Abstract
In this article,1 centered on my work Tramespace I (2012–13), I elaborate on my thoughts about music from philosophical and musical perspectives. Behind my perspectives is the idea of a “metaphysical space in music composition” that acts as an anchor for these ideas. This “metaphysical space” is a major concept in my doctoral dissertation at Columbia University. However, throughout this lecture I do not refer explicitly to this concept. Instead, two consequential aspects of this metaphysical space are emphasized throughout: (1) the act of composition is not just an act of writing and expressing2 but also, as soon as the writing takes place, one must engage him/herself in the act of listening; and (2) because of (1), the aspect of the “self” that is engaging-in-(1) is illuminated. Which leads to the existential question: “who am I (who composes)?”

Introduction
In this article I will discuss my philosophy of music and my compositional process, and show how they are related to my recent work, Tramespace I.

Tramespace is conceived as a diptych for large ensemble. Tramespace I was commissioned by the Stichting Gaudeamus Muziekweek as the result of the Gaudeamus Prize 2011 that I received at the same festival. Before working on Tramespace I, I wrote a work entitled Tramespace, version extrait, as a part of the Projet Tremplin of the Ensemble Intercontemporain and IRCAM. Tramespace II is a work commissioned by Ensemble Intercontemporain, and I am in the process of composing the piece at the moment.

Tramespace is a hybrid word composed of Trame and Espace. Trame means a woof, or a weft, i.e., cross-thread in a textile. A weaver makes a textile by entangling the weft with warp, and on the surface of the textile there is a sense of space (espace) being created. The metaphors of “material that makes up the woof, and the textile itself,” as well as the “space that is made by the textile,” lead me to consider specific compositional processes, particularly of the counterpoint.

About four years ago, I had a conversation with my close friend and Turkish composer Mahir Cetiz on the topic of “compositional space.” Since then I have become increasingly interested in the concept of space in music and I have come to realize that the act of listening to music enables a realization that arrives twofold: (1) listening to music makes me aware of the space that I am sitting in (e.g. concert hall, outdoor field); as well as: (2) my inner space unfolds limitlessly as I listen to the music.

Before I proceed further, it would be helpful to listen to the entire work. This work is written for 18 musicians. The instrumentation of the ensemble is as follows: 2 Flutes, Oboe, 2 Clarinets, 2 Horns, Trumpet, Trombone, 2 Percussionists, Harp, Piano, 2 Violins, Viola, Violoncello, and Contrabass. For the upper four string instruments, scordatura is employed. Furthermore, many instruments in Group B (stage right) have their instruments tuned by a quarter-tone lower. The idea of quarter-tone tuning was important, as it enabled me to explore the harmony in much finer calibration.3 The recording is from the world première of the work by Asko|Schönberg, conducted by Clark Rundell.

Sound example 1 – Integral recording of Tramespace I

I will divide the following article into two large parts. First, I will approach the ideas of Tramespace I from my philosophical perspective. I will extend my thoughts toward the idea of a “spiral form and structure,” which played a major role in my compositional process. In order to arrive to this concept, I will have to go through two intermediate matters, which are:

1. (1) circularity: what is circularity? How should we characterize circularity in music? And,
2. (2) memory: what is memory? How does memory function in tandem with musical circularity?

After a dialectical synthesis of these two matters, circularity and memory, I arrive at the concept of a spiral form and structure. In the second part of the lecture, I will analyze the Tramespace I from this theoretical perspective. In so doing, I investigate how I expanded my own imagination from the problematics of the spiral form and structure, and how I expressed the resultant compositional ideas in the medium of notated music.4

The reason for the structural division of this paper is the following: in thinking about the problematic called music, I believe that there is a limit to thinking through them when relying solely on theoretical knowledge in music. Sometimes there is a moment where a philosophical thought can help solve a compositional problem that theoretical knowledge alone cannot. I have studied under two composition professors who themselves studied under Gérard Grisey. And at Columbia, I have taken composition lessons with Tristan Murail for a year. Certainly, there is a lot to be learned from the Spectral School, and I will not deny at all that my works are influenced by this compositional movement in one way or another. Perhaps one could say that my sense of harmony timbre comes in part from these composers.

I do not believe, however, that the musical idioms of Spectralism alone could solve the entire body of problems that I experience when I compose music.5 Indeed, there are moments in which a philosophical perspective unexpectedly gives me ideas that I have not thought about before. I do not propose the use of compositional techniques of Spectralism dogmatically. Helmut Lachenmann says, “For me, the only credible thing is the consequential confrontation with the aesthetic categories of the explored compositional means.”6 Indeed, I find it very important to evaluate what I learn–analyzing it from as many different angles as possible–so that I can integrate what I learn into my own compositional language.

To me it is indispensable to maintain a balance between philosophical and theoretical thoughts, in order that we may think about music and engage in the practice of music composition. I believe that the focus of attention in the act of composition today lies in how flexible one can accommodate unavoidable musical problems by mobilizing as diverse perspectives as one can gather and internalise.

Part I – Philosophical Perspectives

§1 Circularity
So let us begin with the concept of circularity, a word rich with connotations, not only in Western art music, but also in the music of the world. For example, in Western classical music, the word reminds us of idée fixe of Berlioz’s Symphonie Fantastique, or of Leitmotif in the Ring Cycle by Wagner. Both of the terms describe the phenomenon in which a composer’s musical ideas recur throughout their works.

Figure 1 – Sheet Music for Tout par compas suy composés by Baude Cordier7

We can also think of the canon as a musical form that symbolizes circularity. For example, we know the French folksong Frère Jacques very well; or in medieval music, toward the end of the Ars Nova period called Ars Subtilior, we find fascinating examples of complex canons. Figure 1 reproduces the sheet music for Tout par compas suy composés by Baude Cordier. Written entirely as a circle, the notation itself is a visual manifestation of the circular nature of the canon. The text for the piece reads:

By you placing three whole beats behind me,
You can chase me joyously
so long as you sing with true feeling8

Consider too the circularity of East Asian music. Here the notion of colotomic structure comes to mind. “Colotomic structure” was originally intended to describe a musical structure of Balinese and Javanese gamelan music. It

describe[s] the phrase structure of the gendhing (“piece”). Each major section of a gendhing begins and ends on a gong beat and is further subdivided into subsections and phrases by several other single-note instruments of the gong type; their function is to mark the skeletal melody (adapted and played by the metallophones in unison) at regular metric periods.9

It is a type of musical structure in which a certain number of beats are grouped as a set, and that set is repeated throughout the music. We can imagine a second hand on a clock ticking away its course clockwise, always coming back to the top, numbered 12. Indonesian Gamelan music, or Japanese Gagaku music, to cite another example, are very good examples of circularity as a musical structure.

Note here that in East Asian music, the word “circularity” implies a structure of the music itself, not just a decorative “feature.” To put it simply, circular structure in East Asian music is a phenomenon that the fixed musical materials recur in during a fixed span of time. The reader may know, for example, that the gong ageng in Gamelan music signals the beginning of a cycle. Or that in Gagaku music, not only do certain melodic materials recur, but that the recurrence is supported by the gesture of a kakko (small drum) that accelerates, followed by the booming strike of the gakudaiko (bass drum), which signals the beginning of a new cycle.

§2 Three Preconditions for Hearing Circularity
As we have observed, circularity is a phenomenon that can be heard in the music from all over the world, not just in Occidental music. We thus have one issue to consider: what are the preconditions for us to hear circularity and understand the concept within a given music? We will cover three such preconditions here.

First: there is an element of time being taken into account in the concept of circularity. This may sound obvious. However, in order to understand the concept of a musical circularity, we have to pay attention to the process of: (1) listening to the music (2) then the concept of circularity gradually appears as musical time passes.10 Music is not painting. It must be remembered that because of its characteristics, music should not be understood geometrically.11

Second: musical circularity involves an element of memory. By the word “memory,” I am referring to the sensation of being reminded of things in the past. Listeners retain aural information, and as they keep listening to the music, the aural information stored in the brains enables them to be reminded of the musical situations in the past. The degree of fidelity in being reminded inevitably varies for each listener. For some, it may be the general “ambiance” of the musical situation that might be reminded. While others, especially for those who are musically experienced, may remember and even recall the fine details of such musical situations contained in the aural information from the past. Within a musical structure, the aid of memory in discerning musical circularity is essential precisely because of the sense of being reminded as a point of departure for formal identification.

And last, the third precondition for hearing musical circularity: the logic of circularity must be developed in the listener’s mind. In other words, a certain sonic event must be able to trigger, even subconsciously, a sense of circularity. However, to be clear, no sound possesses this a priori function in itself. Instead, the preceding two preconditions help establish the context in which a sound event signifies the sense of “returning.”

To illustrate this precondition, consider the following chart, which maps a semiotic understanding of circularity in the tradition of the notions of the signifiant and signifié à la Ferdinand de Saussure. In the chart, signifiant and signifié are imbricated, which signifies the “commutability” of signifant and signifié (i.e. what was signifié turns to signifiant) as in the “unlimited semiosis” theorized by Umberto Eco.12

Figure 2 – Semiotic chart of “Circularity”

Suppose that one is listening to Gamelan music and hears the impulse of a low, metallic, resonating sound. Consider:

1. (1) once the sound impulse is recognized as a gong ageng,
2. (2) the sense of “I have heard it before” is proposed as a signified. By the law of semiotic mutability, when this “I have heard it before” is proposed as a signifier, then
3. (3) the sense of “circular” is proposed as a signified. When the sense of “circular” then is proposed as a signifier, then,
4. (4) finally, one is able to sense that because of the gong strike one can perceive the recurrence of the musical materials around the gong.

