat radically extend a concept developed by Godfried Toussaint, these pygmy songs might well have pre-timeline ancestral roots; like the bell patterns that we usually invoke when we talk about timelines, they might express contemporary evolutionary streams through which the multistable temporal nexuses that I suggest define timeline musics as a class of musical expressions have developed. Of course this is highly speculative. But like many of my central claims and tangential asides, I offer it as a small provocation for further discussion—perhaps to expose a future research collaborator?!

About those multistable temporal nexuses, then. My sixth axiom is that

6) [Slide 73] Timeline spaces unfold as malleable, non-hierarchical strata.

A stratified reading, whereby greater expressive “weight” is given to cycle locations where more event onsets co-occur, is a relatively useful way to conceive of metric, or at least cyclic, structure in timeline music. With some caveats though: for one important example that I cannot overstate enought, we must take care not to overdetermine the kinds of event onsets that correspond to the beginnings of articulated sounds. Remember that silence is an event too! As is continuation.

The strata that comprise what I’ll call the basic layers of timeline space include, at the least, the four-count isochronous metric pulse, either a coextensive six-count pulse (in 12-cycle timeline music) or two iterations of tresillo (in 16-cycle contexts), and the timeline itself. There may be more too, but at minimum timeline music requires these three. These strata coexist; they need each other. But—and this is an important point—none of them are hierarchically prior to any of the others: all can and should be able to be conceived as a provisionally stable ground against which the others are measured, depending on where we are turning our attention and why. Furthermore, in the context of performed expression—the only context that matters!—they can morph into one another in manifold ways. For example, a lead drum improvisation, or a dancer’s footwork or arm gestures, can elide the distinction between meter and timeline strata by variously highlighting one or another. We experienced another example in the catá part in our rumba guaguancó example, which articulated aspects of clave and four-count meter through the repeated performance of an additional synthetic layer.

What is at stake in this claim? By asserting that multiple recursive and non-recursive metric layers, meter-like temporal phenomena (like tresillo), and timelines exist in a kind of destratified, democratic, malleable relationship, we can open a space to radically reconceive what syncopation means. If different forces at play in timeline music all swirl together affectively to co-constitute one another, then anything and everything can and should be able to be read as a cognitively stable terrain along which some other force is syncopating.

Here’s a small example [Slide 74]. The top staff shows 12-cycle son clave, the middle staff the six-count isochronous metric cycle, labelled “2-cycle”, and the bottom staff shows the four-count metric cycle or “3-cycle.” If we think of all three of these as non-hierarchical strata, none reducible to any other, then we have an array of differently-directed syncopation relations that we can consider. The first timeline onset, for example [Slide 75], aligns with onsets in both metric cycles, but the last timeline onset [Slide 76] aligns with the 3-cycle but syncopates with the 2-cycle. In contrast [Slide 77], the second and third timeline onsets syncopate with the 3-cycle but align with the 2-cycle. The fourth timeline onset syncopates with both metric cycles [Slide 78]; the phenomenon of moving further away from alignment before returning supports my Zuckerkandlian “wave” thesis. Meanwhile, [Slide 79] every second and third 2-cycle onset syncopates with the 3-cycle, and vice versa: [Slide 80] every second 3-cycle onset syncopates with the 2-cycle. Please note that by “onset” I do not (necessarily) mean the enactment of a performed, sounded event: an onset can exist in the cognitive or imaginary or virtual realm (pick your favorite epistemological framework!).

And of course we need to reverse these terms too: each metric cycle exists in a unique syncopation-relation with the timeline. [Slide 81] Three 2-cycle onsets syncopate with the timeline; [Slide 82] two 3-cycle onsets do so. Again, this is to suggest not that timelines are somehow metric, but that meter is not the only kind of thing that can serve as a comparatively stable framework along which syncopations can be measured.

All of these multiply-directed syncopation-relations hold in 16-cycle timeline space as well.

Here’s an interesting illustration [Slide 83]. The melody of Joe Ford’s “The Vonce,” recorded by Jerry Gonzalez and the Ft. Apache Band, variously expresses the four-count metric cycle, tresillo, and a “2–3” rumba clave timeline. After a two-quarter-note anacrusis, a series of descending half notes culminate [Slide 84] with a striking syncopation with the fourth clave onset. In the discourse of that micro-culture of practitioners in New York and elsewhere known colloquially as the “clave police,” this fourth melody onset would be deemed “cruzao”; cross-clave; wrong. Yet here is Jerry Gonzalez, a bona fide pioneer of Afro-Cuban jazz, articulating that “cruzao” onset not only here, but consistently through this opening melodic statement [Slide 85]. Furthermore, several melody onsets align with tresillo, syncopating with clave onsets or the four-count metric cycle or both [Slide 86]. And then, as would be expected in timeline music, some gestures express what we might call diminutions or elaborations of clave onsets, syncopating with the metric level [Slide 87]. Let’s listen. [Play]

