Resources and materials

Some of the most basic things I need to know when I am beginning a project arise out of and are controlled in relation to numbers: How many? How large? In what order? As I am trying to create a local, a delimited, "history" (as indicated above), consistencies of all sorts, normatives are fundamental. Conventions do not necessarily produce conventionality, and, indeed, it is difficult to produce anything meaningful without them. If a musical work is not saturated with an always-evolving sense of what is probable, listeners cannot grasp or be deeply affected by what actually does occur. As general ideas about what factors and purposes will be germane to a work arise, and as the specification of an overall plan evolves, gaining greater and greater specificity, I needed to know not just that there will be "several" themes, but exactly how many, to understand the implications of that number (rather than another, slightly larger, slightly smaller). That there might be one theme at the basis of a twenty-minute-plus musical work is conceivable, but probably not optimal. But to have, on the other hand, say, 25 themes would have been yet more implausible.

Thus, numbers matter, as does, even, the question: How many numbers? If one is identifying a group of numbers to control some dimension of the piece, how large should that group optimally be? There are, I think, general boundary conditions for many aspects of musical behaviors, and they need to be considered as one sets about attempting to define this aspect of the normative behaviors for a new work. I usually seek an appropriate logarithmic series on the basis of the initial formal ideas and the approximate duration intended. In the case of Angel, the formative series was, as already mentioned: 1.5, 2, 3.5, 5.5, 9, 14.5, 23.5, 38, 61.5, 99.5, 161, 260.5, … This sequence is one that I have used repeatedly because it produces sectional durations (if the numbers are taken as seconds) of plausible and useful size. The primary proportions, which control the final overall plan for Angel, come from this series.

There is a second, extrapolated, series: 1, 2, 3, 7, 11, 18, 29, 47, … This set of integers, also approximately logarithmic, does not (by design) share the nested quality of its related proportional series.

Seeking an integer series that would parallel the logarithmic succession, I took the first 8 members of the original series (0.5, …, 23.5), multiplied them by 10, and then divided by five. Numbers from the integer set determine the number of any item which needs to be established during composition: the number of computer sections (11), the number of transitions and of combinations (2 each), the number of notes in the primary row (18), the number of glissandi in Other (11), the number of iterated chords in Theme 2 (11), and so on. They also define the specific rhythmic values within subsections of the formal plan. I will return to this issue in a moment. Before doing so, it is necessary to look briefly at the issue of tempo. As, in the end, I decided upon five thematic elements, and because a maximally independent identity for each was a basic criterion, each was given a distinct tempo: Theme 1 at MM120, Theme 2 at MM150, Theme 3 at MM60, Theme 4 at MM180, and Theme 5 at MM90. These 5 tempi are related by relatively simple proportions, and, thus, it is straightforward to calculate what equivalences there are from one to the next (e.g., an eighth note at MM 60 = 0.5 seconds, as does a dotted-eighth at MM 90, a quarter at MM 120, five sixteenths at MM 150, and a dotted quarter at MM 180). Such a table was prepared, allowing me to rewrite, as necessary, aspects of the material of different themes in whatever base tempo was appropriate to any section. The musical ideas of each theme could be combined as needed and still retain their characteristic speeds in what was, in effect, a poly-tempoed composite.

The proportional dimensions of each section and subsection were determined by the proportional series (1.5, 2, 3.5, 5.5, …) and the reference speeds were determined by one of five tempi. Within each subsection of a theme, a repertoire of possible rhythmic values was proposed by the integer set (1, 2, 3, 4, …). Proportionality is still evoked here, as at the level of sectional dimensions, but in a flexible fashion. Assume that I was working on Theme 1 (MM 120). If I select the sixteenth note as the basic unit, then I can freely choose rhythmic values from an allowed collection: a sixteenth, an eighth, a dotted eighth, a quarter, a double-dotted quarter, a half note tied to a dotted eight, and so on. But, still within the same base tempo, I could also select a different basic unit (e.g., a triplet eighth or a quintuplet sixteenth) and have available another set, another related reservoir of rhythmic values (in effect, a "tempo within a tempo"). The only caveat is that I would not change the basic reference unit within a tempo too rapidly. I would not normally, for example, alter this value within a subsection, only at subsection boundaries.

