Thinking Waves

1. Of Musical Space, Graphic Gravity, and Black Holes

Sound Object, the exhibition of the score of Marcelo Toledo’s composition Para el Encuentro en los Abismos (For the Encounter in the Abyss), held at Columbia University (from January 22 to February 13, 2004) and curated by Bruce W. Ferguson, presented the score not primarily as a musical score but as a visual aesthetic object.  

Such an experience is bound to have considerable complexity, even when not accompanied by a performance of the composition itself, which, I shall suggest, would further change the nature of the situation.  First of all, a certain reading of the score would still take place in these circumstances.  As musical scores usually do (but in this case to a greater and more significant extent) Toledo’s score contains explicit textual elements, which could be read in the usual sense of the word “reading.”  In this respect, it is not unlike those artworks that use textual elements, although it would be difficult for nearly any viewer to avoid experiencing it as a musical score, even if one could not properly read the music itself.  One can, however, also speak of a more general sense of reading as a signifying process relating a given material object (functioning as a signifier or a set of signifiers) to a certain conceptual or phenomenal content (functioning as a signified or a set of signifieds), or relating the resulting sign or signs to other objects, material or mental.  Such processes appear to be unavoidable in our engagement with visual objects, such as artworks, although their role may be more significant and more akin to the reading of linguistic texts in certain cases, including, it appears, when they involve musical scores.  I am not saying that this engagement could be reduced to such reading processes, however one sees them (along the lines of traditional, say, Saussurean, semiotic, deconstruction, and so forth.)  My point at the moment is to distinguish this general reading from a reading of Toledo’s score as a musical score, although, as will be seen presently, reading it as a musical score cannot avoid this general reading either.  Both types of readings and both types of experience of the score—that of relating to the score as a musical score (on the part of those who do not read music) and that of engaging with it as a visual object—mirror each other.

We begin to sense and experience this mutual mirroring more palpably once our visual perception of the score is accompanied by listening to a recording of the work.  (Playing it live in this setting would introduce yet further complexities, which I shall leave aside here.)  In this case, viewing the score, even without properly reading it, would depart less than it might appear from our experience of music qua music, which (always?) involves a certain phenomenal spatiality, which, one could argue, is represented by the score as a visual object.  One could indeed say that, whether this was deliberate or not on the part of the composer or the curator, the event brought out or enacted this spatial dimension of music.  The title of the exhibition, “Sound Object,” reflects this situation as well.  In principle, one could even see the score, as a visual object, as representing the phenomenal space of the piece as a musical object, that is, the space we phenomenally experience when we hear the piece, actually or mentally (for example, when we read the music of the score).  Indeed, this type of iconicity—that of mapping the phenomenal space of the music, as encoded by the musical notations and “graphics” of the score, by the score as a visual object—appears to have been deliberate on the part of the composer, as his extensive additional notations in the score suggest.  In any event, the phenomenal space in question is given a particular structure and, in its temporal unfolding, dynamics by virtue of the very character of Toledo’s composition.

The composition may also be seen in architectural terms as a composition—a creation—of this space as a space where the encounter in, and with, the abyss appears to be shaped by the image and the very idea of waves (plural, which is important here).  Following Gottfried Wilhelm Leibniz and Albert Einstein, we may even think of this space as itself created and structured, musically “curved” in a certain way, by these waves (or other elements of the composition), rather than thinking in terms of some pre-existing space (similar to Newton’s absolute space) in which these waves would propagate.  Both the image and the idea are also manifest in the visual aspects of the score, which, as I said, contains a complex set of additional—nonmusical or (they cannot be entirely nonmusical for the musicians performing the score) quasi-musical—notations to guide the musicians who would perform the piece, or listeners or viewers who would follow the score.  Thus, a certain graphic space is created or “drawn” as well, as Ferguson observes in his notes for the exhibition.  As he also notes there, to one degree or another, music is “always ‘drawn’ before it is read” (emphasis added).  In this case, however, this space is given a richer graphic architecture, or, one might say, graphic gravity, than usual, although one might think of other cases of such greater graphic gravity as well.  This inscription of “waves” in the score could further indicate the iconic relationships, just suggested, between the score as an abstract visual object and the phenomenal musical space it encodes by its musical notation, but it would be significant even apart from these relationships.  

It is as if the abyss invoked by the composition’s title were conceived of as containing something that signals its existence by (sending) a certain type of waves or a certain particularly organized manifold of waves, coupled to certain (but less prominent) discontinuous elements, without, however, ever manifesting itself as such, in its ultimate nature, in these waves.  In other words, the particular character of waves, imagined by the composer and imaged by the composition, signals the presence of an abyss behind them.  This abyss contains the source of these waves, but the nature of the processes occurring in this source and responsible for these waves is irretrievably hidden in that abyss.  Our encounter in the abyss, which we can thus explore only so far, only up to a certain limit, is our encounter with this (forever?) hidden, assuming that even such words as hidden or abyss, or any other words or conceptions, could still apply to that “hidden.”  I would also like distinguish this conception of the “unconceivable” in the abyss from the idea of chaos, that is, when the latter word refers to disorder or chance, as opposed to “cosmos,” for other conceptions of chaos are possible.  The present sense of the inconceivable would be closer to what the Greeks referred to by aretton or, sometimes, alogon—the incomprehensible.  (Martin Heidegger sometimes invokes this conception in his writings.)  Technically, chaos may be conceived along these lines as well, for example, given that absolute randomness, randomness not containing any order, may be beyond our conception too.  

The image that comes to mind is that of a black hole in physics, which never reveals to us the nature of the processes in its interior, at least its deeper interior, since we may, to some degree, access and conceptualize what happens near the “surface.”  A black hole manifests its existence only by the warped space it creates (through its immense gravity), and by swallowing everything that falls into it and emitting a certain quantum type of radiation, discovered by Stephen Hawking and known as Hawking’s radiation.  As quantum, this radiation requires the application of both discrete, particle-like, and continuous, wave-like, conceptions.  Toledo’s composition also relates its “waves” to the question of the relationships between continuity (wave processes are, by definition, continuous) and discontinuity in music, especially as these relationships relate to modernist music.  “Waves,” however, appear to be given an asymmetrical priority in the composition, as against the discrete elements found there, given less space (in either sense), including in manifesting the abyss behind the image.  As I shall explain, the presence of such elements remains essential, both musically and conceptually.  The image and idea of waves, however, define the conceptual and phenomenal structure of the piece, including as concerns the relationships between this structure and the abyss, although this structure acquires new (quantum-mechanical-like) complexity by being coupled to these discrete elements.

The overall, partly continuous and partly discontinuous, picture of black-hole radiation is immensely complex, as the data we have (related to the black holes found at the centers of galaxies) and computer simulations suggest.  Such a picture is actually complicated even further, because the quantum nature of this radiation make this picture depends on how we set up and arrange our instruments—the observational technology, since such an arrangement always specifically shapes and indeed creates such a picture or set of pictures, sometimes in mutually exclusive ways.  Some of these records are audial.  It was recently reported that the lowest “musical” note (it cannot be heard by us) in existence is physically generated by a rotating black hole.  In any event, the resulting radiational architecture is not unlike that created by Toledo’s composition, including the way it is shaped and created by the twenty-four musical instruments for which the piece is written—the technology of the composition, which would shape our experience of it if we listen to it or even if we read the score musically.  (This dimension of the piece disappears if we merely experience it as a visual object, although not altogether, since the graphic space and graphic gravity of the score depend on different notations for the instruments involved.)  It is as if we were looking, partly simultaneously and partly in sequence, at a set of images of a black-hole’s radiation, emitted from its abyss with twenty-four different telescopes and other devices (visual, radio, and other) showing different, continuous and discontinuous (or mixed), forms of this radiation.  Jointly with the black hole itself, this technology creates for us an encounter with and (phenomenally) in the abyss.

