I enjoyed painting flowers, not bouquets, but a single flower at a time, in order that I might better express its plastic structure. (Mondrian)

A growing interest in Near and Middle Eastern rugs has made me question notions I previously held on what is symmetrical and what is not. In the Anatolian village and nomadic rugs there appears to be considerably less concern with the exact accuracy of the mirror image than in most other rug-producing areas. The detail of an Anatolian symmetrical image was never mechanical, as I had expected, but idiomatically drawn. Even the classical Turkish carpet was not as particular with perfect border solutions as was its Persian counterpart.

A disproportionate symmetry, whether rhythmic or in phrase lengths, characterizes twentieth-century musical development. Webern's Spiegelbild (mirror image) in his last works was integral to his twelve-tone procedure and any imbalance had to do with a slight variation of rhythmic or chordal distribution in its mirror. The post-Webern tendency with rhythm was to effect a compromise between symmetrical and asymmetrical beats. A typical example would be five notes played in the time of four equal beats (5:4), or other similar patterns. Unlike Stravinsky, whose raw syncopation articulated tight harmonic rhythm patterns, post-Webern rhythmic usage stemmed from a twelve-tone polyphonic concept of continuous variation in which a rhythm - not dependent on harmony - varied the motivic shape of the music.

Rugs have prompted me in my recent music to think of a disproportionate symmetry, in which a symmetrically staggered rhythmic series is used: 4:3, 6:5, 8:7, etc., as the point of departure. For my purpose, it "contains" my material more within the metric frame of the measure; while in post-Webern arhythmic language, lopsided acceleration results from the directional pull of one figure to another. What I'm after is somewhat like Mondrian not wanting to paint "bouquets, but a single flower at a time."

There are musical examples where the juxtaposition of asymmetric proportions (all additive) becomes the form of the composition. It is interesting that all three of the composers I will briefly discuss use repetition or reiteration to achieve this additive form and, as a result, are anchored in a fixed pitch-world, whether dissonant or consonant. In one of the movements of Stravinsky's Requiem Canticles there is a continuous play between A and B, where the smaller "border" of A remains unchanged in every detail, while B varies slightly in length when repeated. Varese, in the striking and uncommonly extended "opening" of Integrales, also utilizes an additive scheme, but dispenses with Stravinsky's characteristic juxtaposition of A and B. Like the formation of crystals which so fascinated Varese, this opening statement - created out of three notes - undergoes a continual transformation of rhythmic shapes and time proportions. In Reich's "Four Organs", the rhythmic patterns are more acoustically oriented and are based on the pitch-components of a chord that never changes position. The music begins with a 3+8 pattern in which certain pitches from the basic chord are then varied rhythmically. Reich's first structural rhythmic move is to continue to use the same elevenbeat measure, but now divide it into patterns of 4+4+3; then 4+3+4. What follows is the gradual addition of more beats to the structural frame of now longer measures: 4+3+2+4=13; 4+3+2+2+4=15; then to 18; 20; 23; 24 beats; until Reich does away with the barlines. As the measures grow progressively longer, the oscillation of the recurring pitches can no longer be said to have any marked rhythmic profile.

The moment a composer notates musical thought to an ongoing ictus, a grid of sorts is already in operation, as with a ruler. Music and the designs or a repeated pattern in a rug have much in common. Even if it be asymmetrical in its placement, the proportion of one component to another is hardly ever substantially out of scale in the context of the whole. Most traditional rug patterns retain the same size when taken from a larger rug and then adapted to a smaller one. Likewise, the character of Stravinsky's patterns does not seem to differ if a work of his is either long or short. If we examine asymmetric phrasing - whether in Stravinsky's "hardedged" "Sacre", Satie's "soft-edged" "Socrate", or in Schoenberg's duplication of the irregular prose of "Erwartung" -we find that the partitioning is concentrated enough in time to hear the mosaic-like process of the grid at work. We also recognize a historically reminiscent "conversation" between the phrases. In this regard, these three early twentieth century asymmetric masterpieces were an outgrowth of the symmetrical antecedent/consequence building blocks of the Classical era.