As a musical phenomenon is charted semiotically (as seen in Figure 2), it must be noted that a semiotic understanding of music is inexorably linked to the continuum of time and the function of memory, and that the first two preconditions exist in the background. This logical “unfolding” of signifiant/signifier is possible only when the function of the gong ageng was revealed a priori, or when it was deduced by the listener in the process of listening.

§3 The Impossibility of Circularity?
Given what we have discussed up until now, the reader may wonder how I integrate these thoughts about musical circularity into my music. Rather than existing externally, in fact these thoughts were necessary precisely because they provided me with reasons to be skeptical of the idea of musical circularity. There are two ways in which I projected my skepticism toward musical circularity, both of which are anchored by a particular function of time:

1. (1) Musical circularity as a source of epistemological questions: Historical/ethnographical examples of the idea of musical circularity abound, many of which seem to be reliable upon the first glance. However, precisely because of this apparent reliability, the question becomes, “Can we really trust the examples?” Just as Slavoj Žižek points out regarding the ubiquity of the “meaning” of the Ode to Joy movement of Beethoven’s Ninth Symphony, we are prompted to ask, what if the idea of musical circularity has become an empty signifier, in which “a symbol […] can stand for anything?”13
2. (2) The function of human memory, or the reliability thereof: The semiological understanding of musical circularity seems to put the logical unfolding of such an idea into a complete clarity. However, this unfolding is only possible when the elements of time, and more directly to our concerns, memory are considered. The study of Cordier’s Tout par compas suy composés makes it clear that there is an apparent “link between music and memory.”14 While in the earlier discussion the idea of time was present, we have not sufficiently considered the idea of memory.

§4 Musical Circularity’s Problematics as Epistemological Questions
The problematics of musical circularity contains epistemological questions concerning the faculty of our listening, as well as the compositional process. What do I mean by epistemological questions? In order to answer this question, I begin by citing the following scientific experiment, published by The American Association for the Advancement of Science:

People really do walk in circles when they get lost. This according to Jan Souman of the Max Planck Institute for Biological Cybernetics in Germany. He and his colleagues dropped people in an unfamiliar forest or desert, and tracked them via GPS. Souman says that if the sun or moon was out, the volunteers were able to walk straight ahead. But when it was cloudy, they walked in circles.

What’s more, he says the circles were surprisingly small: just a few hundred yards in diameter. Further tests showed that people don’t automatically default to one direction or another. Rather, Souman suspects that without a fixed reference point like the sun, misleading sensory cues can make us veer slightly left or right without realizing it.15

What kind of questions can we imagine from this scientific experiment? We know a posteriori that the path we have traced appears circular, but is it possible to grasp the feeling of walking in circles in the very situation of walking itself? Or, are not we led to believe that we are walking on a straight line while we are walking, in fact, in circles? To put the question in a more general life-context, how can we be certain of what we know as a fact? How can we know that the ground from which the facts stem out is stable and reliable?

To go back to the discussion of circularity in music, the reality of the characteristics of circularity in music could be grasped a posteriori, but how about during the moment of listening to music? Is it possible to listen to a piece of music and know that it is a circular form as the music goes by? Even if we are to grasp the idea of circularity in any given music, how can we be so sure of it? The core of the epistemological questions lies at this moment of doubt. Epistemological questions are the type of questions that shake the foundations of our knowledge of the world. In so doing, we may discover cracks in the foundations, caused by the realization that something that we believe in comfortably turns out to be far from the truth. And these epistemological questions make us unable to deny such an uncertainty in facts. Beyond incertitude, there may exist the feeling of anxiety, or the projection of refusal against learning and accepting the truth.

However, in an interview with Peter Szendy, Lachenmann states the following, comparing himself with his teacher Luigi Nono:

As for me, I was not Marxist, rather religious – but I was doubtful of everything.16

This quote resonates with methodic doubt in Cartesian philosophy, of which Descartes writes,

for I then was wishing to tend only to the quest for the truth, I thought that I had to […] reject everything that I could envisage the element of doubt, so that in the end something, which was entirely indubitable in my belief, would remain.17

Thus, as Lachenmann implies and Descartes points out rightfully, one’s journey to the truth begins by maintaining a dose of skepticism. Only after the crack is made in the foundation of one’s knowledge may one see the truth. The idea of circularity in music fascinates me precisely because I doubt that such a structure is attainable.

On an initial listening to a piece of music, it would be difficult to hear the concept of circularity in the moment of listening, even if the listener is told the music’s characteristics beforehand. In terms of a priori knowledge, we may be able to speak about how social systems, such as program notes, prejudices, or any other preconceived notions, may affect the logical process of listening to any given music.

§5 Further Elaboration of the Second Precondition: Memory
I would now like to extend some thoughts about memory, which Oxford English Dictionary defines as

the faculty by which things are remembered; the capacity for retaining, perpetuating, or reviving the thought of things past.18

What does memory allow us to do? The function of human memory arrives twofold: to remember things, and to recall them. I will approach this from a psychological perspective.
In the previous section, we discussed how the function of memory partly plays an important role in perceiving musical circularity. However, we have not sufficiently addressed in what capacity the function of memory serves in the process of listening to a piece of music, specifically in the process of discerning the musical circularity that we have been discussing. Therefore, we shall now attempt to circumscribe the path in which sounding music is engrained in the human brain. This circumscription will be carried out upon an anchoring question: compared to sounding music, what is the music like in the memory of a human brain?

The act of memorization is not an autonomous, subconscious process. Behind the process there is the will of the person desiring to remember.19 Indeed, the act of memorization is “an active, inferential process guided by a person’s general knowledge and intuitions about the world and by cues in the present environment.”20 To translate this thought to the act of listening to music, one may, for example, anticipate various musical parameters as the music is being played. If one tries to remember the music, then one may attempt to “recite” the music within oneself. One may try to generalize the sounding information into tangible schemes, such as utilizing the pre-existing concepts and logics (e.g., waltz = triple meter = three beats in a measure) in order to maximize the capacity of information to be absorbed in the brain. Once again, such a generalization, or the ability to recite, presupposes the a priori knowledge of such schemes or the ability to discern different musical parameters.

Gray writes that when one hears a tale, or experiences a life-event, the mind “only encodes into long-term memory only some parts of the available information.”21 Thus we can infer that after hearing the span of music, by the time the music is stored in a long-term memory, there is some loss of information at the final destination.

Thus, the act of memorization in the context of listening to music makes one remember the sounding music in an imperfect state; that is, when the sounding music is fully completed, the music in one’s memory is less than one hundred percent. One may argue that formative musical training, solfège in particular, helps musicians remember the music more accurately.22 Indeed, one may feel content after scoring perfectly in dictation, proclaiming that one did indeed write out all the parts flawlessly. But is that any indicative of memorizing all parameters of music? Is it not that the dictation, for example, deals primarily with the mechanical morphology of pitch and rhythm over time, and not the other musical parameters, such as timbre? What about the last-minute interpretive prerogatives taken by performers? Would the listener remember all the minute rubatos that have occurred?

When we are given the opportunity to internally recall the music, or even perform the music, we do so by retrieving the engrained, “partial” information from our brains. Whenever there are inescapable gaps within the fragmented information, they are filled in by a priori knowledge and logic, which acts as a close approximation of the actual musical event, which we may not recall perfectly. Gray continues:

With repeated retelling, it becomes harder to distinguish what was present in the original encoding from what was added later. Thus, memory of the story or experience is not a simple readout of the original information but a construction built and rebuilt from various sources.23

Figure 3 – Hypothetical process of perceived musical information

Based on Grey’s writing, Figure 3 shows the hypothetical process of how we retain musical information in memory, and how such musical information is “eroded” as time passes. Suppose that we listen to a musical work, either in a live concert situation, or in the situation of listening to a recording. On the first level, assuming that we were fully attentive to the aural information that was presented, we experience all of what was being sounded in front of us as a sound artefact, or an Urtext.24 Once the music, the ephemeral phenomenon of sound, has taken place, the Urtext is supposedly retained in human memory. If Gray’s statement is correct, then we are to assume that we will not be able to retain all of the content in the Urtext that was presented to us. However, we experience the illusion of remembering the piece. This is achieved by a series of logical reasoning, which is provided subconsciously by our brains. As time passes, it becomes increasingly obscure to us as to what parts of the retained Urtext in our memory were the actual Urtext, and which, the logical reasoning.

Now, what can we deduce from this psychological ground? More specifically, what role does this scientific ground play in the context of our listening to music? We can point out the imperfection of human’s capacity in memory. As discussed earlier, there is no chance that human beings can retain 100% of what they experience, and that they can replicate what they have experienced verbally, or recreate such experience physically. This means that it would be impossible for us to retain all the musical information in our brains at all times as we listen to a piece of music. However, as we also have discussed earlier concerning the construction of memories, it would be human nature to create logically a musical construction by supplementing musical fragments that we do remember with certain harmonies, melodies, and instrumentation that would make sense (i.e., relying on a priori knowledge). The ways we arrive at such logical conclusions are dependent upon each of our sensibilities, and it could be thought of as a source of the creativity of us as human beings.