It is possible to read or (here the spectrum of possibilities is especially wide) experience Toledo’s piece as an attempt to imagine and to image the hidden architecture of the abyss, and the title of course also suggests an encounter, say, a meeting with somebody or something, in (i.e. inside) the abyss.  Accordingly, the present reading may be displacing the composition into an encounter with the abyss.  These two readings can, however, be reconciled or made mutually consistent along the lines just indicated, whereby we may be taken into the abyss but only up to a certain point.  At this point we encounter a certain manifold of phenomena, some of them discontinuous, but most especially a certain type of “waves.”  These waves signal the deeper regions of the abyss, regions whose actual nature cannot be accessed (made known, conceived of, or imagined) by any means that are or will ever be available to us.  In the notes accompanying the score, Toledo invokes a “narrow space between the known and the unknown.”  The knowable and the unknowable, the imageable and the unimageable, the imaginable and the unimaginable, the thinkable and the unthinkable?  This narrow space can be seen as the narrow space of the region, filled with waves and more complex (continuous and discontinuous) forms of “radiation,” just invoked, extending a little into the abyss to confront that which is inaccessibly hidden in its depths.

Toledo actually adds, “I believe that in music today that is space of noises and complex sounds,” which, as I shall suggest, may be seen as corresponding to the nature of the space in question here. And then, there is the par [for] of the title—written for our encounter in or with the abyss as we listen to the piece, written so we can have such an encounter, as we have an encounter with the music?  

One may offer the following visualization of the situation.  Imagine a small mountain lake, reasonably rectangular and (for reasons to be explained below) elongated in shape, initially still, with a smooth mirror-like surface.  If a wind slowly begins to blow, waves gradually begin to rise, propagating in a single direction, say, along the long sides of the lake’s rectangular.  If several winds (say, variously channeled by mountains around the lake) are blowing in different directions, a more varied wave pattern appears, at first in a reasonably gradual process.  After a while, the picture becomes quite complex, due to the clashes between and interferences of the waves, depending on the winds or the size of the lake.  This is still mostly a continuous picture overall, although not altogether, since some wave clashes will create (at least perceptually) discontinuous elements.

If one expands the lake to make it big enough to become a sea, subjected to many winds, from breezes to storms, one would get an immense manifold of waves, the type of picture captured by Claude Debussy’s La Mer.  While the work invokes the sound of a sea as well, for the moment, I am concerned with the sound image of waves, also in the sense that Debussy gave this word in his Images.  Physically every sound, however discrete, reaches us as an acoustical wave.  The image, technical or artistic, of this sound, as a wave image, will, however, be continuous, and such an image could be a musical image, generally different from the sound the wave generates, although one can also imagine cases where the relationships between both would be more iconic.  In any event, it is the wave image (“wave object”?) in music that I want to address here.

In the case of Debussy’s work as an image of a sea, we may also encounter the site of the Kantian sublime, for which our phenomenological experience of the sea indeed provides one of the main instances (although the application of Kant’s analysis, and of his very definition, of the sublime to music requires further qualifications).  Such an experience teases us with the possibility of putting it into, or framing it as, a single, if complex, picture or image.  But, unlike the beautiful (which allows itself to be so framed), ultimately what we see or hear always escapes and defeats such attempts.

One finds certain instances of or akin to the Kantian sublime in Toledo’s music in the piece, if not quite in the score as a visual object (although that might depend on the angle of vision when looking at the score).  Overall, however, the Kantian sublime does not appear to be what the composition primarily aims at.  For, unlike the image conveyed (at least to the present author) by Toledo’s piece, the sublime, at least in Kant’s version, would, in the case of waves, deal with the difficulty of encompassing the play of the waves, rather than with the abyss responsible for a certain play of waves, which play may or may not be sublime.  Toledo’s waves are, I shall suggest here, quantum-mechanical or/as black-hole-radiation-like, rather than sublime in Kant’s sense.

For the moment, alternatively to the more continuous wave processes just sketched, we can drop a stone in the lake, which may be seen as a discrete event, initiating a continuous wave process, and the actual picture would contain a kind of discontinuity, a hole.  The process might, again, be conveyed by an actual sound, if we hear the sound of the dropped stone hitting the surface and then that of propagating waves, which is not especially interesting, unless of course we somehow record and amplify the acoustical spectrum (beyond the range of human hearing) of the process.  On the other hand, one can create, compose, a rich musical image of the wave propagation in water even in the case of a single dropped stone, or if we have multiple stones dropped in the lake.  To create a more complex visual picture, one may, as it were, “compose” with the lake by dropping stones of different sizes and weights into it—from small ones (which would not create any perceptible waves and would only leave hole-like images in the surface) to large ones.  One could then observe how these wave and point patterns combine—a kind of abstract expressionism of the play of water.  One could actually think of the dropping and splashing technique in abstract painting, with which one can create a similar interplay of point-like and wave-like elements.  One could also think of writing a computer program that would generate different patterns.  The picture could also be given a musical image, which, depending on a given pattern, would form a wave image or a point image, or combine wave and point images, to which one can give additional spatiality by positioning instruments accordingly.  The question is what kind of pattern or set of patterns—continuous, discrete, or combined—one is aiming to create, and what one is aiming to convey by any set of patterns, or what a given set is able to convey.

On a more discrete side, one could think here of Igor Stravinsky’s Violin Concerto, which contains peculiarly discrete space-like effects, as do certain works of Pierre Boulez.  By contrast (although still close to Stravinsky in its musical spirit, but to Arnold Schönberg in its wave imagery), we may think of Toledo’s composition, written for twenty-four musicians, on this model as well, but now in terms of a wave process initiated—this is the opening of the piece—by twenty-four stones thrown into a rectangular lake.  The process will then produce certain sporadic discontinuities as well, although waves will continue to dominate the piece.  My main reason for thinking of an elongated rectangular lake is that, as a visual object, the score, with its wave notations, can be seen as invoking this picture as well.  

As must be clear from my earlier comments, however, the picture just sketched could only be the first stage in a kind of “quantum-mechanical” or “black-hole-radiation-like” reading of Toledo’s composition itself or of Sound Object that I would like to suggest here.  (To some degree, that may be said about Schönberg, Stravinsky, and Boulez, as well.)  If one thinks about the physical processes responsible for a kind of composing with a lake, as described above, they would present us with a very great complexity, although, in principle, approachable by means of classical physics.  In quantum physics, we would encounter an impenetrable abyss of nature at the ultimate level of its constitution, with only certain manifest effects, continuous (wave-like) or discontinuous (particle-like), manifest, usually in mutually exclusive ways, at or near the surface.  Imagine throwing elementary particles, such as electrons, into a cloud chamber or a silver bromide photographic plate, whereby we could observe and, as it happens, experimentally and theoretically track certain effects, traces, of those collisions, but neither the nature of such (thrown) objects nor of the processes responsible for these traces.  This situation is rather more akin to what we encounter in Toledo’s composition or in Sound Object, and the kind of abyss that we encounter in and through it.  

2.  Quantum Waves and Encounters with the Abyss, Between Physics and Music  

Listening to Toledo’s composition in the setting defined by Ferguson in Sound Object made this particular viewer/listener think about thinking in terms of waves and, indeed, about thinking in waves in physics, first classical and then quantum, which each approach waves, or particles, very differently.  It should be noted that we cannot conceive of entities that are simultaneously both a particle and a wave, or, to begin with, simultaneously continuous and discontinuous.  Classical physics handles this mutual incompatibility by means of two separate types of physics, even though, it may be added, either type of theory would use continuous forms of mathematics, as manifest in the equations of motion for both types of objects (this motion is, in any event, assumed to be continuous in either case).  The difference is defined by the nature of the objects considered, continuous in the case of waves and discontinuous in the case of particles (or composite bodies).

By contrast, quantum physics developed an effective (although physically, mathematically, and epistemologically complex) way of using both wave-like and particle-like features, and certain other physically or conceptually incompatible features, within a single theoretical framework, known as quantum theory.  As will be seen, however, this does not mean that both types of description, in terms of waves or in terms of particles, or, in part as a consequence, even the idea of motion, can apply to the objects considered by quantum theory.  These objects, however, are now seen as the ultimate (micro) constituents of nature (although the theory would also be used in the case of certain macro-objects, whose quantum constitution cannot be ignored in the way it can be in the case of the macro-objects considered in classical physics).