A music in which all properties functioned independently and autonomously from each other was achieved by John Cage in the early 1950s. What Cage did was to compile a table of sound-events - some just a note or two, others arabesque-like figures of different proportions. Included in other charts was information pertaining to the gamut of musical parameters. Through the tossing of coins and consulting the I Ching as oracle, the ordering and subsequent combination of all this material was arrived at. The music was then "fixed" through the same method of tossing coins into a nonprogressive, rhythmic "spatial" notation, not unlike a distance scale on a map. Because this music is subject to the multiplicity of disciplines inherent in its detailed assemblage, its musical shape is only discernible at the moment of hearing - like images in a film. It is not involved with the grammar of design, and is perhaps the only music known to us in which concepts of symmetry/asymmetry cannot be applied.

I was once in Rothko's studio when his assistant restretched the top of a large painting at least four times. Rothko, standing some distance away, was deciding whether to bring the canvas down an inch or so, or maybe even a little bit higher. This question of scale, for me, precludes any concept of symmetry or asymmetry from affecting the eventual length of my music. As a composer I am involved with the contradiction in not having the sum of the parts equal the whole. The scale of what is actually being represented, whether it be of the whole or of the part, is a phenomenon unto itself.

The reciprocity inherent in scale, in fact, has made me realize that musical forms and related processes are essentially only methods of arranging material and serve no other function than to aid one's memory.

What Western musical forms have become is a paraphrase of memory. But memory could operate otherwise as well. In "Triadic Memories", a new piano work of mine, there is a section of different types of chords where each chord is slowly repeated. One chord might be repeated three times, another, seven or eight - depending on how long I felt it should go on. Quite soon into a new chord I would forget the reiterated chord before it. I then reconstructed the entire section: rearranging its earlier progression and changing the number of times a particular chord was repeated. This way of working was a conscious attempt at "formalizing" a disorientation of memory. Chords are heard repeated without any discernible pattern. In this regularity (though there are slight gradations of tempo) there is a suggestion that what we hear is functional and directional, but we soon realize that this is an illusion; a bit like walking the streets of Berlin - where all the buildings look alike, even if they're not.

I'm being distracted by a small Turkish village rug of white tile patterns in a diagonal repeat of large stars in lighter tones of red, green, and beige. Though David Sylvestei*) is right in commenting that our appreciation of rugs such as this was enhanced by our exposure to modernistic Western art, still, this "primitive" rug was conceived at almost the same time that Matisse finished his art training. Everything about the rug's coloration, and how the stars are drawn in detail, when the rectangle of a tile is even, how the star is just sketched (as if drawn more quickly), when a tile is uneven and a little bit smaller - this, as well as the staggered placement of the pattern, brings to mind Matisse's mastery of his seesaw balance between movement

*) Islamic Carpets from the Joseph V. McMzillan Collection. Art Council of Great Britain. 1972.

and stasis. Why is it that even asymmetry has to look and sound right? There is another Anatolian woven object on my floor, which I refer to as the "Jasper Johns" rug. It is an arcane checkerboard format, with no apparent systematic color design except for a free use of the rug's colors reiterating its simple pattern. Implied in the glossy pile (though unevenly worn) of the mountainous Konya region, the older pinks, and lighter blues - was my first hint that there was something there that I could learn, if not apply to my music.

The color-scale of most nonurban rugs appears more extensive than it actually is, due to the great variation of shades of the same color (abrash) - a result of the yarn having been dyed in small quantities. As a composer, I respond to this most singular aspect affecting a rug's coloration and its creation of a microchromatic overall hue. My music has been influenced mainly by the methods in which color is used on essentially simple devices. It has made me question the nature of musical material. What could best be used to accommodate, by equally simple means, musical color? Patterns.

Rug patterns were either abstracted from symbols, nature, or geometric shapes - leaving clues from the real world. Jasper Johns's more recent paintings cannot be placed into any of these categories. Johns's canvas is more a lens, where we are guided by his eye as it travels, where the tide -somewhat different, somewhat the same - brings to mind Cage's dictum of "imitating nature in the manner of its operation." These paintings create, on one hand, the concreteness we associate with a patterned art and, on the other, an abstract poetry from not knowing its origins. We might even question in Johns whether they are patterns at all. When does a pattern become a pattern?

