Smooth and Striated Spacetime, Part 1
In his Darmstadt lectures, the composer and conductor Pierre Boulez presented a framework for conceptualizing musical space and time based on the interplay of two countervailing properties which he referred to as the smooth and the striated.[1] This schema was later taken up by Deleuze and Guattari, who expanded its scope well beyond the terrain of musical form, treating the smooth and the striated as contrasting propensities at work in a wide array of phenomena.[2][3] For some time now I have been using these notions in my clinical practice. In this essay (in two parts), I hope to introduce the concepts of smooth and striated space and time—beginning at their point of origin in Boulez’s music theory—and then explore some of their possible applications in a clinic oriented by the topological paradigm of the late Lacan.
For those without a strong grounding in music theory, the only prior conceptions needed for understanding Boulez’s paradigm are the following:
Pitch is the property that tells us how relatively high or low a musical tone is. A higher pitch (think of the chirping of birds, the flutes in an orchestra, or the whistling of winds through an open terrain) corresponds to a higher frequency—that is, a faster oscillation of the sound wave. A lower pitch (think of the bass in a jazz or rock band, the rumbling of a thunderstorm, or maybe a whale’s song) corresponds to a lower frequency / slower oscillation. Pitch and note, though not identical concepts on a technical level, are often used interchangeably (including in this essay).
Interval is the measure of the difference between two pitch frequencies. A very high pitch and a very low pitch would have a relatively large interval, while two higher pitches, for instance, would have a smaller interval. We tend to conceptualize an interval as representing the “distance" between two notes.
Pitch space is the abstract terrain in which musical sounds emerge and interact, producing the various effects that constitute essential elements of musical form (i.e. melody, harmony, etc.). There are varying maps and models used to represent the spatial world of musical pitch.
In putting forward the concepts of the smooth and the striated, Boulez is trying to conceptualize a principle of musical “relativity.” As he points out, in actual fact there isn’t a single pitch space, but rather an inexhaustibly large set of possible musical worlds. Each of these unique spaces can be characterized by the particular manner in which they subdivide the spectrum of pitch frequencies. The system that has predominated in the Western world—based as it is on the interval of the octave, which is then subdivided into a scale of 12 pitches, each separated by the interval of a half step or semitone—is only one such way to delineate a musical space.
To refer to the subdividing of pitch space, Boulez uses a rather strange term: striation. This word has certain applications in the natural sciences, but to my knowledge, Boulez is the first to employ it in a musical context. In geology, striations are the grooves or furrows that we often observe on the surfaces of rocks or minerals, producing the effect of segmentation and differentiation in what would otherwise be a seamless continuity. If the connection to music seems obscure, simply picture a piano. Notice that the surface of its keyboard is segmented into separate components (the 88 keys, each of which produces a distinct pitch frequency). The structure of the keyboard—like the striated surface of a rock—offers us a clear and concrete representation of a striated musical space.
Now let’s do a thought experiment: picture the keyboard again, but with more than 88 keys. In fact, make it many, many more—but still contained within the same space. In reality, it doesn’t matter if our augmented piano is the same size, but for conceptual purposes, it might help to imagine all those added keys having to become very small in order to be scrunched into the same keyboard space. What does this picture represent? This is another kind of striated pitch space, one wherein the intervals between pitches would be far smaller. This means that within the same overall sound environment (that is, the same spectrum of pitch from low to high) we would have a lot more sonic possibilities than in our standard 88-key piano. Those pitch frequencies that we can’t get on the standard instrument (the ones that fall between the cracks, as it were)? Well, we’d be able to play a lot more of those.[4]
The crucial point to grasp about striation is that no matter how many subdivisions are created, our pitch space will always be a set of discrete elements (that is, distinct and separable locations along the spectrum of pitch). No matter how fine-grained our structure becomes, there will be gaps—unrealizable pitch frequencies that fall between the cracks.
In a second contrasting thought experiment, we can remove the keys entirely and generate a radically different kind of pitch space. In this imaginary scenario, it might be helpful to picture the surface of the keyboard having been transformed into a unified whole, without any subdivisions. Even though we have removed its striations, it still constitutes a pitch space. What does this mean? Well, if we were a small insect walking along this surface, we would find that wherever our little legs stepped, a sound would be produced. In this sonic world, there is a practically infinite set of possible pitch frequencies. The matter becomes clearer if we imagine ourselves to be smaller than an insect. Maybe we are a microorganism oozing across the surface, generating infinitesimally small transformations in pitch frequency as we go, the music oozing with us, as it were.