Thinking in waves or in particles, or in terms of continuity or discontinuity, is of course not restricted to physics and, historically, comes into physics from other forms of thinking about nature or mind.  Physics is a particular and, in modern times (since Galileo), specifically mathematical form of refinement of our everyday thinking.  Thinking in (terms of) waves in art, philosophy, and science extends to the earliest known history of all three, certainly to the pre-Socratic Greek culture, beginning with Homer.  Thinking in waves often helps thinking about thinking itself.  It shapes our image of thinking, in part by virtue of the apparent continuity of the process of thinking, as we experience it in our consciousness, which experience must have been one of the sources of the very idea of continuity.  Consider, for example, this remarkable opening of Percy Bysshe Shelley’s “Mont Blanc,” which harks back to the earliest history of Western literature and philosophy, and (it may be shown) traces it throughout Western intellectual history.

The everlasting universe of things

Flows through the mind and rolls its rapid waves

Now dark—now glittering—now reflecting gloom—

Now lending splendour, where from secret springs

The source of human thought its tribute brings

Of waters,—with a sound but half its own.

Such as a feeble brook will oft assume

In the wild woods, among the mountains lone,

Where waterfalls around it leap forever,

Where wood and wind contend, and a vast river

Over its rocks ceaselessly burst and raves.

(“Mont Blanc,” 1-11; emphasis added)

For the purposes of this discussion (ultimately this is a more complicated matter), one may think of the “things” invoked in the first line as ideas.  Shelley has a statement to the effect that “things are ideas” in his notes.  This statement also aims to suggest, along Kantian lines, that it is not possible to think of anything apart from how it, or something to which it relates, appears to our thinking, whatever that anything itself may be, as a thing of nature or mind, or as something that is knowable or unknowable (e.g. Kant’s things-in-themselves).  Shelley’s description of thinking in waves uses both visual and sound (including musical) imagery and relates both, which relationship persists throughout the poem.  The poem also describes the waves of thought through both visual and sound imagery, or, reciprocally, waves, of whatever kind, in terms of the flow(s) of thoughts, thus creating a complex, interactive interplay of imagery and ideas.

In this respect, it is not unlike Toledo’s composition.  The latter, it is true, does not “speak” of thoughts, but it makes one think of and in waves.  In both cases, moreover, the ultimate source of the waves portrayed is lost in the abyss, and thus is a portrayal of an encounter with the hidden in the abyss.  One can, however, think here of the long history of waves as a theme, image, and, again, a way of thinking in classical music, perhaps in particular, again, in Debussy’s La Mer, and then, of thinking in (terms of) musical continuity vs. discreteness in modernist music, e.g. in Schönberg vs. in Stravinsky.  The situation is of course more complex in all of these cases, and at stake here are differences in the balance of musical continuities and discontinuities.  But such differences are significant and even decisive.

It is worth noting that Shelley’s imagery also includes a play of light reflected in water, in this case apparently the light of the sun, sometimes eclipsed by passing clouds, and, thus, a single source of light, multiply reflected in the movement of waves.  These reflections multiply the source, creating the kind of picture that Jacques Derrida invokes in his reading of Stéphane Mallarmé in Dissemination (Dissemination, tr. Barbara Johnson, Chicago: University of Chicago Press, 1982).  In this type of picture we can no longer quite distinguish between, or rather be able to unequivocally decide, which are sources and which are reflections.  Derrida invokes the image of a crystal chandelier (lustre), which hides its source(s), and which Derrida also links to certain mathematical concepts, specifically Kurt Gödel’s undecidability in mathematical logic (which disallows one to establish either the truth or falsity of certain mathematical propositions).   As Derrida writes elsewhere:

Representation mingles with what it represents, to the point where one speaks as one writes, one thinks as if the represented were nothing more than the shadow or reflection of the representer.  A dangerous promiscuity and a nefarious complicity between the reflection and the reflected which lets itself be seduced narcissistically.  In this play of representation, the point of origin becomes ungraspable.  There are things like reflecting pools, and images, an infinite reference from one to the others, but no longer a source, a spring.  There is no longer a simple origin.  For what is reflected is split in itself and not only as an addition to itself of its image.  The reflection, the image, the double, splits what it doubles.  (Of Grammatology, tr. Gaytri C. Spivak, Baltimore:  Johns Hopkins University Press, 1975, p. 36)

Accordingly, one could think of and describe such situations, or Toledo’s composition, quite apart from any physics, classical or quantum.  Nevertheless, quantum theory, which relates the idea of wave and the idea of particle or, more generally, continuity and discontinuity in an entirely new and unexpected way, appears to offer a useful perspective on Toledo’s composition or on modernist classical music, defined by a number of features analogous (although not simply identical) to those defining quantum physics.  To cite Gilles Deleuze:

There are two sorts of scientific notions. Even though they get mixed up in particular cases.  There are notions that are exact in nature, quantitative, defined by equations, and whose very meaning lies in their exactness: a philosopher or writer can use these only metaphorically, and that’s quite wrong, because they belong to exact science.  But there are also essentially inexact yet completely rigorous notions that scientists can’t do without, which belong equally to scientists, philosophers, and artists.  They have to be made rigorous in a way that’s not directly scientific, so that when a scientist manages to do this he becomes a philosopher, an artist, too.  This sort of concept’s not unspecific because something’s missing but because of its nature and content. (Gilles Deleuze, Negotiations, tr. Martin Joughin, New York: Columbia University Press, 1995, p. 29)     

I am not altogether sure whether one should even use the word “inexact” here.  Toledo’s “quantum-mechanical” or “black-hole-physics” musical encounter with the abyss engages this type of (scientifically) “inexact” but rigorous conceptuality, which, however, also has its musical genealogy and significance, which I shall address later.  

In invoking quantum mechanics, I am thinking, first, about Erwin Schrödinger’s version of it as a wave mechanics, as opposed to Werner Heisenberg’s so-called matrix version of the theory, based, initially, on the idea of particles but, ultimately, arriving at the epistemology of the abyss in question here.  Heisenberg’s discovery came first in 1925, while Schrödinger independently arrived at his wave mechanics several months later in 1926.  It is worth noting that Shelley wrote “Mont Blanc” at the time (1816) of the transition from the particle (corpuscular) theory of light of Newton to the wave theory of light in optics, which led to the abandonment of Newton’s corpuscular view of light by classical physics.  This alternative is, however, different from the way quantum physics handles both ideas and types of phenomena in approaching all objects that it considers, be they radiation, such as light, or matter, such as electrons, which would be separated by classical theory.  Quantum theory returned a particle-like character to our handling of light in physics, but it retained a wave-like character of this handling as well.  The cost one might have to pay is the impossibility of ascribing either character to light itself, or other elementary physical entities, as opposed to certain phenomena resulting from the impact of such entities upon the world that we can observe.  In other words, unlike in classical physics, in quantum mechanics any quantum object can, through its impact on such instruments, manifest its existence in either particle-like or wave-like observable phenomena, although, it appears, not its ultimate character as a quantum object.  The ultimate (quantum) constitution of nature, thus, becomes hidden in the abyss, while the existence of this abyss itself is manifest in the wave-like and the particle-like character of the phenomena, individual and collective, that we can observe.  These phenomena, thus, arise through the impact of certain, in principle, unobservable and ultimately inconceivable “objects” and “processes,” which we call “quantum,” on the world we can observe.  It follows that such terms as object, process, or quantum, may not be applicable either, any more than any other terms, hence my quotation marks.  My usage of the term “quantum object” in this discussion must be understood with this qualification in mind.

This view of quantum physics, the view that I shall adopt here, is closer to that of Werner Heisenberg or Niels Bohr.  Schrödinger’s wave vision aimed to replace this inconceivable abyss with the picture of waves as representing, even if approximately (his views concerning this point were complex), the ultimate nature of physical reality.  In other words, Schrödinger attempted to (re)think nature at the ultimate level in (terms of) waves, in accordance with the classical view of radiation, such as light, which he wanted to extend to particle-like objects as well.  Such objects would be reconceived as manifestations of the more fundamental underlying wave-like processes.  By contrast in Bohr’s or Heisenberg’s view quantum mechanics becomes an encounter with the abyss of nature at the ultimate level of its constitution.  At the same time, however, this view involves and sees as necessary both ideas or forms of intuition or visualization, waves and particles, although, inevitably, making the applications of both more complex, including as concerns symmetries and asymmetries in using them.  As will be seen, similarly to their role in Toledo’s composition, thinking in waves will be given the primary significance in signaling the existence of this abyss of nature.  For the moment, both particle-like and wave-like phenomena that we observe would be seen as arising under different and, again, mutually exclusive circumstances from the black-hole-like abyss of nature at the ultimate level.  By contrast, we would not be able to apply either idea, that of waves or that of particles, or conceivably any idea we can possibly have (the idea of “process” included), to this abyss itself or to the processes that occur there and lead to the observable phenomena in question.