§6 Synthesis: Toward the Spiral Structure
Let us now go back to thinking about circularity in a piece of music as we integrate our speculations about how human memory works. Earlier I quoted Gray, who wrote about how we could not process matters in the way a cassette tape recorder could, and I elucidated the notion of the imperfection of our human minds. Indeed, it now seems impossible to realize musical circularity in its purest sense, i.e., to remember the phenomenon perfectly, and to replicate the phenomenon perfectly. Earlier I demonstrated how the Saussurian chart of the notion of circularity is an event on the theoretical level, and, indeed, it is utopian in conception.

Until now, we have focused on such idiosyncrasies of human memory from the perspective of the listener. As for the composer’s perspective, the questions become: “what is the implication of such idiosyncrasies for composers? Is there a way of making such conditions of our memory to our advantage in the praxis of composition?” For me, the further question arises: “if musical circularity was not the structure to employ in my music, what would be the alternative?”

Suppose that the information concerning musical circularity would reach my ears 100%, and I would retain 65% of it in my memory. The rest, 35%, would be filled-in based on logical reasoning as well as a priori experiences. This “landfill” information certainly helps the repetition of musical information in my memory, and helps grow (or at least suggests) the logic of musical circularity in my mind. Considering such “landfill” information alone, however, I could conclude that the succession of such information helps connect the focus from musical circularity to the musical narrative.

Figure 4 – The idea of musical circularity as eroded by the flow of time

Let us suppose that we are listening to a music whose structure embodies circularity. If we recall the earlier discussion of memory from the psychological perspective, in order to remember the musical information embodying circularity, we have to utilize logical reasoning. As time progresses, we become more and more uncertain as to which part of the remembered musical information is the logical reasoning “landfill,” and which part is actually remembered, because memory is a construction, not the reproduction of reality. Figure 4 shows such a process. Here it should be remarked that as we go from one stage to another, time (which is required to hear such circularity) passes. The passage of time means that our perception cannot be fixed with the sensation of deepening, which is the causal effect of circularity; instead, the manner of listening tends to be led toward a linear, diachronic listening.

Thus we arrive at the point of emergence of what could be termed a spiral structure. It is the logical progression from what would be the purely circular structure. This logic was not given a priori; rather, it was the result of skepticism toward musical circularity: “can a purely musical circularity be possible?” As I compared what I hear now with what I heard then, I found myself in between two opposing dynamics: (1) I am listening to music whose structure would be described as circular; yet, (2) I was, and I am, on the inescapable flow of time that is not circular. In other words, at the moment I recognize the resemblance of the past and present, nothing is circular. Thus my proposition in Tramespace I is: I shall construct a musical structure, in which the musical materials “try” to follow the circular structure, but as the circular structure is perceived in the teleological continuum of time, it spirals instead.

Indeed, in my work Tramespace I, the idea of spiral structure manifests itself in different guises throughout the piece. From this point on, therefore, I will discuss Tramespace I in terms of formal scheme, rhythmic language, as well as pitch organization.

Part II – Theoretical Perspectives

“La perception est cognitive et le compositeur est son premier auditeur.”25 (Perception is cognitive, and the composer is the first listener of his own music.)

This quotation by Fabien Lévy is often mistaken as a statement that justifies composers writing for themselves and nobody else. I, however, interpret it as suggesting I should be always responsible for every note that I write. What I shall discuss presently are the compositional techniques I utilized in Tramespace I, which to my best abilities closely correlate with my perception. However, in order to arrive at such techniques, I had to follow the philosophical paths as elucidated in the first part of this lecture. And while I was doing so, I was always aware of my teacher’s maxim.

With this in mind, I shall present an analysis of Tramespace I.

§1 Form
Overview
My approach to formal schemes in music can be stated as follows: I follow a dialectical process between the side in which I willfully make decisions and the side in which the temporal materials are decided a priori. I tend to resist the idea of automatic writing, in which I remove myself from the act of listening, and make prescribed decisions (e.g., “for this section I would use 55 measures because that happens to be the golden ratio of the entire proportion”). Yet, I also find it inadequate to make seemingly impulsive decisions one after another, for such a process does not make the organic structure explicit. I like to set a large goal or an objective, and then to think about how to achieve it. Sometimes there are situations in which I have to change something in my compositional system so that the musical goal is preserved. Sometimes the compositional system that I devised for the musical goal might lead me to an unexpectedly different place altogether. Indeed, this dialectical process is also in conflict with what I imagine and the actual musical events that take place.

In order to think about the form of Tramespace I, I first set the following goal: the piece itself will be a transition from the sonority of white noise to the sonority of what could be perceived as pitch collections. It is easy enough to imagine how it sounds if I were to set such a process in the short span of time of 5 seconds, for example. If the flute plays such a process using overblowing (muraiki), I would hear it as a distinct musical gesture.

However, if I were to execute such a musical gesture for a long span of time (in the end the piece became 18 minutes long), I felt that there would be a need for some kind of structural support, just as in piers that support a big bridge. These musical piers have some kind of dramatic elements, and they reinforce the elements of musical narrativity. For this reason, I decided to divide the process of transition from the “white noise” sound to “non-white noise” sound into five sections.26

Figure 5Figure 5 – Overall sectional divisions of Tramespace I

§1.1 Section I
In Section I (mm. 1–72), white noise-like sound governs the overall soundscape. Here I paid particular attention with regards to the density of the soundscape. By “density” I mean a horizontal/chronometrical density; that is, how much of the sound event there would be in a given amount of time. Of course, it is possible to consider “vertical/chordal density,” as in the thickness of a chord. However, as far as this section is concerned, I considered that such vertical/chordal density assumes a secondary role; the denser the music is in terms of the horizontal/chronometrical density, the more likely that the individual notes align vertically and are played as a chord.

As I shall discuss below, I wrote this piece more in a manner that was horizontal/contrapuntal than vertical/chordal.

At first, I thought of the following three ideas concerning the structure of the section:

1. (1) Section I would be composed in cycles.
2. (2) The 8th-note ratio per cycle of the section would take the contour of the zigzag and diminish in nature: 5 : 3 : 4 : 2 : 1.
3. (3) The beginning of each cycle would be demarcated by the aural signal of a septuplet, principally played by a brush scraping the head of a bass drum. Sometimes there are other instruments, such as contrabass playing the bridge of the instrument, to imitate the scraping as well.

As a matter of fact, I initially wanted to set the form of the entire piece in the ratio of 5 : 3 : 4 : 2 : 1, but the ratio warped as I continued to write the piece. There are a few reasons for this. First, when I started to write the piece, there were some parts that were worth reconsidering in terms of the formal scheme of the piece as a whole. Second, as I revisited and tried to hear the work as I was writing it, there were parts that I needed to shorten, or conversely, to enlarge in order to completely exhaust the compositional system that was in use. Even in this Section I as well, I had to alter the duration of each cycle, significantly warping the 8th-note ratio from the original 5 : 3 : 4 : 2 : 1. However, I still kept the original ratio in mind, and at least the “impression” of the ratio can be seen from the actual example of the duration of each cycle, which follows the similar zigzag contour.

Figure 6 – Durations of the five cycles in Section I

Thus, I set out to write the “initial fragment,” which turned out to be about half of this section, or mm. 6–36.27

Sound example 2 – Measures 6–36

However, after writing the piece, I realized that the fragment was too short. So I did two things: first, I wrote an introduction, which is two pages long on the score. In order to accentuate the hearing of the measure ratio even more, I decided to apply some kinds of musical impulse when each of the duration in the measure ratio starts.

Sound example 3 – A small interlude-like section, measures 37–40

Second, after composing a small interlude-like section (mm. 37–40), I decided that the “initial fragment” would be repeated. As I mentioned earlier, in this section the horizontal/contrapuntal density plays an important role. The “initial fragment” I wrote had a lot of white noise passages and as far as the density was concerned, it was rather high. By the time I decided to repeat the initial fragment, I wanted Section I to start in a rather sparse soundscape, accumulating density toward very high by the end of the section. Thus, in the repeated “initial fragment” from mm. 41–72, I left the sketch intact. Instead, at the mm. 6–37, I took out some notes and passages systematically, so that the section as a whole would have the impression of a gradual accumulation of density.

Therefore, the two subsections at mm. 6–37 and mm. 41–72 come from identical material, except that the former sounds decidedly sparser than the latter.28 As far as the appearance of the score is concerned, it would be very easy to confirm that these two subsections are related. However, on the level of perception, it is perhaps difficult to discern that these two subsections are, in fact, modified repetitions of each other. This seemingly paradoxical phenomenon gives rise to the “epistemological-questions-as-statements” (as elucidated in the previous section), as well as ontological interests in music.

§1.2 Section II
We now arrive at Section II (mm. 73–110). As discussed in the previous paragraphs, I used a lot of white noise-like sound, paid careful attention to the horizontal/contrapuntal densities, and created a sonic texture that resembles a tapestry of some kind. I imagined that the longer this kind of soundscape continued, the more I desired pitches. Hence we can see a latently dialectical relationship between white noise and pitch. For this reason, in Section II, I decided to employ a gradual transition from the white noise-like soundscape to the soundscape with audible pitches. How to do this is discussed in a later section, “Concerning Timbral Morphology.”

Sound example – 4 Section II, beginning

§1.3 Section III
Section III (mm. 111–173) interrupts the flow of the music up to this point, and serves as a sort of loud “interlude.” In terms of the pitch organization, whereas the pitches in use in the previous two sections were limited and sparse, both vertically and horizontally, there is a sudden and progressive development in unfolding the pitches in this section.