This is a recording that I played on; the band Sonando covering Jerry Gonzalez’s version. I like this recording very much, but notice how in Gonzalez’s original recording the distinction between stratified layers is reinforced by the melodic interpretation: the melodic gestures that orient toward the metric cycle are played legato, while those that articulate tresillo or clave are played with a crisper, more forceful articulation For this reason, Gonzalez’s recording serves a wonderful didactic purpose. [Play]

Let me make one more speculative aside here. [Slide 88] Taken together, the 12-cycle son clave timeline and six-count and four-count isochronous metric cycles articulate nine of the twelve 12-cycle positions. The same configuration with the rumba clave timeline articulates ten onsets [Slide 89]. The standard pattern timeline adds one onset further [Slide 90], op<11>, leaving op<1> the only location unarticulated [Slide 91]. A topic for future study might be the degree to which that unarticulated second cycle position, in essence conferring agogic accent status [Slide 92] to the triply-articulated first order position, reinforces our sense that is indeed the cycle-beginning. I’m not prepared to make this claim with any confidence, but it is, shall we say, interesting.

To turn to a malleable conception of non-hierarchical strata opens up some potent conceptual possibilities. First, we can sidestep the sticky question of whether tresillo, as the 16-cycle correlate to the 2-cycle in 12-cycle timeline music is or is not metric. If strata are malleable, they can fold into one another: a metric cycle becomes a tresillo cycle, or vice versa. Second, since any stratum can be attended to as a comparatively stable ground against which we might measure other layers of musical activity, we can consider proliferating ranges of possibilities for understanding improvisational provenance: where some performed event derives, or how it is functioning within larger contexts. My invocation of an array comparatively stable grounds, depending on how one shifts one’s perspective, suggests multi-stability, but this is a multi-stability that can be (and is, by enculturated listeners and performers) attended to as multiple co-occurring, co-constitutive temporal strata rather than rapid shifting between levels in fluid acts of figure–ground reversal. Third, this states outright what many earlier studies suggest: that phenomena like off-beat cycles, repeated non-isochronous temporal gestures (like tresillo), and timelines are more than surface features undergirded by a steady metric grid. They interact multi-directionally as equal contributors to the ongoing improvisatory structure of timeline music. It is also important to reinforce that my usage of “syncopates with” is intentional and rhetorical: I am deliberately eschewing the more common “syncopates against,” in part to soften the implication of hierarchy between the figure–ground implications of the latter, in part to reinforce the communal and cooperative ideal of African and diasporic African musical practice such as Kofi Agawu describes so eloquently.

[Slide 93] Returning to our growing list of axioms, [Slide 94]

7) The concept of call and response plays a legislative role in the constitution of timeline spaces.

With two caveats, though. First: I think the terms antecedent–consequent better capture the co-constitutive relationship between performed events, in most cases. This is because the terms antecedent and consequent, and the concepts they represent, need each other for their identity: antecedent is that which comes before a consequent, and gains its identity as such retroactively; consequent is that which follows an antecedent. More important, this is not a binary process: not call à response à call à response (except in special cases, like congregational response singing or salsa moñas), but concatenated expressions: a call engenders a response that itself becomes a new call, engendering a new response, and so on. This democratizes musical process, since every performance layer and every improvisational utterance can be a new call, if taken up as such by a response gesture. This last point is important—a call becomes meaningful as such through the ways it is taken up in a response. There are many ways that that taking-up can occur: it can be mimetic, for example, or it can proceed through what we might call gestural “completion” (say, a melodic or rhythmic gesture that seems to metaphorically answer a question left open by the call). There are very often micro-stylistic constraints that suggest certain ranges of responses to a given call. For example, beginning with the basic tumbador–segundo melody in Cuban guaguancó matancero [Slide 95; sing], a variation on what we sang together earlier, a tumbador elaboration might enact a brief dialogic chain, where one of a small handful of response gestures is expected, any one of which invites one of a small number of next responses. If tumbador plays this elaboration [Click; sing], a likely segundo response is this double onset [Click; sing], which becomes a new call to invite a next tumbador response [Click; sing].

But second, even this more nuanced account oversimplifies the dialogic nature of timeline spaces in two ways: it focuses overly on a <this, then-this…> reading of musical process as an ordered string, and it overdetermines a single cause as that which brings a response into being. Calls and responses are components of complex ecologies, and we should always be wary of ascribing causal status that reduces away this complexity. They operate transversally too. A now-ongoing gesture impinges affectively on co-occuring gestures, which can transform those gestures even while itself being transformed by them. An individual participant is affectively attuned, to borrow psychologist Daniel Stern’s term, such that she will respond to the affective implications of a temporal or transversal “call” in a way that expresses a range of internalized, embodied, pre-cognitive response behaviors. A great deal of timeline music goes by at a rapid-fire pace, and turning to affect in this way reveals some possible ways of getting around the intention problem, as well as the entrainment problem as it relates to what is or is not cognizable. If as Brian Massumi describes, affect is the “dark precursor”—Gilles Deleuze’s term—that precedes cognition, the space between action and reaction, or, better, between event and action, is the pre-cognitive space of context formation; where what William James calls “something doing” happens.