Accordingly, this interpretation also distinguishes quantum objects, as the ultimate objects in question in quantum mechanics, from the manifest physical phenomena we physically consider, somewhat analogously to the Kantian difference between phenomenal and noumenal (things in themselves) objects, but ultimately more radically.  In particular, as I said, such terms as things in themselves or objects, or noumena, cannot rigorously apply to “quantum objects,” which are, in this sense, unknowable even as unknowable and inconceivable even as inconceivable.  Phenomena in quantum mechanics are defined in terms of what we observe in measuring instruments impacted by their interactions with quantum objects, rather than in terms of quantum objects.  Such phenomena, as opposed to quantum objects, can be either wave-like or particle-like, but never both together simultaneously.

Both types of phenomena are and must be used in quantum mechanics, which, however, need not use such different and mutually incompatible phenomena simultaneously.  Instead, it can and must use either one or another at different junctures, by experimentally arranging our, in turn different, measuring instruments for different types of interactions between them and quantum objects, thus leading to mutually exclusive types of phenomena as manifest in these instruments—wave-like or particle-like.  Bohr’s famous term “complementarity” originally referred to this situation, a situation in which certain experimental arrangements and types of phenomena manifest in them are mutually exclusive and, hence, not applicable simultaneously, and yet, while thus alternatively used, both are necessary for the proper functioning of the theory.  The “trick” enables one to avoid a contradiction that would arise if one were simultaneously to ascribe mutually incompatible properties, such as both wave-like and particle-like properties, or more generally, continuous and discontinuous properties, to the same entities.  Instead, it depends on what kind of experiments we want to arrange.  Eventually, Bohr extended the application of the term to his overall interpretation of quantum mechanics and the epistemology just sketched, primarily because this interpretation and the abyssal character of the ultimate constitution of nature defining it, or defined by it, are correlative to the unavoidability of certain mutually exclusive situations of measurement or phenomena in quantum mechanics.

One may, accordingly, be skeptical as to how well suited Schrödinger’s wave program may be for quantum physics, even though Bohr and Heisenberg’s view has been (and remains) controversial in turn.  For some, Einstein and Schrödinger, among them, it has been outright unacceptable in view of its radical epistemology, which uncircumventably prevents us from reaching nature at the ultimate level of its constitution.  By contrast, the problems of Schrödinger’s approach were primarily related to certain physical difficulties.  Schrödinger by and large abandoned his attempts to rethink quantum mechanics as a form of wave theory, although not his hope that one day this may be possible by means of an alternative theory.  His thinking of the ultimate nature of the physical world in terms of waves and his thinking in waves itself are, however, interesting in their own right, including as an attempt to relate continuity and discontinuity in terms of (underlying) continuity, since he needed to account, in terms of waves, for the discrete, particle-like, features of quantum physics.  It is as such that it can be linked to Toledo’s work, although, as I said, I shall also and ultimately more fundamentally relate the latter to Bohr and Heisenberg’s view as well. Both are encounters with and in the abyss as unavailable to our thinking.  Bohr sometimes invokes the image or, as the case may be here, the unimaginable of the abyss.  This view, however, gives waves a fundamental significance.

It is worth noting that at some point Schrödinger thought of the possibility of some kind of intermediate objects, which would be neither particles nor waves, and yet would combine some aspects of both.  This is not an easy task and perhaps is ultimately impossible, given that such properties are, in general, incompatible.  In any event, this is quite different from the irreducibly inconceivable black-hole-like “objects” in the abyss, as conceived by Bohr and Heisenberg.  At an earlier stage of his thinking, concerning the difficulties of quantum theory, Bohr, too, was trying to conceive of, to imagine, or, as he said at some point, “dream-up,” all in vain, some pictures of what is going on at the quantum level.  He, however, abandoned such attempts in the wake of Heisenberg’s discovery of quantum mechanics (which preceded Schrödinger’s wave mechanics), in favor of the view outlined here.  

In this view, to recapitulate, we are confronting the unimaginable, inconceivable processes in the abyss at the ultimate level, the processes which, however, can produce, as their effects, certain phenomena that we can imagine and describe, as just explained.  Ultimately, in this interpretation, all manifest wave-like phenomena appear as multiplicities of individual and, as such, discrete manifest phenomena, thus making all (strictly) elementary phenomena in question individual and justifying calling the theory of such phenomena quantum.  I emphasize “manifest” because the discrete character of such elementary phenomena does not change the abyssal, inconceivable (in either discrete or continuous terms) nature of quantum objects.  The latter and the processes involving them are, in their interaction with measuring instruments responsible for these phenomena, but are physically different from what is manifest in measuring instruments, impacted by them, and thus defines the phenomena in question.

It is crucial, however, that these multiplicities also exhibit a particular, wave-like, type of organization or pattern, in other words, order, in certain circumstances, but, equally crucially, not always.  The contextual nature (in some circumstances but not in others) of the emergence of this pattern, uniquely peculiar to the behavior of nature at the quantum level, reveals this behavior to be profoundly strange and even mysterious.  The situation may be explained by considering briefly the so-called double-slit experiment, a kind of archetypal quantum-mechanics experiment, containing most essential features of the situation.  In particular, it may be shown to reflect the probabilistic character of quantum-mechanical predictions and to be equivalent to Heisenberg’s famous uncertainty relations.  The latter make it in principle impossible to perform a simultaneous exact measurement of such variables as position and momentum, which is always possible, at least in principle, in classical physics.

The well-known arrangement consists of a source; a diaphragm with a slit (A); at a sufficient distance from it a second diaphragm with two slits (B and C), widely separated; and finally, at a sufficient distance from the second diaphragm a screen, say, a silver bromide photographic plate.  A sufficient number (say, a million) of quantum objects, such as electrons or photons, emitted from a source, are allowed to pass through both diaphragms and leave their traces on the screen.  Two set-ups are considered.  In the first, with both slits open, we cannot, even in principle, know through which slit each quantum object passes.  In the second we can, either in practice or, importantly, in principle.   In the case of the first set-up, a “wave-like” interference pattern will emerge on the screen, in principle regardless of the distance between slits or the time interval between the emissions of the particles.  The traces, once a sufficiently large number of them are accumulated, will “arrange” themselves in such a pattern, even when the next emission occurs after the preceding particle is destroyed after colliding with the screen.  This pattern is the actual manifestation and, according to, at least, the present interpretation, the only possible physical manifestation of quantum-mechanical “waves.”   If, however, in the second set-up, we install counters or other devices that would allow us to check through which slit particles pass, the interference pattern inevitably disappears.  Merely setting up the apparatus in a way that such knowledge would in principle be possible would suffice.  

These facts are indeed extraordinary and difficult to confront, and such locutions as mysterious, incomprehensible, or paradoxical are hardly surprising, and to some degree justified.  It is as if quantum objects could know, individually or, even more strangely and in the context of quantum-mechanical waves more significantly, collectively, whether both slits are open or not or whether counters are installed or not.  Attempts to conceive of the situation in terms of physical attributes (whether particle-like or wave-like) of quantum objects themselves appear to lead to unacceptable or at least highly undesirable consequences.

One might, however, also see the situation, with Heisenberg and Bohr, as indicating the impossibility of ascribing any physical attributes to quantum objects themselves or to their behavior.  In this view, in considering individual marks on the screen we may rigorously speak of them only as particle-like effects or, in certain circumstances, as wave-like effects, and not as traces left by collisions with classical-like particle or wave objects.  We are dealing with two different and mutually exclusive types of effects of the interaction between quantum objects and measuring instruments upon those instruments under specific physical conditions.  Such circumstances are, then, in turn mutually exclusive or complementary, which is, on this view, the only meaning of the wave-particle complementarity, arguably the most famous among the complementarities found in quantum mechanics, but only one among others.  But the fact that these circumstances are always mutually exclusive also allows us to avoid the difficulties of reconciling them.  In other words, we can take advantage of this mutual exclusivity, but at the cost, unacceptable to some, again, both Einstein and Schrödinger among them, of placing quantum objects and processes themselves forever out of reach even of our thought—in the inconceivable abyss.   This view, accordingly, retains a certain mystery at the heart of quantum mechanics, insofar as the ultimate nature of the processes responsible for the data in question is ineluctably beyond the reach of our knowledge or even conception.