Sound example 5 – Section III, beginning

§1.4 Section IV
Section IV (mm. 174–232) is the section that best exemplifies the spiral structure discussed in the previous section. Each sequence starts with a loud musical impulse, then in the course of the sequence a dynamic decay takes place. If we compare the length of each sequence, the section can be divided into three components. In the first component, the length of the sequence becomes longer; then, in the second component, shorter; and, in the final component, the length becomes longer again. This control of parameters is related to my ongoing fascination with spiral structure, but it also has something to do with my personal desire to experiment. By “experiment,” I mean an experiment on the limits of perception of sound. For example, can I hear differences in the lengths of adjoining sequences, chronometrically or subjectively, by gradually lengthening or shortening the sequences? Or, does the existence of clearly audible impulses actually obfuscate the perception of chronometrical length of a sequence?

Furthermore, in this Section IV, I brought back the musical materials (portamento motifs in woodwinds section) from Section II. There are two reasons for this return–formal and thematic. Formally speaking, the spiral structure has something to do with the idea of an imperfect return. I thought that representing Section II in a different context would latently illuminate the imperfection of human memory. Thematically speaking, both the second and fourth sections deal with an idea of timbral morphology, which I shallfurther exploit in the coming section.

Sound example 6 – Section IV, beginning

§1.5 Section V
In the concluding section of the piece (mm. 233–261), it was my intention to create a kind of surprise in terms of sonic transformation. As I mentioned before, in this piece as a whole I attempted to form a transition from noise-like sound to a sound in which one can clearly perceive pitch. As I arrived at Section V, I made a conscious choice in the instrumentation that has not been used in the piece up until this point: i.e., two percussion, piano, and harp, that are playing the instruments in the ordinary way, yielding collections of pitches that have a relatively minimum degree of noise.

These four players play certain successions of chords, which I divided into two groups. The chords in Group A carry the overall characteristics of loud, accented, and dry. Conversely, the chords in Group B are quiet and sustained. To foreshadow the discussion of pitch organization, I can say that the chords in Group A are all derived from the chord in ms. 166, whereas the chords in Group B are derived backwards from ms. 258, i.e., the last chord. I will examine the way I derived the chords in detail below.

Sound example 7 – Section V (beginning)

§1.6 Summary
To summarize the formal scheme: the five sections in this piece indicate a kind of arch-like relationship. That is, Sections I and V are dialectically related in terms of the absence or presence of noise-like elements. Sections II and IV are related in terms of compositional strategy, namely morphing. Thematically these two sections present similarities with one another, as the portamento motifs of woodwinds in Section II are recapitulated in Section IV.

Figure 7 – Formal scheme of Tramespace I

I should like to point out that this arch-like form was the result of the dialogical compositional process. When I first set out to write this piece, I did not have the compositional idea for it, except that I wanted to do a transition of some kind. Taking such a compositional idea as a point of departure, I investigated how such a transition is possible on a larger scale. Not only did I think about it theoretically, but also I reread the music, just as Lévy mentioned the importance of the composer being the first listener of his/her own music.

§2 Three Pre-compositional Materials
As I have mentioned regarding the formal scheme in this piece, the following question arises: from whence do these musical material originate? Thus from here I will mention three pre-compositional materials that I was inspired by, or that I found useful. These three materials are: SDIF files of the inside-the-piano sound; the concept of “infinite hemiola”; and intervallic techniques for developing the pitch organization in my piece. These three groups of materials all resonated with my interests in the spiral musical structure on a philosophical level.

§2.1 SDIF Files – Inside-the-Piano Sound
SDIF (Sound Description Interchange Format) Files are the resources that give me insight and inspiration during pre-compositional research.29 I consider the act of research (that is, “search again and again”) indispensable in the praxis of music composition. When Berio was the adjudicator for the Toru Takemitsu Composition Award in Tokyo in 1999 stated, “I consider the words creativity and research synonymous. I think that creativity without the anchor of the research is futile.”30 SDIF files allow me to look deeper into the various types of sounds that I am using. Certainly, I like to seek the full potential of the timbre. One of Lachenmann’s famous mottos is, “Viola in my life,” which is a quotation of the eponymous work by Morton Feldman. In the motto, Lachenmann sought new contexts in which his unique way of utilizing the instruments could be situated, and consequently, he equally created his own musical language.31 I, too, was at first fascinated by the music of Lachenmann, particularly how he effectively recontextualized out-of-ordinary playing techniques in his music. I also tried to write pieces in a similar fashion; however, I came to the following realization soon afterwards: music has to have not only visual but also aural elements that should appeal to those who listen to it. To appeal aurally means that each of the sounds that is produced as a result of the playing technique has to be integrated into the fabric of a certain musical grammar. In order to do so, I have to reconstruct the musical grammar itself on my own. That realization arrived six years ago, when I first enrolled at Columbia University as a doctoral student.

To reconstruct a musical grammar so that the contemporary playing techniques and their resultant sounds appeal to those who listen—not as shock value but rather as a profound musical experience—has been a project of mine. In order to do so, I began research on how such resultant sounds possess inner possibilities for compositional activities. I started to pay close attention to inside-the-piano playing techniques. I believe that the full potentialities for inside-the-piano playing techniques are still yet to be discovered. The reason for this is that there is still a sense of taboo associated with the interior of the piano.32 Of course, I am hoping that the situation has improved a bit. I am hopeful there are more composers who have a better understanding of how to use such inside-the-piano techniques in ways that they do not harm the instrument, and that more pieces are written in such a manner. In any case, because the inside-the-piano techniques and the resultant sounds still provoke audiences in such ways, I consider that there exist latent notions of political, or even violent, elements of the inside-the-piano sound.

Thus I started the process by recording the sound of the parallel grinding of the low A and A# piano strings with a guitar pick. I made the SDIF-format file using the software AudioSculpt, available from IRCAM. Afterwards, using the software program OpenMusic (also available from IRCAM), I was able to analyze the sounds from various perspectives, such as spectra, frequency traces, amplitudes, and resonances.

Figure 8 – SDIF analysis on Open Music, screenshot

I decided to incorporate the data I gathered from the SDIF files into my composition. But I found the process of incorporating the data into my compositional process—for example, applying the SDIF data for pitch contour directly into the melodic line of my music—too much of a cliché. However, the idea of intentionally misreading musical parameters occurred to me. As seen in Figure 8, as I was evaluating the data for the amplitude transition in my OpenMusic patch, I decided to apply the data for duration of notes. That is, the louder a sound is (the higher the amplitude value is), the longer the note will be. This idea of misreading turned out to be very useful.

The following is an example of misreading: in Section IV, concerning construction of the spiral form, let us suppose that I wanted to control the duration of each cycle, which is to be gradually shortened or lengthened. If I were to shorten or lengthen the sequence in an improvisatory manner, for example plus or minus two beats per sequence, I could do so. However, seeing the thoroughness in timbral control, I felt that deciding the duration of each sequence in a manner that is not “thought out,” was going to put the whole discourse out of balance. This had to be avoided, considering the long-term trajectory of this spiral structure. Indeed, I was reminded of the criticism posed by young Pierre Boulez to the serial music of Arnold Schoenberg. In order to avoid such insufficiencies, I needed to situate a core idea that would affect the decision-making process in my music.

Figure 9 – Graph sketch based on the SDIF analysis

Therefore, graphs such as Figure 9 were very useful to me. First, I attempted to see if the list of values provided by the SDIF files would be useful in my compositional process. If the sequences of values satisfied my musical imagination, then it would be used as is; however, the raw materials were rarely useful. I needed to engage in a dialogical interaction with the raw materials and change the parameters so that the rigidly mechanical raw materials and my personal “interpretation” would be dialectically synthesized.

For example, in the sketch shown in Figure 9, there are few points of interest for the timing of my work. There is a section in the example where I traced the graph with the red line, as is shown in Figure 10.

Figure 10 – Tracing the contour out of the graph sketch

This zigzag line has the characteristic of going slightly upwards, then slightly downwards, then going once more up, then falling drastically down. Based on the shape provided to me, in a rather deformed and approximated manner, I wrote out the sequence of numbers as: 14, 17, 20, 23 and so forth. These numbers became the lengths of the sequences in Section IV. In fact, Section IV starts with a sequence of 17. In the music it is possible to trace the number by counting the amount of eighth notes.

Figure 11 – The “Separation” sketch for Tramespace I33

At the moment the sketch was written, I was associating the word “séparation” with Section IV. That is, while Section III had the general characteristic of the musical materials meshing with each other, giving rise to a “well-integrated” sound, I wanted to disintegrate such a unified sonority in Section IV, just like the debris in the ocean. By using the pseudo-periodic characteristics of the spiral structure, I thought such a sensation of disintegration could be emphasized more strongly.

Furthermore, if I mirrored this shape on a vertical axis, the following shape could be formed:

Figure 12 – Mirroring the shape on a vertical axis

This new shape has the characteristic of starting from the lowest point and thereafter spiking up to the highest point. After this, the line goes down to the intermediate point of the two extremities, then to just above the intermediate point, finishing with just below the intermediate points. If I am to numerically express the order of height of the points, 1 being the lowest and 5 being the highest, it would be:

xxxxx1 : 5 : 3 : 4 : 2

Then if I rotate the order to the left, I obtain the order:

xxxxx5 : 3 : 4 : 2 : 1

This order of height in the new shape of Example 12 was used as the 8th-note ratio in Section I.