Before leaving the subject of numbers, it should be mentioned that, as composition proceeds, matters become more complicated. Most significant, perhaps, is that the restricted set of proportional numbers in the series guiding the size (in seconds) of sections and subsections could not be maintained when it came to the planning of the Domain version. The reasons for this will become clear below. Further, it is the case that the most precise and consistent use of the integer series is to be found in the thematic elements themselves. As indicated, their separate identities are perceptually crucial, because they become the source of (almost) all else. But as they are extrapolated, in, for example, a Transition or a Combination, other factors may override the rhythmic precision of extracts, though primarily at their edges (that is, at the points where the extract is made to fit within the new context).

The pitch resource for Angel was extrapolated from a basic 12-tone row: C, Gb, F, Ab, C#, A, Bb. E, D, G, Eb, B (cf. Ex. 7). I evolved an 18-pitch series (Ex. 6) by slight permutations and extension such that there were sub-units of 4 (C, Gb, F, Ab) + 7 (A, C#, Bb, E, B, D, G) + 7 (Eb, A, E, F, Ab, C#, G) = 18. Thus, the parsing of the total row was provoked and guided by four numbers from the integer set (3, 4, 7, and 18). This pitch resource was highly disjunct, containing only three half steps. For other compositional purposes, I extracted conjunct sets from the basic 18-note series maximizing the small-interval criterion. Performing metaphoric surgery by longitudinal sections (Ex. 7), three new sets of pitches were obtained: at the top: Ab, A, Bb, B, A, Ab, G; in the middle: Gb, F, E, G, Eb, E, F; and at the bottom: C, C#, D, C#. The content of these more conjunct strata (Ex. 8) became the source of several intermediate groupings which, in turn, allowed a new conjunct composite of 56 pitches: 11 (C#, C, B, A, Ab, G, Gb, F, E, G, Ab) + 11 (A, Bb, B, C, C#, D, Db, Gb, F, E, G) + 11 (Eb, E, F, Ab, A, Bb, B, A, Ab, G, F) + 11 (E, Eb, G, E, F, Gb, C#, D, C#, C, G) + 12 (Ab, A, B, Bb, A, Ab, F, E, Eb, G, D, C#). Both of these resources (the disjunct and the conjunct) were transposed to all of the 12 chromatic levels and utilized in ways that maximized the distinctions between the five thematic elements.

The ensemble was conceived (though not, in the end reliably used) as strata defined along familial lines (woodwinds, brass, percussion, strings). In order to achieve a consistent and well-balanced sonority and to maintain maximal agility, I avoided double reeds, choosing 2 flutes (both doubling piccolo), clarinet, bass clarinet, horn, 2 trumpets, trombone, three percussionists, 2 violins, viola, cello, and contrabass. I considered enlarging the ensemble to 18 (a favored integer) by adding a contra-bassoon and a second trombone, but decided that 16 was the more practical alternative.

It is regrettable that the contemporary chamber ensemble (The Ensemble Intercontemporain, founded by Pierre Boulez at his admirable research center, Ircam, is the most notable exemplar) has been conceived as a flexible resource, a reservoir able to provide all the likely smaller combinations of instruments needed (string quartet, woodwind quintet, etc.). The widespread existence of this sort of ensemble established an extremely heterogeneous timbral mix, an "ideal" which renders the fusion of multiple instruments into clear and consistent harmonic entities difficult if not unmanageable. I resist the "one-of-a-kind" model, choosing, rather, to allow the simulation of well-matched choirs.