The wave-like effects, again, only appear by virtue of a collective and specifically sequential (one by one) accumulation of particle-like effects in a large number of trials when both slits are open and no counters are installed; otherwise, the sequential collectivities are particle-like in character, that is, random without a wave-like distribution pattern.  It is clear, however, that wave-like effects or phenomena are decisive in defining the nature of the phenomena in question in quantum mechanics and of quantum mechanics itself.  Even if one sees all elementary quantum-mechanical phenomena as individual and discrete in the sense just explained, what makes the situation peculiar and quantum mechanics (including its particular mathematical character, very different from that of classical physics) necessary is the presence of a particular type of pattern or organization of, or, again, order in, multiplicities of such phenomena.  This pattern is wave-like, and it is the emergence of this type of pattern in certain circumstances and its absence in other circumstances that necessitates quantum mechanics.  Were it not for these patterns, one would not need quantum mechanics (classical physics would do), and it would not take the particular shape it takes, including Schrödinger’s famous wave equation.

The situation is, again, peculiar and peculiarly, uniquely, quantum-mechanical, beginning with the emergence of a certain (relatively) ordered multiplicity consisting of events each of which is random as an individual event, which emergence, it also follows, is itself enigmatic or mysterious.  One has an exact reversal of the situation that obtains in classical statistical physics or, to begin with, in our general thinking about randomness and chance.  In this case, individual events are subjects to regularities (such as the laws of classical, Newtonian, mechanics), while chance arises in the case of multiplicities of objects or, as in tossing a coin, factors that we cannot properly trace for practical reasons.  If we could, we would be able to predict such events exactly and randomness would disappear.  In quantum mechanics these are collective (sequential) occurrences that exhibit order, at least a certain order (this order is still that of statistical correlations) but an order nonetheless.  By contrast individual events are irreducibly random.  In other words, we must view differently both randomness and order alike, and how they combine.  It is order, however, even if in combination with chance, rather than chance or chaos, that is primarily manifest and allows us to infer the inconceivable or, to return to Greek terms, aretton or alogon, in the abyss that is responsible for something nature allows us to observe.

But to have order—even though in combination with disorder, rather than only disorder (“noise”), and by virtue of this combination, invoked in Toledo’s notes for the composition—to make us sense the abyss behind it and even know that it must be there is also fitting in the case of Toledo’s composition, or of art in general.  Indeed, this situation also brings physics closer to art.  While keeping chance alive, art creates order to have, and make us have, an encounter with, and in, the abyss and with the incomprehensible in the abyss.

Music, however, especially that part of modernist tradition to which I relate Toledo’s music here, may be especially close to quantum mechanics.  This special proximity arises because both link the relationships between continuity and discontinuity (“bits”) to the relationships between order or information (in-form-mation) and disorder or randomness, “noise.”  This is perhaps why, in speaking in his notes for the composition of “that narrow space between the known and the unknown,” Toledo adds, “I believe that in music today that is space of noises and complex sounds.”  Thus, it is not a space between noises and complex sounds but a space containing both or, again, created their interplay.  Waves, musical or quantum, are, then, patterns of this complexity, which are, thus, also information patterns whose bits are, at least in quantum mechanics, irreducibly random, once each considered in its own term.  We can predict and describes a pattern, but cannot predict any of its bits, or, again, describe the process by which the pattern (or each of its bits) is created.  This remarkable “architecture” can, it appears, only arise from the abyss that itself (i.e. whatever happens in its interior) exceeds any possible conception, whether in terms of order or randomness, continuity or discontinuity, or any interplay between and among them, or, again, anything else we can conceive of.   In other words, order and information can arise, as waves in a sea, and can be transmitted.  It does not, however, appear possible to know or even conceive of how this happens (there is certainly no continuous process of this emergence or transmission conceived on the model of classical wave physics), and, at least in quantum mechanics, this impossibility appears to be necessary for this order or information both to arise and to be transmitted.   Toledo’s composition is a portrayal of this kind of “space”—the space created by waves, waves of “particles” or “bits” (notes?), arising from the abyss.      

This essential role of certain wave patterns in quantum mechanics may justify Schrödinger’s view, or rather his thinking in waves, even if within different limits and along different lines of argumentation (such as those of the present argumentation), which is perhaps why his thinking had a certain appeal for Bohr (although not for Heisenberg).  As I said, Schrödinger would prefer physics or our minds to be able to reach further, if not quite all the way, in our description of the underlying physical reality.  He was reluctant to accept this “reality” to be irreducibly inconceivable (including even by means of any concept of “reality”) and, thus, to have an insurmountable epistemological discontinuity, an unbridgeable abyss between nature and knowledge.

By contrast, Toledo’s “For the Encounter in the Abyss” appears to me to portray and to be, and to engage us with, an encounter with the abyss and with the waves and certain patterns of waves through which this abyss sends us the message of its existence, but gives us no information as to the nature of its existence.  As will be seen, however, certain discrete musical elements play a role in the compositions as well, giving it a certain (albeit asymmetrical) complementarity in this respect.  Toledo’s music is, of course, not quantum mechanics, since the latter must have a rigorous mathematical formalism that would predict the numerical data obtained in experiments, to accompany such concepts as waves or particles, however these are conceived.  Toledo’s composition, however, or, again, Sound Object, conceives of and images, and allows us to experience, something akin to a certain type of quantum-mechanical thinking, as here described.  This thinking gives thinking in waves a special, dominant, role in the phenomenal space in which an encounter in and with the abyss takes place.  This abyss, whose existence is signaled by waves, or whatever happens in this abyss shapes this space, but can never be included in this space.  It is outside any thinking that is or can be available to us.  One might say that, rather than quantum mechanics, Toledo’s composition portrays a certain type of thinking of the quantum-mechanical type, wherever such a vision occurs.  It also engages us with this type of phenomenal architecture and makes us experience this type of vision, the vision of waves sent from the abyss, regardless of whether we think of quantum theory in the process, or whether we even know anything about it.      

It is of some interest that Schrödinger did not play an instrument and, insofar as we know, was not an especially musical person.  Heisenberg, by contrast, played piano, as did Bohr (Heisenberg well, Bohr badly), and Einstein, whose thinking was close to that of Schrödinger, played violin, which gives us more continuous sound than piano.  These facts pose an intriguing question concerning the differences in the ways of thinking between those founders of quantum physics who played musical instruments vs. those who did not.  Thinking in different domains, however, or products of this thinking, say, physical theories and musical compositions, may share their features whether the author of a given physical theory is musical or not.  On the other hand, certain cultural paradigms and different ways of thinking, such as continuous or wave thinking vs. thinking in (terms of) discontinuities may have their  “Zeitgeist” (I am uneasy about this, often abused, term) effects and, thus, lead to a different structure or architecture of thinking and works that results in a given domain.  I wouldn’t go so far as to argue that the fact that both Vienna and Berlin, where Einstein worked at the time, were the major centers of symphonic music played a role in some physicists’ inclinations toward wave theories in both places, given that both Einstein and Schrödinger were earlier ardent supporters of quantum discontinuity.  One cannot, however, altogether discount that such factors played a role.

Schrödinger’s particular education in mathematics and science in Vienna, and the influence of the contemporary Viennese culture (as interesting at the time in philosophy as it was in music, literature, and the arts) upon his thinking in physics may have played their role in the development of his ideas.  Both Ludwig Wittgenstein, whose brother was a concert pianist, and Kurt Gödel, arguably the greatest mathematical logician of the twentieth century, were Schrödinger’s Viennese contemporaries.  A more definitively established, and arguably uniquely significant, influence was Louis de Broglie’s work on the so-called matter waves, the work based on his idea of attributing a wave character to particle-like objects, such as electrons, just as earlier Einstein was compelled to ascribe a particle character to wave-like objects, such as light.  Several among Einstein’s other ideas, including those based on de Broglie’s work, also stimulated Schrödinger’s thinking in (terms of) waves.  The shift towards preference for the wave theories in both cases, that of Einstein and that of Schrödinger, is itself noteworthy, and the influence of Einstein’s work on Schrödinger’s ideas concerning his wave mechanics is unquestionable, just as is his influence upon Schrödinger’s earlier predilection for quantum discontinuity.