As for the exact usage of this ratio in Section I, it would be as follows: after finding the ratio, I felt that the numbers of the ratio appeared to be too rigid, thus I deformed them further to derive the ratio of 7 : 4.5 : 5.5 : 3 : 1.5. I then multiplied each number of the ratio by 3, deriving the sequence of numbers: 21 : 13.5 : 16.5 : 9 : 4.5. These numbers then became the unit of the eighth notes in Section I, starting at ms. 6.34

§2.2 The Infinite Hemiola
The infinite hemiola first emerged when I was writing a miniature work for the Projet Tremplin, a competition for awarding commissions to young composers that was jointly held by IRCAM and Ensemble Intercontemporain. By definition, a hemiola is a rhythmic technique in which a triple meter is forcefully established in the larger context of the duple meter or vice versa, creating a sort of “floating” sensation until the two opposing metric schemes meet vertically.35

Figure 13 Infinite Hemiola, Rhythmic Scheme (First Page; click the picture to view in detail)

This rhythmic scheme fulfills the following conditions:

1. (1) It embeds multiple hemiolas in contrapuntal layers. Specifically, there are three rhythmic layers against each other.
2. (2) As one layer of subdivision ends its process, another layer begins promptly.
3. (3) As (1) and (2) carry themselves out, the time axes of the hemiolas are also displaced. Thus in this musical situation, once the hemiola process starts, at the very beginning of the hemiola process, there is no “meeting point,” i.e., no pulse that shares its attack with any other pulses.

Why did this hemiola-like pulse group interest me? For two reasons, that are dialectically opposed:

1. (1) Because of the meeting points in the proper hemiolas, they are recurrent and circular in nature. Thus this infinite hemiola, too, attempts to be recurrent and circular as well. In other words, it goes infinitely close to the notion of circularity.
2. (2) However, as it seems to promise circularity, the infinite hemiola in fact does not provide the meeting point at the end. This means that while the infinite hemiola gives the impression of promised circularity, it provides a more linear musical path.

These two notions seemed to fit well with my philosophical thoughts as elucidated in the first part of this paper. As a result, this hemiola-like pulse group served as a solution to the issue of musical circularity, or the spiral structure of my music at the local level.

There is one more reason why this infinite hemiola attracted my attention. When I was writing the extracted version of Tramespace for Projet Tremplin, I was making very conscious choices in terms of the disposition of the instruments on the stage. I thought that by applying the pulse scheme of this infinite hemiola to the instruments on the stage, and if I had the musicians play very dry, staccato passages, then I could make the spatial relationship of the instruments on the stage clearer. The following sound example demonstrates how this infinite hemiola is realized, in a cross-rhythmic fashion:

Sound example 8 – An excerpt from Tramespace – extrait pour Projet Tremplin

I was not satisfied with this, however, because the way I utilized the rhythmic model of the infinite hemiola was too literal and monotonous. I was afraid that once I perceived the soundscape as a sort of “effect,” then I was going to lose interest in it rapidly. I thought that there were other ways to derive different sonic materials out of the rhythmic scheme.

First, I re-read the rhythmic scheme of the infinite hemiola. Out of the five lines, at any given moment there are four voices at the most that are sounding simultaneously.36 Instead of hearing the simultaneous pulses as four distinct voices, I imagined how it would be expressed as one single layer. In the sketch shown below, there is handwriting of mine that has “generalized” the rhythmic model into one line, as expressed in (b.).37

Figure 14 – Engineering the rhythmic figure based on the infinite hemiola sketch

If I write this “reduced” line on a single instrument, the perception of the rhythm as the infinite hemiola may become secondary; however, I found it fruitful as a composer to be able to have the mindset to be able to deduce further materials from a core idea.

After obtaining the reduced rhythm, I attempted to utilize it in my piece. In doing so, I tried to utilize a technique that could be considered half-canon, half-heterophony.

Figure 15 – Onishi, Tramespace I: Violin I, measures 174–17738

Figure 15 comes from the Violin I part, mm. 174– 177. It shows one of the consequences of utilizing the reduced rhythm of the infinite hemiola in Tramespace I. After deriving the reduced rhythm in Example 14 (b.), I felt that I needed a rhythmic passage that was slower in pace. I therefore decided to tie every other note as in Example 14 (c.). Thus I was able to obtain a rhythmic passage that is slower, yet retains the irregular characteristics of the original reduced rhythm. The rhythmic figure obtained in Example 17 (c) was then utilized in the violin passage as figured in upper stems, which indicate the location and pressure of the bowing action.

The rhythmic strategy in the strings in Section IV is, therefore, the superposition of the derivatives of the “reduced rhythm” that originally comes from the infinite hemiola scheme. Once a layer (that is played by one instrument) is made, it is metrically displaced in the other string parts. The result is the counterpoint of lines in which the simultaneously sounding materials are self-similar,39 and at the same time globally show the characteristics of a canon.

§2.3 Concerning Pitch Organization
General Remarks
In Tramespace I I was interested in connecting the important compositional element of pitch organization with the problematics of musical circularity, just as I had with the other compositional elements. However, the perceptual properties of pitch offer a unique problem in linking the two, because the function of time in pitch organization is far more elusive than in other compositional elements. Because rhythmic and formal organizations are based on the horizontal plane (a time-based understanding), the problematics of musical circularity, which are largely based on the function of memory (thus the notion of time, both psychological and chronometrical), are an obvious fit with rhythms and forms. Pitches, however, seem to present a different notion of understanding because they reveal themselves instantly. Once they are sounded, they are heard and recognized in a flash of time, an Augenblick. Consider the following passage when played.

Figure 16 – Ascending C major scale (repeated)

I am likely to hear this as a circular passage, but only over time; furthermore, the lowest pitch C serves as a referential point. This is reinforced by the fact that the lowest pitch C is preceded by another C that is an octave higher. But to say that it is because C is always the fundamental and the departing point of a tonal system would be only half-correct. Any pitch can be a fundamental pitch if there is a proper context given. No pitch alone has an a priori function. The context is given in the continuum of time, which means that speaking of pitch organization without giving consideration to temporality would be quite incomplete.

Consider the following chord.

Figure 17 – A chord

By “viewing” the chordal structure of Figure 17, we can find that this chord is a result of the repeated superposition of intervallic sequences from bottom to top: Perfect 4th (or 5 half steps apart), Minor 2nd (1 half step apart), Minor 3rd (3 half steps apart).40

However, if someone played this chord to me for the first time, I would only recognize it as a sound object in which there are multiple notes that are sounded simultaneously. The exact intervallic relationship in the chord would only be recognized, if at all, a posteriori.41 A posteriori implies the recalling of an action from the past, which means that I would have to spend time with this chord, instead of the chord revealing itself to me at once. Again, the act of discovering such an intervallic relationship is only possible when we consider the temporal domain of perception.42

However, while it may only be in indirect ways that we are able to hear circularity or a spiral structure in pitch organization, it is possible to construct a solid, consistent discourse if we adhere to a certain systematic method. And this discourse would bring a kind of stability to the listening experience. Here I am reminded of the brief quotation from Lord Polonius in William Shakespeare’s Hamlet, Act II Scene II: “[…] Though this be madness, yet there is method in ‘t.”43 It is this madness that I am interested in attaining.

§2.3.1 Pitch Morphology: Section III
As far as developing the pitch materials in Tramespace I, I used the following procedures. First, I sought a collection of pitches that would serve as the basic material. By “sought,” I mean that I analyzed the pitch- and spectrum-content of a recorded sound of the lowest strings of the piano being scraped by a guitar pick. As seen in Example 18 (a), there are a lot of pitches that could be obtained. However, I did not use all of them in my piece. The reason is that this is raw data, and with my aesthetic, I must choose several among all these pitches. Without doing so, there is no point of doing research. I subtract pitches in one way or another, deducing several chords that may have potential. From there, once I take the chord I like the most, I can then proceed to further refinement of the pitch materials.

The main chord that is generated in Section III by the brass instruments is generated from the following pitch scheme in Example 18(a), obtained by partial tracking analysis of the SDIF file containing the sound of the lowest two piano strings scraped with a guitar pick.44 From this, I removed the duplicated pitch in unison,45 providing me with the pitch collection of Example 18(b). Thereafter, I arrange them vertically to create a chord, as in Example 18(a). As I allow myself octave transpositions, I come up with the chord seen in Example 18(c2). By the time I derive the (c2) chord, one may argue that this chord is far from the analytical result of the SDIF file. However, I am interested in the dialectical transformation of musical material; that is, the synthesis of the analytical result that is mechanical on one hand with my human desire to interact with such a raw material, on the other hand.

Figure 18 – Extracting pitch materials from SDIF file (click the picture to view in detail)

§2.3.2 Pitch Morphology: Section V (mm. 233–261)
First Attempt
As I mentioned earlier, in Section V the chords are divided into two chord families. Chord Family A is characterized by the forte and staccato-sounding chords. Conversely, the chords that are played piano and sustained, repeatedly, characterize chord Family B. I will focus on Chord Family A.
The chords in Chord Family A are all derived from the following chord:

Figure 19 – The foundational chord of Chord Family A, i.e. Figure 19(c2)

I was initially interested in making permutations of this chord by borrowing the idea of circularity. If I express the chord numerically, where 0 represents the pitch G ¼#, and each number represents the distance from this lowest pitch by quarter-tone step:46

xxxxx[0, 5, 14, 17, 26, 35, 43, 63]

then, once again, each number represents the quarter-tone step, I can express the intervallic relationship in this chord as:

xxxxx(5-9-3-9-9-8-20)

My initial idea was to rotate the order of the intervallic numbers, where the initial number will go last. I would obtain the following orders for the intervallic numbers:

xxxxx(9-3-9-9-8-20-5); (3-9-9-8-20-5-9); (9-9-8-20-5-9-3); etc.