This persistence of pattern which runs throughout the art of the East is due to the inclination of the craftsmen to let well alone. (A. F. Kendnck and C. F. C. Tattersall) Hand-woven Carpets, London 1922)

Why Patterns is a composition for flute, glockenspiel, and piano consisting of a large variety of patterns. The work is notated separately for each instrument and does not coordinate until the last few minutes of the composition. This very close, hut never precisely synchronized, notation allows for a more flexible pacing of three very distinct colors. Material given to each instrument is idiomatically not interchangeable with that of the other instruments. Some of the patterns repeat exactly -others, with slight variations either in their shape or rhythmic placement. At times, a series of different patterns are linked together on a chain and then juxtaposed by simple means.

The most interesting aspect for me, composing exclusively with patterns, is that there is not one organizational procedure more advantageous than another, perhaps because no one pattern ever takes precedence over the others. The compositional concentration is solely on which pattern should be reiterated and for how long, and on the character of its inevitable change into something else.

I enjoy working with patterns that we feel are symmetrical (patterns of 2, 4, 8, etc.) but present them in a particular context:

Example 1 is characteristic of a vertical pattern framed by silent beats; in this instance the rests on either end are slightly unequal. Linear patterns are naturally more ongoing, and could have the "short breath" regularity of Example 2 or anticipate a slight staggered rhythmic alteration such as in Example 3. Another device I use is to have a longish silent timeframe that is asymmetrical; in this instance, with a quixotic four-note figure in the middle:

or a symmetrical silent frame around a short asymmetric measure: (A. F. Kendrick and C. F. C. Tattersall) Hand-woven Carpets, London 1922)

Why Patterns is a composition for flute, glockenspiel, and piano consisting of a large variety of patterns. The work is notated separately for each instrument and does not coordinate until the last few minutes of the composition. This very close, but never precisely synchronized, notation allows for a more flexible pacing of three very distinct colors. Material given to each instrument is idiomatically not interchangeable with that of the other instruments. Some of the patterns repeat exactly -others, with slight variations either in their shape or rhythmic placement. At times, a series of different patterns are linked together on a chain and then juxtaposed by simple means.

The most Interesting aspect for me, composing exclusively with patterns, is that there is not one organizational procedure more advantageous than another, perhaps because no one pattern ever takes precedence over the others. The compositional concentration is solely on which pattern should be reiterated and for how long, and on the character of its inevitable change into something else.

I enjoy working with patterns that we feel are symmetrical (patterns of 2, 4, 8, etc.) but present them in a particular context:

Example I is characteristic of a vertical pattern framed by silent beats; in this instance the rests on either end are slightly unequal. Linear patterns are naturally more ongoing, and could have the "short breath" regularity of Example 2 or anticipate a slight staggered rhythmic alteration such as in Example 3. Another device I use is to have a longish silent timeframe that is asymmetrical; in this instance, with a quixotic four-note figure in the middle:

Repetitive chordal patterns might not progress from one to another, but might occur at irregular time intervals in order to diminish the close-knit aspect of patterning; while the more evident rhythmic patterns might be mottled at certain junctures to obscure their periodicity. For me patterns are really self-contained sound"groupings that enable me to break off without preparation into something else.

In String Quartet there is an almost obsessive reiteration of the same chord - dispersed in an overlay of four different speeds:

The rhythmic structure of the block consists of four uneven bar lengths with four permutations that incorporate the instrumentation of the quartet. I must caution the reader not to take the barlines here at face value. This passage becomes rhythmically obscured by the complicated nonpatterned syncopation that results. Only after rehearsals, and by following the score, could I catch an individual pattern as it crisscrossed from one instrument to another.