Schrödinger also did important work in acoustical theory and other subjects, including vibration of a string, where he dealt with waves and with representation of the physical phenomena, mathematically encoded in a particular way (wave equations), that needed to be thought in terms of waves.  Accordingly, it is not surprising that musical analogies and terminology are found throughout his work.  Conversely, atomic theories, especially earlier one, based on the quantum and, hence, in part discontinuous “planetary,” Keplerian, models of electrons moving around nuclei in atoms were seen as a new music of the spheres, an image borrowed from Kepler, the image that persisted even when these models proved to be problematic.  Arnold Sommerfeld, one of the founders of (pre-quantum-mechanics) atomic theory, spoke of spectra emitted by atoms (electrons actually) as the “real celestial music of the atom.”  Einstein turned the image around when he famously referred to Bohr’s 1913 theory of atom as “the highest musicality in the sphere of thought.”

This history is of much interest and significance, including in the present context, even though the ultimate origins of Schrödinger’s thinking leading to his wave quantum mechanics must be seen as in turn lost in the abyss.  I cannot, however, address the subject here, apart from a few most essential points.  Schrödinger arrived at his wave mechanics in a series of complicated steps, involving a complex reciprocal traffic between the physics of wave processes and the formalism he was developing.  This traffic also led him to his attempt to develop a form of phenomenal representation or visualization (the usual translation of German Anschaulichkeit [intuition] in this context) of these processes.  This visualization, complex even in the classical case, was bound to be extremely complex and incomplete, if possible at all, in the case of quantum theory, given the peculiarities of the data in question in quantum mechanics, in particular the discontinuous aspects (which gave it the name “quantum”).  For, as I have indicated, these data needed to be accommodated, in one way or another, by any form of wave theory.  

This is an important point, both for quantum theory itself and for my subject here, since Toledo’s music may and must be positioned in relation to more discontinuous modernist music, say that of Stravinsky, similar to that which defines Schrödinger’s theory vis-à-vis more discontinuous thinking about quantum physics.  Schönberg’s music is wave-like, too.  It may, however, be argued to be more classical in nature.  That is, it is closer to classical rather than quantum wave thinking, as the latter is understood here and as it, I argue, is manifest in Toledo’s music, with waves arising in our encounter with the abyss, which is itself, as an abyss, nothing like waves any more than particles, or anything else.  One might even consider from this perspective the relationships between Schönberg’s music and Kandinsky’s art of a certain period (roughly around 1911-1913), which is, however, a separate subject.  A more quantum-like view of his music is possible, and some of Adorno’s commentary on Schönberg may point in this direction.   Be that as it may, the significance of these themes themselves (waves and particles, continuity and discontinuity, the conceivable and inconceivable abysses we face, and so forth) is crucial for the emergence and the history of modernism in science, philosophy, and the arts.

Schrödinger was a classicist, and he hoped that the problems of quantum theory could be handled by physically and epistemologically classical means, and especially by classical field theory, which, as a wave theory, inspired him most.  This theory accounts for electromagnetic waves, such as light (composed of electromagnetic waves of very high frequency).  It was developed in the work of Faraday, Maxwell, and Hertz, and then, more radically, Einstein in his relativity theory.  (The latter may be seen as classical in epistemological terms, and was so seen by Einstein himself, in contrast to quantum mechanics.)  It has great richness and complexity in terms of mathematics and physics, and in terms of the kind of picture of waves they provide, especially once one moves to relativity.  Nothing less than the richness of Debussy’s La Mer would do to help us think of the wave phenomena in question in classical electromagnetism.  Could not, then, this type of complexity be sufficient to provide an underlying continuous picture for quantum physics, even given its manifest discontinuity at the level of observable phenomena, which, on this view, would only mask the underlying wave structure or texture, available, at least in principle, to some quantum-level microscope?

The problem that Schrödinger defined for himself was as follows.  Could one find a wave-type equation that would describe the physical processes at the quantum level, such as the behavior of an electron in the atom, and that would enable us to predict the results of quantum-mechanical experiments?  What would the (wave-like) character of the processes corresponding to this equation be, given the peculiar features of quantum physics, which correspond to the outcome of experiments and which, accordingly, must be retained—in particular, discreteness and a-causality and indeterminism?  Can a more wave-like theory, specifically, a form of field theory, solve “the quantum riddle,” as Einstein put it, hoping indeed that such would be the case one day?  Would such a solution allow us to capture the ultimate reality of nature in (terms of) a wave-like picture?  Or would it instead retain the hidden in the abyss and make it manifest in some complex wave picture or set of pictures (some possibly mutually exclusive), rather than in terms of a set of mutually exclusive wave and particle phenomena?  Bohr and Heisenberg thought the “riddle” solved by means of this latter view, in part by retaining the riddle, the mystery of that which is hidden in the abyss, as an irreducible part of the theory, not something Einstein and Schrödinger, both inclined towards a more realist theory, were ready to accept, however.  But, as I have stressed throughout, the abyss of quantum nature would not manifest itself apart from the wave-pattern in question, which gives at least a certain type of wave thinking a special significance.

Could, then, these processes be sufficiently captured or related to, even while thus hidden, in terms of wave theories, thinking, pictures, and so forth, as against the quantum-mechanical combination that, apparently irreducibly, retains mutually exclusive wave and particle, continuous and discontinuous, features within it?  Perhaps they could not.  On the other hand, as we have seen, the presence of the wave picture, the wave-like pattern of manifest effects of quantum objects upon the world, and, hence, thinking in waves were all crucial for discovering the existence of the abyss in the ultimate constitution of nature.  These waves are the primary signals of this abyss.

Initially, Heisenberg and his coworkers (Max Born and Pascual Jordan) saw their matrix quantum mechanics as, in their words, a “true theory of a discontinuum,” while Schrödinger saw his wave mechanics as, in his words, “a step from a classical point-mechanics towards a continuum theory,” which would also represent quantum reality accordingly (Schrödinger’s emphasis).  As it happens, while they do look different (one as a theory of a more discontinuous, while the other of a more continuous, type) at first sight, both theories depend on the continuous mathematics (in fact a particular form of continuous mathematics, that of the so-called Hilbert spaces), and are, as I said, mathematically transcribable into each other.  Neither, however, may mathematically represent any physical or phenomenal reality, continuous or discontinuous, and, as I argued here, such a reality at the quantum level may not be representable by any means, any concept of “reality” or “abyss” included.  The relationships between our mathematical and phenomenological intuition of, especially, continuity are a complex matter, which critically affect our handling of mathematical theories in physics, since their interpretation as physical theories inevitably depends on our phenomenological intuition, such as that of waves and their propagation.  As Hermann Weyl astutely observed in his book The Continuum:  “The conceptual world of mathematics is so foreign to what intuitive continuum presents to us that the demand for coincidence between the two must be dismissed as absurd.  Nevertheless, those abstract schemata supplied us by mathematics must underlie the exact science of domains of objects in which continua play a role” (Hermann Weyl, The Continuum: A Critical Examination of the Foundation of Analysis, tr. Stephen Pollard and Thomas Bole, New York: Dover, 1994, 108).  This point acquires a special poignancy in quantum mechanics, specifically in the case of the wave-patterns found in the experimental data.  These patterns require both “the conceptual world of mathematics” with its intuition to be properly predicted and our general wave-like intuition.

As I have stressed here, however, the fact that certain wave patterns are found in this case is decisive, for otherwise neither quantum mechanics itself nor the epistemology of the abyss of the present interpretation would be necessary.  In this sense (and keeping this sense of waves in mind), no waves, no abyss!  Or rather, no thinking in waves, no abyss, since it is this thinking that enables us to infer the existence of the abyss, or, to begin with, to perceive wave-patterns in quantum experiments or anywhere.  But then, physically and epistemologically, the discontinuity and, correlatively, individuality or even singularity of elementary phenomena is crucial as well, again, both for the theory to function and for the abyss to appear.  Indeed we link our thinking in waves, including waves as continuous phenomena, to quantum physical and epistemological discontinuities.  Accordingly, no discontinuity, no abyss either!  In a certain sense, the continuity of waves or quantum continuity in general is primarily phenomenological, while quantum discontinuity is primarily epistemological, although it is also phenomenological insofar as one must think in discontinuous terms and ways, including in thinking (through) this epistemology.  A complex, multi-symmetrical and multi-asymmetrical situation!