Two editorial algorithms, SPLITZ and SPIRLZ, were used extensively in the computer component of Angel (discussed in detail later), though not — at least explicitly — in the composed instrumental materials. "At least explicitly" because the underlying ideals of these two complementary algorithms are quite general, and they do influence my "by-hand" composing at the level of local strategies even when I do not systematically invoke them. These algorithms have been extensively described elsewhere (FN: ICMC Proceedings, The Hague, 1986 and Form and Method: Composing Music, Routledge Publishers, 2002). In brief, SPIRLZ samples a source so as to produce cycles of extracted segments progressively moving outwards from a designated starting point within the source. A segment is extracted, for example, just after a starting point somewhere within the source, then prior to it, then again after but further out from the starting point, etc., in a continuing, back-and-forth alternation, just as though a blade were spiraling out from a center towards the outer limits of the source. Each cycle's values become progressively longer (or shorter) so that the effect of waves of alternating pulsations occurs. Depending upon how the algorithm is applied, it can re-cast its source material in a variety of ways, which tend either to converge on an undifferentiated blur of tiny segments, or expand outwards from such a fined-grained beginning towards progressively larger (and more recognizably sized) segments -- as though a flower were opening its petals. The output is monophonic, because in it, all of the extracted segments are placed end-to-end in a single stream.

SPLITZ fragments its source in a different way, cutting it into a proportionally defined set of segments that is dealt with as two, quasi-contrapuntal output layers: the odd half-pass and the even half-pass. While the odd half-pass segments are arrayed chronologically, with silences of continually increasing size interposed between them, the segments from the even half-pass are presented in reverse order, with the magnitude of interposed silences continually shrinking. What results is a two-layered mosaic of discrete fragments spread out in time as widely as the composer wishes. The plot can be thickened, along with the texture, by adding a second, or third (or more) set of half-passes. Depending upon the degree to which the algorithm is invoked to expand the temporal dimensions of the source, a hall-of-mirrors impression can result.

Both algorithms are called "editorial" because they do nothing to the original source other than to cut it into constituent segments and rearrange these segments, in a particular way. The essence of these processes is to enlarge the time domain over which a source casts its influence, and to superimpose a new formal ideal over the structural character of the original source. One obtains what might be thought of as structural modulation.

Because of how the above-mentioned algorithms work, and because of the fact that their influence is also to be found in the nonalgorithmic aspects of my composition, it follows that there is considerable importance attached to the textural definition of thematic sources (in the present case, the thematic elements). For example, if a source has a mirrored nature, neither SPIRLZ nor SPLITZ transformations would produce characteristically contrasted outputs (because, if mixing very early and very late sections of the source results in an undifferentiated succession (SPIRLZ) or overlap (SPLITZ), the structural profile of the algorithms is undercut, rendered anemic).

As a result of this procedural pre-condition, which came into existence with Archipelago, commissioned by Ircam and realized in 1982-83, I began to plan the textural character of each section and subsection of anything which is to serve as a source for algorithmic processing. There is also a strong, built-in predilection for on-going, even narratively evolving formal ideals. It must be mentioned that while "texture" in its most elementary sense refers to familiar ideals such as contrapuntal, homophonic, or "pointillistic" behaviors, there is another important quality inherent in texture: musical character, the expressive or characterizing ideal the music is meant to serve. (One is always thinking of each thematic component as an aspect of the total resource that the overall piece may call upon.) So textural planning includes a number of considerations: Is the music conceived in horizontal or vertical terms? Is it temporally patterned or discursive? How does it use register? Are elements unadorned or ornamented? Are sharp distinctions drawn between subsections, or do they exhibit continuously evolving character? How is the pitch resource mapped into the textural potential? What sort of energies does the music arouse, and hence, what expectations? and so on . . .²

By that preparations of this sort are valuable in my work, I do not imply that other composers fail to consider a similar wealth of possibilities, rather that my work habits have caused me to think about this aspect of my composition in a formal way.