In Spring of Chosroes for violin and piano, the "pattern" of one section consists of heightening the effect of the plucked violin figure (encompassing three pitches) by not establishing any clear-cut rhythmic shape except for its constant displacement within the quintuplet. This allows for five permutations, which are then juxtaposed in a helter-skelter fashion as the series continues. The use of three pitches against five uneven beats created, in my ears, a crippled symmetric constellation of "eight" as I was writing it. Aginst the violin's pattern, the piano has an independent rhythmic series of the same three pitches, played in a symmetric unit of four equal beats to a measure. This functions as still another deterrent to the natural propulsion of the quintuplet.

If my approach seems more didactic now - spending many hours working out strategies that only apply to a few moments of music - it is because the patterns that interest me are both concrete and ephemeral, making notation difficult. If notated exactly, they are too stiff; if given the slightest notational leeway, they are too loose. Though these patterns exist in rhythmic shapes articulated by instrumental sounds, they are also in part notational images that do not make a direct impact on the ear as we listen. A tumbling of sorts happens in midair between their translation from the page and their execution. To a great degree, this tumbling occurs in all music - but becomes more compounded in mine, since there is no rhythmic "style", a quality often crucial to the performer's understanding of how and what to do. I found this just a true in my music of the fifties - where rhythm was not notated, but left to the performer.

The attempt to find a suitable notation for musical ideas has been a major preoccupation of Boulez throughout his career. There is a significant revision of Le Soleil des Eaux that comes to mind. The original version is in the "Klangfarben" manner: brief segments of the musical line distributed from one instrument to another. The revised version has the individual instruments follow through with a more continuous line. The notation of the earlier version looks "good" in the manner of the times; but the revised score sounds better - rather, it sounds like Boulez. In contrast, my notational concerns have begun to move away from any preoccupation with how the music functions in performance.

It is difficult to describe what characterizes notational imagery."If we could suspend for just a moment all the reasons we think distinguish one era from another - and briefly glance at the pages of the last movement of the Hammerklavier, or a florid bar or two from Chopin, or any work of Webern's - we will observe that these pages do not visually resemble the music of their contemporaries. The degree to which a music's notation is responsible for much of the composition itself, is one of history's best kept secrets.

The following example from Trio for violin, cello, and piano might best illustrate this, as well as my increasing move away from the practicality of how the music will come off in performance. Starting with the metronome indication ( = ), there is a difficult coordination problem between the three instruments. The performers must pace seven beats into six equal ones, and subdivide another rhythmic idea in which each pitch of the four-note piano chord, and the separate notes of the double-stops in the violin and cello are all of different, finicky durations. This "machine" goes on for thirty-six measures, with other problems developing along the way. Technically, the music is both idiomatic and playable; but depends, to a taxing degree, on the performer's concentration.

Many composers and theorists will disagree with the almost hierarchical prominence I attribute to the notation's effect on composition. They would argue that new musical concepts, resulting in innovative systems, necessitated changes in notation. This is then referred to as, say, a new "piano style", as Leibowitz did in discussing an important early piano piece (Op. 11, 1909) by Schoenberg. This interpretation cannot be refuted, but some room should be left open to question it. My speculation over how a ,,notational look" may have contributed this, since at that time I remember I was dangling between various procedures that I knew didn't apply to my music. And as yet, I had not met the painters whose tactical solutions were to contribute so much to this problem that confronted me. In no way do I want to imply that youthful arrogance ignored all that I could learn from the study of composition. But my approach, which was not conscious at the time and only revealed itself many years later, was: work first, study later. Recently I was in a bookstore in Berlin where the clerk, not understanding English, found it impossible to help me find certain German books on rugs. A distinguished man intervened, and from our conversation it turned out that he too was an avid rug enthusiast. He then took out of his pocket sheets of paper with singlespaced columns of his countless rug books in many languages. "Could it be possible for me to see your rug collection while I'm in Berlin?" I asked with some hesitation. "I haven't collected any rugs as yet, only books about them. You see, I first want to learn all there is to know about them", he replied. I remember Boulez saying something similar: "I must know everything before I step off the carpet."