3.  The Highest Musicality in the Sphere of Thought

I am now going to translate or transfer (with some important differences) this situation, first to modernist music in general and then specifically to Toledo’s composition.  What are “quantum” features of modernist music?  First of all, there is discontinuity, both in terms of sound and, more significantly, conceptually, whereby we encounter seemingly and deliberately unmotivated interruptions of established, but usually brief, continuities or switches to new continuities, usually equally brief.  This structure leads to a certain a-causality (i.e. to breaks in expected causal chains set up by preceding developments) and heterogeneity even in relatively “linear” chains.  Beyond this heterogeneity, one also finds a more radical (than in the previous tradition) heterogeneity in the relationships between parallel lines or rather simultaneously occurring musical events, which introduce dissonant (in either sense) conflicts and clashes within a given piece.

It appears that, analogously to quantum theory, these elements are reflected more readily in discontinuous, discrete types of musical compositions, such as (simplifying the situation) those of Stravinsky and those composers who follow him along these lines.  This type of music seems to depart more radically from the preceding nineteenth-century (“Romantic”) tradition than more continuous wave-like modernist music, including and in particular, that of Schönberg.  I, again, simplify the case.  Schönberg’s music contains certain discontinuous elements, while Stravinsky’s works contain wave-like aspects, and some compositions may indeed be seen as belonging to the wave-music paradigm.  Also, the “Romantic” tradition, at least from Beethoven on, contains or anticipates such modernist elements.  It does not, however, take these elements to the limits we encounter in modernist music, and these limits are most decisive here.  By contrast, some earlier pre-modernist wave-like music, such as, again, that of Debussy may be seen as a culmination of the wave-like paradigm, taken it to the limit of its complexity, although certain of Debussy’s compositions, such as, for example, A Toy Box and his string quartet in G-minor, may be seen as closer to Schönberg, and even to Stravinsky.

It is worth qualifying that “decisions” (choices, imperatives, and so forth) of taking one’s music, or literature and other art, or indeed physics, in one direction or another, toward a greater or lesser departure from the tradition in music is a complex aesthetic and cultural matter.  One can “decide” to be, or cannot help being, more traditional.  One might think, for example, of Richard Strauss as opposed to Schönberg or Stravinsky, or, in literature, Thomas Mann as opposed to Kafka, Joyce, or Musil, or in physics Einstein and Schrödinger (at least they tried to be), as opposed to Bohr and Heisenberg.  Thomas Mann himself addressed this very question, not surprisingly via modernist music, in Doctor Faustus, in part with the role of the piano, essentially a percussive instrument, in the Romantic tradition, from Beethoven on, which is important in the context of Toledo’s music.  It may also be noted that (the waves of) Debussy’s La Mer play in the novel a role analogous to the one the composition plays in the present discussion.  The novel also addresses at length the role of visual aspects of musical scores.  I cannot, however, consider these subjects here, except for registering that one indeed finds in Beethoven’s music arguably the most significant earlier anticipation of certain modernist elements, such as the interruptive discontinuities in question here.  Schönberg provides the main inspiration for Thomas Mann, and along with Adorno, some major help in Mann’s treatment of music in the novel.  By contrast, his music, with its, in Joyce’s word, chaosmic abysses, is a great source of ambivalence and unease for Thomas Mann.

The success or failure of either more traditional or more radical approaches (and the balances are complex in nearly all major cases) is, however, differently determined in art and science.  And then there is always the question, especially in literature and the arts:  A success or failure, among whom?

Now, the question to which Toledo’s music appears to respond is whether and to what degree the radical modernist elements in question could be recaptured or even radicalized within a wave paradigm.  Specifically, could these elements be recaptured or extended by a certain complex interplay of waves, continuities, perhaps of a very particular type, or set of types, putting them to work against one another, even though certain discontinuous elements are retained as well?  Must they be retained, thus also posing the question how far one is able to reach on this road by waves alone?

The situation, accordingly, is not unlike that to which Schrödinger tried to respond with his wave mechanics in quantum physics, as considered above, but as we have seen, with some crucial differences as well, ultimately bringing Toledo’s musical thinking closer to that of Bohr and Heisenberg, while retaining the role of waves in his music.  First, Toledo’s music does not appear to aim to restore anything to the classical (i.e. in this case “Romantic”) tradition, but in fact to depart from it and even from preceding modernist tradition.  In this respect, one might indeed want to be closer to Stravinsky, perhaps closer than to anyone else, and Toledo’s music may be.  Secondly, the question here is, again, that of relating to that which is forever hidden in the abyss though the image of waves, of waves as effects of that hidden, as opposed to representing or even remotely conceiving of that hidden in terms of waves, and in this respect this thinking is closer to Bohr and Heisenberg than to Schrödinger.

Indeed, as I have argued all along, in both cases, especially Bohr’s, thinking in waves was just as crucial, in terms of physics and epistemology alike, as it was for Schrödinger.  It is, however, an epistemologically very different type of thinking in waves, especially and most crucially, by virtue of epistemological discontinuity, and hence an encounter with the abyss, the quantum abyss of nature it leads to.  In short, at stake is how we conceive of the conceptual architecture of waves and the epistemology in which this architecture participates.  In this sense, we may, again, say that while the elementary physical phenomena they consider are discontinuous, although sometimes organized in certain wave-like patterns, the space of their thinking, including and in particular as it relates to the abyss and the epistemological discontinuity of the abyss, is largely governed by thinking in waves.  Toledo’s composition, I argue, represents an analogous phenomenal space, and the space of its musical thinking itself is defined in the same way.  In this sense, as (conjoined) encounters with an abyss, both are closer to the present, rather than Schrödinger’s, view of quantum mechanics.

As I said, it is especially close to the way one thinks conceptually or, one might say, in quantum mechanics and how this thinking encounters its abysses in this case.  Quantum mechanics is a mathematical and phenomenological music written “for an encounter with the abyss,” and it is as such that it offers us “the highest musicality in the sphere of thought,” unless it is offered to us by twentieth-century music itself.  It is fitting that Einstein applied this phrase to Bohr, who was first to enter such an encounter, with the help of the ideas and work of Einstein himself and Planck, both of whom, however, preferred a different type of music in physics, wave-like but without the abyss.  

We may now return to our possible imaging and imagining of the black-hole radiation and the parallel between them and the phenomenology of Toledo’s composition, from the, conceptually and epistemologically, more complex perspective developed by the preceding discussion.  As I mentioned at the outset, the already complex, partly continuous and partly discontinuous picture of the black-hole radiation is complicated yet further and, given the quantum aspects of this radiation inevitably, depending on how we use and arrange our observation technology, instruments that shape and create this picture.  If, however, we take the epistemology of quantum mechanics, as here considered, into account, a rather different and still more complex phenomenal space emerges, when we think through the actual imagery received from such instruments, each designed for different aspects of this radiation, and working in complex regimes, including, given the mutual exclusivity of certain arrangements involved, complementary regimes.  For, as we have seen, depending on how we set the experiment, we can have either a “wave” picture or a “particle” picture of such radiation (using “wave” and “particle” in the sense considered here) but never both together.  Now we can have both images, say, as photographs, and look at them (or picture them in our minds) simultaneously.  We can also represent such pictures in a certain type of music, say, one part of it written for strings and another one for percussions (or, as Toledo does, we can also do so by making our instruments sound like their counterparts), to be performed and listened to at the same time.  We can also have a musical score reflecting such contrasting pictures, for example, again, corresponding to the scores for different instruments.  However, we can never, in principle, encounter both types of situations or, accordingly, both types of pictures, simultaneously in any single experimental situation, in any given experiment actually performed on a given quantum object.  It is worth noting that we can actually set up more multiple complementary situations by performing more complex experiments even in the case of an elementary particle, let alone of a composite object, such as a black hole.  By the same token, some of the wave-like processes involved now represent conceptual or phenomenal entities (rather than actual physical processes), which signal the abyss responsible for what can be observed or even in principle imagined, especially by way of visualization, such as these very entities.