Using this order of intervallic numbers, I reapply these orders to pitch, thus constructing the following succession of chords:

xxxxxChord 1: [0, 5, 14, 17, 26, 35, 43, 63]
xxxxxIntervallic relationship: (5-9-3-9-9-8-20);

xxxxxChord 2: [0, 9, 12, 21, 30, 38, 58, 63]
xxxxxIntervallic relationship: (9-3-9-9-8-20-5);

xxxxxChord 3: [0, 3, 12, 21, 29, 49, 54, 63]
xxxxxIntervallic relationship: (3-9-9-8-20-5-9); etc.

The following chord progression is thus generated:

Figure 20 – Chord progression, first attempt

However, I was not satisfied with this chord progression, because in it the bottommost and topmost notes remain stationary, giving an impression of a double pedal. It gave a rather static impression, and I felt I was in need of a chord progression in which each chord varied more vividly from the others.

Second Attempt
Thus I made a modification to the permutation system so that the topmost note would be varied at each chord change. This I did by inserting an imaginary pitch, which would enable me to obtain an imaginary interval. After obtaining the numerical expression of the original chord:

xxxxx[0, 5, 14, 17, 26, 35, 43, 63]

I decided to insert an imaginary note above the top pitch D#. For this I decided upon G 1/4#, because it was the bass pitch of the chord.47 I now obtained the following pitch heights, in which the bold-italicized number is the fictitious pitch:

xxxxx[0, 5, 14, 17, 26, 35, 43, 63, 72]

Using this new set, I derived the following intervallic relationship, in which, once again, the fictitious interval would be expressed in bold-italic:

xxxxx(5-9-3-9-9-8-20-9);

As before, I rotated the order of the intervallic relationship. Whatever the first number was in the previous order would be the last number of the subsequent order. And I reapplied the order to the pitch according to the order of the intervallic relationship; however, since I would be consistent in that I wanted to have chords of 8 notes each (as in the case of the original chord), I would not use the last number in the order of the intervallic relationship. I thus obtained the following succession of chords. The crossed out number indicates the fictitious pitch and/or the number in each of the order of the intervallic relationship that was not used:

xxxxx[0 5 14 17 26 35 43 63 72]
xxxxxIntervallic relationship: (5-9-3-9-9-8-20-9);

xxxxx[0 9 12 21 30 38 58 67 72]
xxxxxIntervallic relationship: (9-3-9-9-8-20-9-5);

xxxxx[0 3 12 21 29 49 58 63 72]
xxxxxIntervallic relationship: (3-9-9-8-20-9-5-9); etc.

The following chord progression was thus generated:

Figure 21 – Chord progression, second attempt

This was a more satisfactory chord progression, as it gave me a varied contour on the top line of each chord, rather than being stationary on D#. As a result, I found the chord progression to be more “moving” than “stationary.”

In addition, what fascinated me was the relationship between the generative process of this chord progression and the actual aural result. I do not claim to assert any hierarchical relationship between these two, however; rather, I am referring to my interest in how these two actually seem to interact. Specifically, as far as the generative process is concerned, I came closer to the idea of the spiral structure. In the first attempt, the simple rotation and reapplication of the intervallic relationship in order to generate the chords was more of a circular structure, as the intervals were simply reused with a mere rotation. However, I saw that not only the chord progression was unsatisfactory, but also it contradicted my compositional approach to the other parameters. With the second attempt, I deliberately made sure that the rotation of the intervallic relationships would be varied, first by the assertion of the fictitious pitch/interval, and second by the omission of the last number of each intervallic relationship. As a result, while the rotation of the intervallic relationship was still a matter of “circularity,” the rest of the processes in the second attempt reinforced the idea of spirals, in which the departing and ending points of the proposed “circle” did not meet.

Once the chord progression was obtained, I proceeded to further treat each of the chords in the progression. Specifically, since I wanted the trajectory of the chord progression go downwards, I decided to displace certain notes by an octave (cf. Figure 22). By this process, the aforementioned process in the spiral structure qua intervals, is destroyed; however, the collection of pitches remains.48

Figure 22 – Modification to the chord progression of the second attempt via octave displacement

§2.3.3 Motivic Interests In Pitch Morphology
Since SDIF files provide pitch information (via partial tracking analysis) over time, inevitably I am concerned with pitch organization in terms of the horizontal, chronometrical treatment of pitch. I here give a small example of this from Section II of the piece. From measure 86, the woodwinds play passages that contain pitch bends. The aforementioned SDIF file hinted at this portamento motif. There, I sketched out a projected phraseology of the raw material.

In Figure 23(b), I immediately found that there were two fragments, both of which have pitches B ¼# and C, back to B 1/4#.

Figure 23 – Phraseology sketch out of the SDIF analysis

This oblique motion served as a hint for the pitch-bending material of the woodwinds in Sections II and IV, as seen in Figure 24.49

Figure 24 – Tramespace I. Alto Flute, Oboe, and Bass Clarinet in measures 86~91

To conclude discussion of the pitch organization of Tramespace I, I would suggest that to think about pitch is to recognize and be aware of the latent notion of time that affects the pitch manipulation system/process, as well as that gives any pitch a certain contextual understanding. It is also important to consider the relationship between the methodology of pitch manipulation and its perceived result; however, I personally feel that it is important to go beyond binary value judgments. As Lerdahl and Jackendoff argue:

But accessibility per se is not a measure of value. Some of the world’s greatest art is highly inaccessible; most of the ephemeral art is readily accessible. Both tonality and contemporary techniques have produced acknowledged masterpieces; both have produced trash.50

§3 Concerning Timbral Morphology
As I have mentioned in the description of the form for Tramespace I, in Section II I decided to employ a gradual transition from a white noise-like soundscape to a soundscape with audible pitches. In doing so, the idea of a “pivot chord” appeared essential. That is, to take advantage of the transition from one soundscape to another, I decided to choose a sound in which the elements of non-pitched sound and pitched sound coexist. Specifically, the sound I chose was that of the marimba and vibraphone, played with wooden dowels, starting at ms. 73. Thus, while the strings are playing jetés in the register beyond the fingerboards, which contains a lot of noise-like element with high, indeterminate pitches, the sustained sounds of marimba and vibraphones are introduced. Then the woodwinds overlap the percussion with breath tones, which reinforce the sense of pitch, and finally the soundscape changes into the pitched soundscape with a hammer on the frame of the piano at ms. 86.

Sound example 9 – Measures 73~93

In Section II, the idea of timbral morphing was represented by the shift from noise-like sound to non-noise sound, using the idea of a common chord. As I discussed earlier, by a “common chord,” I mean that I selected the sound of the vibraphone/marimba with wooden dowel as a sound that embodies both polarities. In Section IV, the idea of morphing is manifested in two ways. First, what I shall refer to as “dynamics morphing” occurs in this section. As the sequences are played from one to another, this so-called “diminuendo sequence” (i.e., a sequence starting with a strong impulse, then making a diminuendo toward the end of the sequence) gradually morphs into a “crescendo sequence” (i.e., a sequence that starts quietly, or with a less-strong impulse, then intensifies toward the end of the sequence).

As was the case in Section II, the timbral morphing takes place in Section IV, but in a much more involved way. The timbral character of the impulse that initiates each sequence is exploited.

Figure 25 Impulse Characters at the Iteration Points in Section IV

As we see in the chart of Example 25, in each occurrence of the impulse, the presentation of the gesture, both in terms of notes and the instruments that play it, varies from sequence to sequence.
Another important aspect of the timbral morphing in Section IV is that there is a stratification of the micro-impulses. These micro-impulses are played by the percussion, piano, and harp. Each sequence, at least about half of Section IV, begins with a strong impulse, as we have seen in the chart. Behind these strong impulses and other musical materials in the section, the pulsations of these micro-impulses can be heard in the background. The following Figure 26 provides a sketch that served as pre-compositional material.

Figure 26 – Sketch on Sound Types

In the sketch, I have written out some sound types that are similar to one another—that can be played by percussion instruments as well as the piano and the harp. These sounds are arranged according to two principles:

1. (1) from long to short reverberation/resonance
2. (2) from dull to sharp attack of the sound51

As a result, as these sounds are played in the particular arrangement outlined in the sketch, the perceived characteristics of sound changes gradually. Here I am reminded of Lachenmann’s technique of Klangtypen.52) Section IV is constructed in such a way that these micro-impulses are perceived before and after as the orchestration varies from sparse to dense. In the final form, the micro-impulses’ timbral arrangement is carried out differently from the sketch shown above, even using the timbre that was not included in the sketch (e.g., muffled metal tubes at ms. 222 and onward). I consider, however, the sketch such as this one, which explores and exploits the possible timbral trajectory, is essential in order to make myself fully aware of the timbre-types that I am working with.

Final Remarks
In conclusion, I would like to add the following four thoughts.

(1)
While I have covered a range of topics concerning my philosophical and compositional perspectives on Tramespace I, I would like to prevent readers from thinking that these compositional methods are dogmatically critical, such that listening for such methodologies is the only way to properly evaluate my work. Certainly, putting my compositional processes through these compositional methods was important to me. However, on the level of listening to my work, I rely on the strength of the imagination of each listener. Instead of a dogmatic approach, as in, “You have to listen this way and not other ways,” I prefer the approach of alternatives, as in, “There was this way of listening, as well.”