This strange keyless (!) keyboard that we’ve conjured is a representation of what Boulez calls a smooth space. Unlike striated spaces, which are always constituted by discrete elements, smooth spaces are continuous.[5] That is to say, in a smooth space, the number of points (i.e. pitch frequencies) in the space goes to infinity.
These references to the infinite and the infinitesimal should alert us to the fact that a smooth space (in an absolute sense) is unlikely to ever be concretely realized. Even with the aid of electronics, we can only imperfectly approximate a continuous spectrum of pitch. As a concept, smooth space functions more like an abstract principle through which the complementary term in the dialectic, striation, takes on its precise significance. When we approach this dialectic in its totality, we have an interplay of two heterogeneous coalescences of spatiality and subjectivity: one in which a subject moves through a space (or, more radically, is themself a space) that is discernibly punctuated by some system of demarcation, and another in which the subject-space lacks any such system. In a striated space, the possible moves—no matter how numerous—are already delineated; in a smooth space, the next move is always uncharted.
For Boulez, the smooth-striated dialectic is applicable not only to space but also to time. The significance of smoothness and striation for musical time has not so much to do with questions of speed (i.e. fast or slow tempos) but rather the degree to which a regular tempo is operative at all. Accordingly, pulsed time is governed by a reliably steady beat, at whatever velocity. This periodicity lends itself to forms of temporal mapping like meter (the structuration of cycles of beats into measures, which will contribute to the articulation of musical form). On the other hand, amorphous time refers to a quality of musical temporality that is not reducible to any detectable patterns of regularity or metricality; musical events occur, but without fixed duration. As should be evident, pulsed corresponds to striated, and amorphous corresponds to smooth.
There are a great number of subtleties in Boulez’s framework that I won’t attend to here, but suffice it to say that with this simple dialectical structure applied simultaneously to space and to time, a vast array of musical phenomena can be elucidated. Part of the explanatory depth of the model arises from the fact that the degree of striation or smoothness in a given pitch space need not be paralleled in the temporal field that is operating in the same musical work. That is, we can easily conceive of smooth pitch environments in conjunction with striated/pulsed time, or the inverse (striated pitch spaces coupled with amorphous/smooth temporality). While these latter conditions are interesting from a musical perspective, I have not found them very useful when approaching clinical matters. Instead, when employing the smooth and striated in the clinic, I have tended to work from the notion not of space and time but of spacetime,[6] positing that the subjectivity of the human being can be rendered as a dynamic spatiotemporal process wherein a transformation in the temporal is correlated with a complementary transformation in the spatial and vice versa. In fact, to put it this way is to falsify things a bit by upholding a separation between space and time. It would be better to say that the spacetime of the subject is a dynamical system in which its spatial or temporal characteristics are not meaningfully distinguishable because they ultimately refer to a single process.
What exactly do the smooth and the striated bring to the clinic, though? What does it mean for us if, as Deleuze and Guattari say (in an excellent encapsulation of Boulez’s system) “in a smooth space-time one occupies without counting, whereas in a striated space-time one counts in order to occupy”? Or again, that “all progress is made by and in striated space, but all becoming occurs in smooth space?”[7]
In part 2, I am going to try to answer these questions, bringing these concepts to life in the clinic and putting them into dialogue with the late work of Lacan.
Notes:
1. Boulez, P. (1971). Boulez on music today (S. Bradshaw & R. R. Bennett, Trans.). Harvard University Press.
2. Deleuze, G., & Guattari, F. (1987). A thousand plateaus (B. Massumi, Trans.). University of Minnesota Press. (Original work published 1980)
3. Deleuze, G. (2007). Two regimes of madness: Texts and interviews 1975-1995 (A. Hodges & M. Taormina, Trans.; D. Lapoujade, Ed.). Semiotext(e).
4. In fact, there are musical systems that resemble this mythical piano that we’ve just imagined. In my music school days, I studied one such microtonal system in a class taught by Joe Maneri, who at that time was doing a lot of composing using a 72-note scale (remember, the Western system employed by everybody from Beethoven to the Beatles is a 12-note scale)!
5. Credit is due to my colleague Benoît Le Bouteiller for helping to clarify the correspondence between the smooth/striated dialectic and the mathematical concepts of continuous and discrete functions.
6. Here I am of course thinking of this term’s use within physics, wherein, following the implications of Einstein’s theory of general relativity, space with its three dimensions and time as the fourth are rendered inextricable, together forming the spacetime continuum.
7. Deleuze, G., & Guattari, F. (1987). A thousand plateaus (B. Massumi, Trans.). University of Minnesota Press. (Original work published 1980)