It is this more complex phenomenal and conceptual architecture, and the encounter with and in the abyss, rather than a set of pictures or sound records obtained by our physical instruments, that is rendered or (Paul de Man would say) allegorized by Toledo’s composition, which is also an occasion for our encounter with and in the abyss.  Toledo’ twenty-four instrumental scores, which in turn follow and combine complex discontinuous and continuous patterns, may in part reflect such visual or audial imagery (phenomenalized in the process).  It is as if we had a set of complementary images or forms of music for twenty-four instruments, with further complementarities added within each instrumental part.  (As I said, we can have multiple, rather than only dual complementary situations in physics as well.)  The overall image or image/un-image (while still a “sound image” it can, it follows, no longer be only a “sound image”) is that of the conceptuality arising in the quantum physics of black holes, most especially because only this image/un-mage allows for an encounter with and in the abyss, never imageable or imaginable.

Let me reiterate here that, as is the rest of this essay, the comments just offered must be seen along the lines of Deleuze’s argument, cited earlier, concerning “essentially inexact yet completely rigorous notions that scientists can’t do without, which belong equally to scientists, philosophers, and artists.  They have to be made rigorous in a way that’s not directly scientific, so that when a scientist manages to do this he becomes a philosopher, an artist, too.”  It is this “inexact” rigor, that is, philosophical or artistic, including musical rigor (I am, again, not sure that one should use “inexact” here), that is at stake in the present argument, which, again, could be made apart from quantum physics.  The point is, however, that this type of conceptual and epistemological architecture is crucial to quantum theory as well, where of course it has its mathematical (exact) counterpart, to which, however, it cannot be reduced.  

The musical genealogy of this type of music or, again, this type of epistemology and conceptuality, including musical thinking in waves, is a more complex matter, which I shall put aside here, although Schönberg’s work and ideas do come to mind, as is also suggested by Doctor Faustus.  Writing this type of music in waves is a tall order, assuming that discontinuity helps and, it appears, even primarily defines the direction in question, as just explained.  Or does it?  This is, again, a difficult question.  Would, for example, Deleuze’s concept of the new Baroque, extending and transforming the historical and specifically Leibniz’s Baroque into twentieth-century art, including music, allow for a more continuous if radical modernism?  Deleuze’s concept of the Baroque, as developed in his The Fold: Leibniz and the Baroque is based, at least in large part, on the idea of continuity, and indeed, along and interactively with the fold itself, on the idea and image of waves.  So are most of Deleuze’s key concepts, such as rhizome, smooth space, or the fold.  Deleuze and Félix Guattari open their A Thousand Plateaus with a peculiarly pictorial musical score for the piano, but written in continuous lines joined into a rhizome, by Sylvano Busotti (A Thousand Plateaus, tr. Brian Massumi, Minneapolis: University of Minnesota Press, 1994, p. 3), which could be related to the graphic architecture or graphic gravitation of Toledo’s piece.  One might see Deleuze as primarily a philosopher of continuity, the “Schrödinger” of twentieth-century philosophy, who takes the idea to its most radical limit thus far.  Deleuze’s primary mathematical source of  “inexact concepts” is Bernhard Riemann’s topology and differential geometry, also a primary source of Einstein’s theory of gravitation, known as general relativity.  This mathematics is primarily that of continuity, which of course takes nothing away from its mathematical or conceptual, including philosophical, greatness.

Deleuze’s philosophical ideas are important, including in this respect, for Toledo’s music, and both, it follows from this discussion, pursue a similar task of a more continuous, wave-like, modernism, or radical modernism or post-modernism (in the sense of extending the radical aspects of modernism, as explained above).  One can speak here of wave-rhizomes, or rhizome-waves.  Deleuze himself invokes Boulez (especially via his Folds upon Folds, inspired by Mallarmé’s folds, a concept of continuity, at least as read by Deleuze) and Stockhausen.  Mallarmé of course was an inspiration for Debussy’s music earlier, and, as I mentioned above, was an important (here relevant) figure for Derrida, including, it is worth adding, in terms of Mallarmé’s engagement with music and dance.  This is an immensely rich cultural landscape.  For the moment, all these figures serve Deleuze as examples of “new harmonies” of the Baroque in music, to borrow the title of Paul Klee’s famous painting, which can be interestingly considered here as an example of modernist continuity.  Klee is himself the key example of the new Baroque, of the Baroque line—or wave—for Deleuze.  How far could the Baroque continuities, even those of the new Baroque, take us?  To cite Deleuze's conclusion in The Fold, which gives this idea the dimension and the harmony, the new divergent harmony of the new Baroque:  

To the degree that the world is now made up of divergent series (the chaosmos), or that crapshooting replaces the game of Plenitude, the monad is now unable to contain the entire world as if in a closed circle that can be modified by projection.  It now opens on a trajectory or a spiral in expansion that moves further and further away from a center.  A vertical harmonic can no longer be distinguished from a horizontal harmonic, just like the private condition of a dominant monad that produces its own accords in itself, and the public condition of monads in a crowd that follows the lines of melody.  The two begin to fuse on a sort of diagonal, where the monads penetrate each other, and modified, inseparable from the groups of prehension that carry them along and make up as many transitory captures.

The question always entails living in the world, but Stockhausen's musical habitat or Dubuffet's plastic habitat do not allow the difference of inside and outside, of public and private, to survive.  They identify variation and trajectory, and overtake monadology with a “nomadology.”  Music has stayed at home: what has changed now is the organization of the home and its nature.  We are all still Leibnizian, although accords no longer convey our world or our text.  We are discovering new ways of folding, akin to new envelopments, but we all remain Leibnizian because what always matters is folding, unfolding, refolding.  (The Fold: Leibniz and the Baroque, tr. Tom Conley, Minneapolis: University of Minnesota Press, 1993, p. 137; emphasis added)

One could certainly locate similar movements in Toledo’s music, by way of wave-image (to allude to Deleuze’s subtitles, “movement image” and “time image,” of Cinema 1 and Cinema 2, both, again, indebted to Riemann’s ideas).  On the one hand, as I have stressed throughout, the “abyss” of Toledo’s composition, as made apparent or rather (it can, again, never appear in any form, is irreducibly “unappearable” and “unthinkable”) felt though the play of waves or of waves and discontinuities, even explosions, with which the composition begins and ends, suggest something else.  It suggests a radical (epistemological) discontinuity akin to that of black holes or quantum mechanics, which, I also suggested here, also involves giving a different, quantum-mechanical, conceptual epistemological meaning to continuities and specifically waves, so that they become the waves sent from the abyss and enable our encounter with this abyss.

If one wants to invoke contemporary philosophy, this aspect of Toledo’s music would be closer to the thought of Derrida, for whom, we recall, Mallarmé (read along more Derridean lines) is a major figure as well, than to that of Deleuze.  One could, however, also think of such figures as Friedrich Nietzsche, Jacques Lacan, Maurice Blanchot, and Paul de Man.  Or one could, again, think of Heisenberg and Bohr in quantum physics, especially if one takes into account the crucial role of thinking in waves in their thinking about things quantum—waves as the signal that we have encountered the quantum abyss of nature, waves as our encounter with this abyss.  What is crucial, however, is that, as the wave image linked to the unimaginable in the abyss, this music brings itself closer to the most radical thinking of the twentieth and now the twenty-first century in science, philosophy, and art.

This is perhaps “the highest musicality in the sphere of thought,” invoked by Einstein.  There is a justice in using this as the last proper name here, since there would not be the physics of black holes or, even though he never fully accepted it, quantum theory, without the highest musicality of his thought.  Nor perhaps could this thought exist without music, without its waves, in which Einstein thought while doing physics or playing his violin (the picture of his thought while doing either would be quite fantastic).  

…its rapid waves

Now dark—now glittering—now reflecting gloom—

Now lending splendour, where from secret springs

The source of human thought its tribute brings

Of waters,—with a sound but half its own.

The other half comes from the world outside, but the ultimate source is in the abyss, the entrance into which, and a little space beyond, only a little but enough to get a sense of the abyss, is allegorized by the image of the Ravine of Arve, “dark, deep Ravine, … many-colored, many-voiced vale” (12-13).  But the abyss of what?  Of nature?  Of mind?  Perhaps of both, or neither—of something else altogether.