I suspect this is all at the heart of how groove works Note, for example, how it builds on one of Anne Danielsen fundamental contextual principles: [Slide 96] “All of the instruments, including the vocals…[form] small but significant rhythmic gestures that are linked in every direction. The groove has become an intricate fabric of sharp percussive sounds in which one sound brings on the next: the texture of the music has changed from horizontally divided layers of sound to a rhythmic patchwork.” I would enjoy further discussion around this possibility, as a kind of ontological framework for thinking about groove studies.

My last axiom is slightly more audacious:

8) [Slide 97] Timeline spaces evoke circularity in a non-trivial way.

This is the focus of another area of my current research, in which I’m examining Brazilian popular and folkloric music in order to develop a conception of circularity as a real organizing force—this through factors such as multiple non-coinciding metric cycles, harmonic motions that obscure or evade cycle beginnings, textual allusions, and much more.

Willie Anku once asserted that “African music is perceived essentially as a circular concept rather than linear.” This because much African drum/dance music involves repeated cycles that loop back around on themselves in a way that expresses a different sort of temporal unfolding than, say, a Hegelian teleological or deterministic or organicist process. We can debate how true this; I suggest that its truth-status is not particularly important. But if this is our position—that repeating cycles such that the tail of one strand is fed into the mouth of the next amounts to a circular ontology—then most, if not all, metric music is circular, in which case there is nothing special about claiming timeline music to be so. Kofi Agawu cautions us not to overdetermine circularity as a cosmological or metaphysical given, and I think we should heed his call to sobriety; after all, a circular reckoning whereby an end-directed impetus gives way to a next beginning is exactly the claim the Zuckerkandl makes of meter in Western art music. Still, there are other compelling reasons to think of timeline music as circular. An important one manifests when we turn back to the notion of non-hierarchical strata, none grounding any other. I’m thinking here of David Locke’s offbeat cycles, for example, which provide fundamental entrainment orientations that need not reflect a grounding “on-beat”: the on-beat can be the off-beat to the offbeat! Many of these strata have unique directional profiles, with accumulations of directed motions that point to non-coinciding beginning points, a consideration of which can help us downplay a cycle downbeat as any more of an arrival than any other temporal location. This, by the way, is where Jones and Brandel and other early Africanists, with their oft-criticized transcriptions with staggered barlines and Stravinskian odd-meter phrases, turn out not to be as wrong as we sometimes assume. This is abetted by thinking about timepoints themselves; or rather, thinking away from timepoints and toward the more temporally elastic spans of time that n-cycle notation is at best merely a synecdoche for. There are other criteria, but let me focus on these as I turn to one final example, from Brazilian samba.

[Slide 98] Here is a transcription of an excerpt from Cartola’s “Alvorada,” a well-known samba composition. The top staff is an abstraction of an underlying tamborim pattern; this is one of a handful of such patterns from which a performed tamborim part derives via a series of improvised variations. The second staff shows the part that was actually played during this excerpt from the recording. I will discuss the relationship between the prototype and played part in a moment. The third staff shows the alternating muted and open tones of the surdo or bass drum. The fourth staff shows the vocal melody and harmonic progression in lead sheet notation. In the fifth and sixth staves we see the cavaquinho and sete coro parts respectively: the high-pitched guitar that outlines the harmonic progression in a rhythmic pattern that also transforms the basic tamborim shape from the first staff, and the seven-string guitar that provides a melodic counterpoint to the lead vocal. All of these layers express cycles in differently overlapping ways. [Play]

In “Alvorada,” we should also read as “away from,” or what I will describe as “upwardly-directed,” certain groupings of event onsets that syncopate with a metric substrate, and as “back to” or “downwardly-directed” those groupings that reorient back alongside it. For example, we can measure the abstract tamborim pattern against—or, better, with or alongside—an 8-count (2-cycle) metric grid [Slide 99]. Four event onsets coincide with 2-cycle OPs in this reading; five onsets syncopate with it. These are shown by circles and squares respectively. Furthermore, the upward-directed events form a series that points ultimately to the first downward arrival; this is shown by the curved arrow above the system. Turning from the 2-cycle to the 4-cycle [Slide 100], a different pattern of relationships is revealed, with two contiguous points in the cycle articulated by event onsets in the tamborim pattern, and two not. Again, these are highlighted by circles and squares, with a curved arrow to show the large move from upward to downward motions. Importantly, the downward-directed arrival at OP<2> of the 4-cycle does not coincide with the tamborim pattern’s return to alignment with the 2-cycle [Slide 101].