(2)
Sometimes there are composers who do not say much about their music, because they want their music to speak for itself. There are even those who consider the act of saying this and that about their music taboo. As though supporting their argument, Ludwig Wittgenstein ends his Tractatus Logico-Philosophicus with,

7. Whereof one cannot speak, thereof one must be silent.53

Scholars, such as Martin Puchner, argue that Wittgenstein did not stop there.54 I share the same belief as theirs. Music cannot be spoken of; nevertheless, precisely because of that, one must continue to try to speak of it. I certainly will not be able to speak of what a piece of music offers in its entirety. But beyond the imperfection of words about music, I believe that verbalizing about musical phenomena ultimately helps composers reflect upon their own acts in composition, improve themselves as composers, and even as human beings.

(3)
One of the reasons I continue to compose and perform music is to answer the eternal philosophical question, “What is my self?” Of course in answering such a broad and open-ended question, I have to take the element of culture into account. However, I consider it futile to define my own cultural identity by some surface, CV-like reasons such as “Where I was born” or “What nationality I have.” Preexisting cultures cannot be defined so easily; furthermore, it is a life’s work to come to a grasp of one’s own cultures, while we already have them.55 One thing we can say is that any culture is constantly evolving over time. Therefore, to believe ardently in a definition of a culture is merely an indication of the fact that only one facet of the culture is understood. In fact, at that state of understanding, it is plausible to say that the culture is misunderstood.

(4)
Having lived in the United States for the past sixteen years, I would suggest that if I have an American sensibility, then it would be that I believe in the freedom of listeners. I do not demand that others think in the same way as I do when I listen to music. In fact, the diversity of thoughts on one thing affirms the existence of humanity in the over-saturated society of today. Early on I discussed the topic of memory, where we human beings supplement whatever we remember with our a priori experiences as well as with logical reasoning. We can perhaps say, then, that such an act of supplementation relates to our imagination and creativity in the end.56 In The Death of the Author, Roland Barthes writes the following:

In this way is revealed the whole being of writing: a text consists of multiple writings, issuing from several cultures and entering into dialogue with each other, into parody, into contestation; but there is one place where this multiplicity is collected, united, and this place is not the author, as we have hitherto said it was, but the reader: the reader is the very space in which are inscribed, without any being lost, all the citations a writing consists of; the unity of a text is not in its origin, it is in its destination; but this destination can no longer be personal: the reader is a man without history, without biography, without psychology; he is only that someone who holds gathered into a single field all the paths of which the text is constituted.57

With this, Barthes implies that the true value of a work of art resides, not in the work itself, but rather in the aesthetic sensibilities of the readers, spectators, or listeners. This implication empowers them; however, it also gives tremendous responsibility to attain freedom. It is this idea of fulfilling responsibilities, as composers, as listeners, or as human beings, that is foundational to approaching freedom, as utopian as it may seem.

Acknowledgements
I thank Rozalie Hirs, composer and chief editor of The Ear Reader, for inviting me to contribute an article on the webzine. She expressed an interest that I discuss the work Tramespace I. I am glad to have written this document, because as I have mentioned in the footnote, it was already my plan to include the discussion of the work in my doctoral thesis at Columbia University.

At first I was going to write the article in English, but I was given an opportunity to deliver a lecture on my music at Toho Gakuen University in Tokyo, Japan in March 2014. I decided that my lecture would be on Tramespace I. I thank Hitomi Kaneko, Associate Professor in Music Composition at Toho Gakuen University in Tokyo, Japan, for her warm invitation.

The script for the lecture was thus written and delivered in Japanese. The Japanese text is thankfully unpublished. I say “thankfully” because as I translated the script in preparation for The Ear Reader publication, I found myself rewriting the text rather than simply translating it. Japanese words, as far as I observe, have widely different connotations within them than the equivalent English words do. Therefore, I had to supply more words in order to accurately describe what I meant in the lecture. Then, as I read the original Japanese text, I finally realized that even in Japanese I was not thorough in my words. I am glad to have been given the opportunities to improve my text in this way.

I also thank the readers who read the draft of this article and provided me with many invaluable feedbacks. Music theorist Scott Gleason, together with composer Ashley Nail, read the document very attentively, and provided me with grammatical corrections, as well as very potent questions that ultimately helped improve the document. Composer Mahir Cetiz, whose name appears in the early part of the article, was also one of the readers whose thoughtful feedbacks are reflected.

Finally, I thank my parents for their support over the years. I happened to be visiting their home in Hokkaido as I was in the final editing process of this article.

Nanae, Hokkaido, Japan
19 August 2014

Yoshiaki Onishi, Composer
Department of Music
Columbia University
New York, NY, USA

Works Cited

Barthes, Roland. “The Death of the Author.” UbuWeb Papers..

“Colotomic Structure.” In Grove Music Online. Oxford Music Online: Oxford University Press.

Descartes, René, and Eugène Durand. Discours De La Méthode Avec Une Notice Biographique, Une Analyse, Des Notes, Des Extraits Des Autres Ouvrages, Et Un Exposé Critique Des Doctrines Cartésiennes, Par L’abbé Eugène Durand [in French]. Second Edition ed. 1 vols Paris: C. Poussielgue, 1901.

Eco, Umberto. Drift and Unlimited Semiosis. Distinguished Lecturer Series. Bloomington, Indiana: Indiana University, Institute for Advanced Study, 1990.

Gray, Peter. Psychology. 5th ed. 1 vols New York: Worth Publishers, 2007.

“Helmut Lachenmann in Conversation with Catherine Milliken.” Berliner Philharmoniker Digital Concert Hall, 2011.

Hirshon, Bob. “Walking in Circles.” The American Association for the Advancement of Science http://sciencenetlinks.com/science-news/science-updates/walking-in-circles/.

Lachenmann, Helmut. “Typologie Sonore De La Musique Contemporaine.” Translated by Michel Pozmanter and Martin Kaltenecker. In Écrits Et Entretiens, edited by Martin Kaltenecker, 36 – 59 p. Genève: Contrechamps, 2009.

Lachenmann, Helmut, and Martin Kaltenecker. Écrits Et Entretiens. Genève: Contrechamps, 2009.

Lerdahl, Fred, and Ray S. Jackendoff. A Generative Theory of Tonal Music. Cambridge: MIT Press, 1996.

Lévy, Fabien. “”Le Compositeur, Son Oreille, Et Ses Machine À Écrire”.” Unpublished, 2014.

“Memory, N.” [in English]. In Oxford English Dictionary. Oxford University Press.

Onishi, Yoshiaki. “Tramespace I.” Berlin, Germany: Edition Gravis, 2012.

Puchner, Martin. “Doing Logic with a Hammer: Wittgenstein’s “Tractatus” and the Polemics of Logical Positivism.” Journal of the History of Ideas 66, no. 2 (2005): 285-300.

“Sdif Sound Description Interchange Format Documentation.”

Sultan, Agathe. “La Harpe Et La Forge; Poétique De L’ars Subtilior.” In “De Vrai Humain Entendement” : Etudes Sur La Littérature Française De La Fin Du Moyen Âge : Offertes En Hommage À Jacqueline Cerquiglini-Toulet Le 24 Janvier 2003, edited by Jacqueline Cerquiglini-Toulet, Yasmina Foehr-Janssens and Jean-Yves Tilliette. Recherches Et Rencontres, 45-64 pp. Genève: Droz, 2005.

Tokyo Opera City. “［武満徹作曲賞］１９９９年度審査結果・受賞者の紹介.”

Wittgenstein, Ludwig. Tractatus Logico-Philosophicus. 2010.

Žižek, Slavoj. “‘Ode to Joy,’ Followed by Chaos and Despair.” In The New York Times. New York, 2007.

“ピアノ内部奏法の問題.” 音ヲ遊ブ (Japanese Contemporary Music).