My first lucky encounter with a painter who was to become crucial to my music occurred soon after meeting John Cage in the latter part of 1950. Cage knocked on my door and announced that he had just met an extraordinary young artist and that "we're going down to his studio". The artist was Robert Rauschenberg. While looking at a large black painting in which newspapers (also painted black) were glued to the canvas, Rauschenberg jokingly suggested I buy it. "How much do you want for it?" "Whatever you have in your pocket." I had about seventeen dollars and change - which I happily gave him, and which he happily accepted. We put it on the roof of Cage's old Ford and off we went. I'm looking at it now (thirty years later) as I write this. After living with this painting and studying it intensely now and then, I picked up on an attitude about making something that was absolutely unique to me. To say that the Black Painting could be relegated to "collage" simply did not ring true. It was more: it was like Rauschenberg's discovery that he wanted "neither life nor art, but something in between". I then began to compose a music dealing precisely with "inbetween-ness": creating a confusion of material and construction, and a fusion of method and application, by concentrating on how they could be directed toward "that which is difficult to categorize".

Soon after meeting Rauschenberg I met Jackson Pollock, who asked me to write music for a film about him that had just been completed. I was very pleased about this since it was just the very beginning of my career. Pollock lived way out on Long Island and only came to the city sporadically, making it difficult to establish a real continuity to our relationship. In thinking back to that time, I realize now how much the musical ideas I had in 1951 paralleled his mode of working Pollock placed his canvas on the ground and painted as he walked around it. I put sheets of graph paper on the wall; each sheet framed the same time duration and was, in effect, a visual rhythmic structure. What resembled Pollock was my "all over" approach to the time-canvas. Rather than the usual left-to-right passage across the page, the horizontal squares of the graph paper represented the tempo - with each box equal to a preestablished ictus; and the vertical squares were the instrumentation of the composition.

As I came to know Pollock better - especially from those conversations where he would relate Michelangelo's drawings or American Indian sandpainting to his own work - I began to see similar associations that I might explore in music. I must point out here that the intellectual life of a young New York composer of my generation was one in which you kept your nose glued to the music paper. Wolpe was intimate with many painters and constantly spoke of other things besides music. Varese, too, was a composer with vast interests in other areas. Unless you came to know creative people in other fields, your own intellectual and artistic development was not the same. How a painter - who walked around a canvas, dipped a stick into a can of paint, and then thrust it in a certain way across the canvas - could still talk about Michelangelo was, and still is, baffling to me.

My edification in offstage references continued during my friendship with Mark Rothko. On numerous occasions we went together to the Metropolitan, where his favorite haunts were, surprisingly, not the painting galleries, but the Near Eastern collection and especially a small room of Greco-Roman sculpture. Rothko always followed through his reaction to something that would catch his attention with a brief, reflective commentary. I remember his absorption one afternoon with the Greco-Roman sculpture: "How simple it would be if we all used the same dimension the way these sculptures here resemble each other in height, stance, and the distance between one foot and the other." Rothko was leaning toward a possible answer in the more subliminal mathematics of his own work. And I agree artistically. It seems that scale (this subliminal mathematics) is not given to us in Western culture, but must be arrived at individually in our own work and in our own way. Like that small Turkish "tile"rug, it is Rothko's scale that removes any argument over the proportions of one area to another, or over its degree of symmetry or asymmetry. The sum of the parts does not equal the whole; rather, scale is discovered and contained as an image. It is not form that floats the painting, but Rothko's finding that particular scale which suspends all proportions in equilibrium.

Stasis, as it is utilized in painting, is not traditionally part of the apparatus of music. Music can achieve aspects of immobility, or the illusion of it: the Magritte-like world Satie evokes, or the "floating sculpture" of Varese. The degrees of stasis, found in a Rothko or a Guston, were perhaps the most significant elements that I brought to my music from painting. For me, stasis, scale, and pattern have put the whole question of symmetry and asymmetry in abeyance. And I wonder if either of these concepts, or an amalgamation of both, can still operate for the many who are now less prone to synthesis as an artistic formula.

Until about ten years ago I wrote often about music. I no longer do. The writing was usually polemical in content. In recent years I do not want to argue with talent. I want to be thankful for it regardless from where it comes.