If we turn, as we should, to the played tamborim part rather than the prototype, we will find that these relationships not only adhere, they are transformed and intensified. Let’s look at the relationship between the played tamborim part and the 8-cycle metric grid [Slide 102]. In this excerpt, each “away-from” series of onsets is played in accord with the abstract model [Click]. After each “back-to” arrival, however, the ensuing figure is transformed on each iteration [Click]. The first (measure 2) reproduces the prototype. The second (measure 4) enacts a temporally extended transformation of this new figure: [Click] the four-onset “downward side” is stretched to form a 3:2 cross-rhythm against the cyclic grid. A loosely transformative process continues: [Click] a third four-event gesture is played in measure 6, where after the initial downward-directed arrival a delayed second onset suggests perhaps a repeat of the 3:2 cross-rhythm, but instead [Click] enacts the beginning of a four “away-from” sixteenth-note figure culminating on a new consonant arrival. And then in measure 8 the final two-onset “back-to” move from measure 7 is repeated two times [Click]. All of these variations are part of the improvisational flux of the performance.

[Slide 103] The surdo part is straightforward; oscillating between muted and open tones. That open tone is a big expansive sound, marked in consciousness among practitioners as an arrival point; a back-to. So the muted tone marks an up or away-from, the open tone a down or back-to, and together they define a cycle half the length of that articulated by the tamborim. Since the muted tone is very soft, we might simply hear it as the end of the open tone’s duration. The surdo stratum demonstrates that timbre and event endings also play roles in articulating away-from and back-to motions.

[Slide 104] Melody and harmony in “Alvorada” form an assemblage that articulates another kind of away-from/back-to relation. The harmonic motion ii–V–I unfolds a progression away from, then back to tonal stability [Click]. The harmonic rhythm reinforces the cycle beginning point on the downbeat of measure 3. This is important, since this downbeat is not saliently articulated by other strata, so its status might be called into question without this harmonic arrival. The first melodic phrase, likewise, skips along a series of chordal arpeggiations prolonging the appoggiatura that resolves at the end of the word “beleza” [Click]. Note that this resolution is displaced from the metric cycle but aligns with the tamborim onset; in fact, all beginnings and ends of the melodic line coincide with onsets in the tamborim prototype [Click]. In contrast to this flow of rhythmic alignments, the sete coro part, playing a melodic counterpoint to the sung melody, provides frequent downbeat departures or arrivals that syncopate with the rhythmic impetus of the tamborim–voice assemblage [Slide 105]. Non-alignments with the tamborim part are circled in this example.

In “Alvorada” we experience multiple away-froms and back-tos or ups and downs. These are powerful metaphors: up (or away-from) equates to a sense of non-equilibrium, suspension, or floating that is followed by equilibrium, resolution, or groundedness. Each double motion unfolds periodically, articulating different cycle lengths and different n-levels within a given cycle length. These are, importantly, non-coincident with one another: multiple back-tos point to multiple arrivals, which engender different away-froms. But they all articulate cycles. [Slide 106] Here are the cycles I have illustrated thus far, articulated by no fewer than six away-from—back-to motions, and represented as upward and downward arrows to reflect that these are differently-directed motions, not simply event onsets. [Slide 107 to 113; note syncopations with timeline in sete coro part].

Among other things, all of this reinforces a sense of circularity by directing one’s attention away from any single arrival as a point of cyclic reference. This is not to say that there is no reference point for the cycle—there is, always, a downbeat—but that it is experientially problematized by the nexus of non-coinciding event onsets and accents that traverse its cyclical unfolding.

I’m going to wrap up here, because I’m most interested in opening up a discussion about all of this. What I have presented are eight axioms, by which I mean eight propositions for formulating a robust theory of timeline music, the verity of which will play out through further consideration, analysis, development, and refinement. Much of this involves pushing against sedimented ideas about what is or is not entrainable, what does or does not get to count as figure or ground, and what figure and ground even mean. It also demands that we become sympathetic to paradoxical constructs, like the notion that something can be both meter and not meter at the same time. The productive aporias that I believe animate timeline spaces might well derive from certain flavors of paradoxical African metaphysics, like the plural identity of the Yoruba orisha Eleguá, who guards the crossroads, is both young and old, both female and male (and in-between; an originary story for non-binary gender identity!), policeman and provocateur, helpful friend and devious trickster. That Deleuzo-Guattarian “and” is important, since it is through an expressive and creative “and” that identities proliferate, plurality is celebrated, and binaries melt away. [Slide 114]