_______________________________________________

1. This article is based on a lecture given at Toho Gakuen University in Tokyo, Japan on March 4, 2014. In the lecture I read from a script written in advance. This English text is a translation by me of the original Japanese text. The translation is faithful to the original Japanese text; however, wherever deemed necessary, I have edited, reordered the concepts to be presented, and elaborated the text for an improved readability. This article will become the basis for Part III of the written part of my doctoral dissertation at Columbia University. [↩]
2. In Japanese, the word for “expression” is 表現(Hyō-gen), that is: Hyō: to represent, and gen: to appear, or make something appear. [↩]
3. Pitch organization of the work is discussed later in the article. [↩]
4. While the analysis of Tramespace I in this paper is broad, it is in no way exhaustive. I will pick up on some of the compositional strategies where the rubric of circular/spiral musical structure is well-manifest. [↩]
5. Or any other compositional “–isms” could solve all problems at hand. [↩]
6. Helmut Lachenmann and Martin Kaltenecker, Écrits Et Entretiens (Genève: Contrechamps, 2009). “Pour moi, la seule chose crédible est la confrontation conséquente avec les catégories esthétiques des moyens de composition explorés.” All translations are mine, unless otherwise noted. [↩]
7. The music can be found here [↩]
8. Agathe Sultan, “La Harpe Et La Forge; Poétique De L’ars Subtilior.” In, De Vrai Humain Entendement” : Etudes Sur La Littérature Française De La Fin Du Moyen Âge : Offertes En Hommage À Jacqueline Cerquiglini-Toulet Le 24 Janvier 2003. Jacqueline Cerquiglini-Toulet, Yasmina Foehr-Janssens, and Jean-Yves Tilliette, eds. Recherches Et Rencontres (Genève: Droz, 2005).,53. The original Medieval French reads: “Trois temps entiers par toy poses / Chacer me pues joyeusement / S’en chantant as vray sentiment.” [↩]
9. “Colotomic Structure,” in Grove Music Online. Oxford Music Online (Oxford University Press). Accessed on March 13, 2014. [↩]
10. Which, along with other musical information, often obscures the perception of circularity. [↩]
11. Note, however, that while I say that music is not painting, this is also not to say that in painting one can understand all of what painting offers to viewers instantaneously.  Suppose that someone goes to a museum and comes across a painting.  If there were an interesting element that was in the painting, he/she would stop and look at the painting for a while.  As time passes, he/she may start to discover how such an element is interacting with the surrounding elements.  It is true that in art appreciation, where one can physically see things, such a structural analysis could be executed faster than one could with music.  Even so, it is undeniable that the element of time plays an integral role in the two- and three-dimensional arts. [↩]
12. Umberto Eco, Drift and Unlimited Semiosis, Distinguished Lecturer Series (Bloomington, Indiana: Indiana University, Institute for Advanced Study, 1990). [↩]
13. Slavoj Žižek, “‘Ode to Joy,’ Followed by Chaos and Despair,” in The New York Times (New York 2007). [↩]
14. Cf. Sultan, “La Harpe Et La Forge; Poétique De L’ars Subtilior.”, Pages 52-53. [↩]
15. Bob Hirshon, “Walking in Circles,” The American Association for the Advancement of Science [↩]
16. Lachenmann and Kaltenecker, Écrits Et Entretiens. Page 196 – “Quant à moi, je n’étais pas marxiste, plutôt religieux – mais doutant de tout.” [↩]
17. René Descartes and Eugène Durand, Discours De La Méthode Avec Une Notice Biographique, Une Analyse, Des Notes, Des Extraits Des Autres Ouvrages, Et Un Exposé Critique Des Doctrines Cartésiennes, Par L’abbé Eugène Durand, second ed., 1 vols. (Paris: C. Poussielgue, 1901). 55. [↩]
18. Oxford English Dictionary, “Memory, N.” (Oxford University Press). In this article, the word “memory” is used in the context of the definition shown in the text. [↩]
19. Certainly, the exception would be a traumatic experience, where the drastic change in the life-event is suddenly “forced” on a person. [↩]
20. Peter Gray, Psychology, 5th ed., 1 vols. (New York: Worth Publishers, 2007). 329. [↩]
21. Ibid., 329. [↩]
22. However, what does it mean to be “more accurate?” This notion is already problematic. It presupposes that what is being played live is exactly what is being written on the piece of paper. However, as is evidenced in the MIDI recording of musical works, the literal, mechanical execution of sheet music often renders the musical experience “non-humanlike.” [↩]
23. Gray, Psychology. [↩]
24. Of course, strictly speaking, the word Urtext means an original, or a text in its earliest version, against which the later versions can be compared. [↩]
25. Fabien Lévy, “”Le Compositeur, Son Oreille, Et Ses Machine À Écrire”,” (Unpublished, 2014). [↩]
26. In our discussion of timbre, I intentionally avoid saying “pure pitched sound” or something of the like, because with the exception of the sine wave, all sound contains one form of harmonic spectrum or another, implying that all sound could be placed in the dynamic and subjective continuum of the perception of noise. [↩]
27. Of course, at the time I wrote the fragment, it was mm. 1–32. [↩]
28. Mm. 37 and 72 should be considered “finishing measures,” as these measures both carry the aural idea of static sound without apparent changes in density and dynamics. These two measures, therefore, are not part of the 8th-note ratio scheme, and, as discussed earlier, in the case of ms. 37 it is considered part of the “interlude.” These two measures contrast in character. Ms. 37 reads: “calme et statique” (calm and static), and ms. 72 reads: “nerveux” (nervous). [↩]
29. For more information on SDIF Files, the following site provides a good introduction: “Sdif Sound Description Interchange Format Documentation,” [↩]
30. Tokyo Opera City, “［武満徹作曲賞］１９９９年度審査結果・受賞者の紹介,”, Translation by Y. Onishi [↩]
31. “Helmut Lachenmann in Conversation with Catherine Milliken,” (Berliner Philharmoniker Digital Concert Hall, 2011). View in particular the first five minutes. “Viola in my life,” as said by Lachenmann, can be found at 1:36. [↩]
32. This notion is prevalent especially in Japan. But this is not just a problem of the conservative piano owners for the concert halls, but also possibly the ignorant composers who do not know how to safely use the instrument so that no damage is caused. Read, as an example: “ピアノ内部奏法の問題,” 音ヲ遊ブ (Japanese Contemporary Music) [↩]
33. The sketch reads: “Tramespace I Section IV , “Separation” – each number signifies an iteration value that expands or contracts gradually. [↩]
34. Cf. Figure 6. The decimal number was rounded up. [↩]
35. My interpretation of hemiola goes beyond triple/duple meters, as it employs finer subdivisions of five, for example. [↩]
36. When I generated the infinite hemiola rhythmic scheme, one rule was that a new group of pulsation must start right after the previous one ends. [↩]
37. More precisely, when I reduced the multiple layers into one, I also simplified the rhythmic line whenever feasible. This process is necessary, as it touches on the issue of the metaphysical space of me as a composer, i.e. composer as listener. I do not think of this as a compromise, or degrading factor in my compositional process. Instead, this dialogical process between what is being written and what has been thought, is something I consider to be very important. [↩]
38. Yoshiaki Onishi, Tramespace I, (Berlin, Germany: Edition Gravis, 2012). Used with permission. [↩]
39. i.e. one of the characteristics deduced from the concept of heterophony [↩]
40. I noted the distance in half steps because in the chordal example some pitches are enharmonically spelled so as to avoid unnecessary accidentals. [↩]
41. Even before I recognize it as a “sound object,” which I am using to describe what I heard, there existed the sound that was presented to me. In other words, at the moment I recognized the sound as the “sound object,” it is already an a posteriori event. [↩]
42. One may, however, rightfully object to this notion by citing an example of how major and minor triads in Western Classical Music present, respectively, “happy” and “sad” (or something of the like) affects. However, it is imperative to consider the cultural sedimentation, which gives us the ability of recognize such affective differences. [↩]
43. Lachenmann cites this quotation on occasion. [↩]
44. Note that this file lasts about two seconds long. The entire passage A is therefore to be played during that duration, which is quite fast. Each of the notes that came out of the SDIF file’s partial tracking analysis is given in quarter notes for improved legibility. [↩]
45. I left the duplicated pitch by octave intact. [↩]
46. Thus, in this case, G ¼# would be found at numbers 0, 24, 48, 72, etc. [↩]
47. While I may not hear this fictitious pitch in reality, I imagined that placing an imaginary G ¼# would be appropriate because it is a harmonic spectrum of the fundamental pitch. [↩]
48. One may claim that going through different structural processes is redundant. However, I do prefer this kind of process over ones where the notes are not contextually established. In other words, I do not believe that I could have come to the conclusion I had (i.e., the chord progression obtained by the Second Attempt, as well as its octave-displaced final product), had I not gone through this process. [↩]
49. As a side note, the rhythmic organization of this portamento motif is also generated by the tie-treated rhythms obtained by the infinite hemiola’s reductive line. [↩]
50. Fred Lerdahl and Ray S. Jackendoff, A Generative Theory of Tonal Music, (Cambridge: MIT Press, 1996), Page 301. Here Lerdahl and Jackendoff are referring to the inaudibility of methodological perception in serial music. While I do not claim that my compositional processes are purely serial, this quote appears to be relevant. [↩]
51. Note that this is not an arrangement of sounds by the similar materials that vibrate in order to produce these sounds. Furthermore, the last one, where the guitar pick is used to grate the pleats of the low piano string, is different in terms of the manner the sound is pronounced, but I put it in the continuum of the morphing sound type. This particular sound is used at the very end. [↩]
52. Helmut Lachenmann, “Typologie Sonore De La Musique Contemporaine,” in Écrits Et Entretiens, ed. Martin Kaltenecker (Genève: Contrechamps, 2009 [↩]
53. Ludwig Wittgenstein, Tractatus Logico-Philosophicus, (2010). Page 90. [↩]
54. For example: Martin Puchner, “Doing Logic with a Hammer: Wittgenstein’s “Tractatus” and the Polemics of Logical Positivism,” Journal of the History of Ideas 66, no. 2 (2005). 297. “[…]And so it [manifesto] considers its own speech acts to be the last necessary speech acts, the final word before the real action begins. Every manifesto thinks that it is the last manifesto, that it has expressed everything that needs to be expressed. The bitter truth that all avant-garde manifestos only led to more manifestos is repeated in Wittgenstein’s own oeuvre: after having said the last word, Wittgenstein had to begin all over again.” [↩]
55. This is the thesis of the first tableau of the Ten Ox-Herding Pictures, instructive materials used by Zen practitioners to awaken their awareness of such issues as existences. [↩]
56. Both words imagination and creativity are pronounced as sōzō-sei in Japanese. [↩]
57. Roland Barthes, “The Death of the Author,”, UbuWeb Papers. (Translated by Richard Howard) [↩]