When we cannot count the individual notes in a group any more, they surpass the group. If someone wants to join in and play with our group, I say, well, fine - but five in a group is good, six is already dangerous. And seven: with seven the mass begins. Because then completely different relationships among human beings begin to act. I said, when we cannot count the elements any longer, so this depends on the speed: naturally when the pace is too fast we can't count them any more and thirteen notes may just appear like a gesture, one doesn't feel like counting them. Or there are too many events all happening at once, like a swarm of bees; when you perceive the swarm as a shape, it becomes a single entity. If we see a tree, we don't count the leaves, but are still able to tell a pine tree from a beech. It is an effect of the elements, but there is something else, the shape, the overall form, that characterizes the mass.

I still remember a moment during my musical education in the conservatory at Cologne when I was given a composition exercise. At the time I had no thought of ever becoming a composer. I showed it to my teacher: it was two measures of a great many notes in a small amount of time. And he said, who is going to hear that? Who can hear these notes? You don't control what you are writing. You see, what's the point in writing notes if people can't hear them? I said, oh well, I don't want you to count them. He said, well, what do you mean? Look at all these notes here. I said, I want this: frrp! He said, well just put one note; be precise, it's better.

So it was unthinkable. Some time later, in order to justify what I was interested in, I did an analysis of the music of Debussy which was later made into a radio broadcast, and the programme was subtitled 'Remarks on statistical form'. I was thinking about trends in registers of certain textures, about increasing and decreasing trends in density and in the character of density overall, of gradual shifts in predominating colours, from dark to bright, or from metallic to muted colours, and so on. This was a completely new language of musical analysis; I spoke of textures being perforated - I used the word textures as opposed to structures - and of rising or drifting shapes, sawtooth shapes of masses. It is easy to imagine, you can draw them.

When I was composing GRUPPEN for three orchestras, I had a little room in Switzerland for three months, and there was a small window in front of my desk through which I could see the incredible shapes of the mountains on the other side of the valley. There are quite a few groups in GRUPPEN which follow exactly the shape of these mountains: I became quite expert in drawing the outlines during that time. I would take such a shape and divide it vertically into musical measures of equal duration, fundamental durations, let's say they were whole-notes. Then I would add horizontal lines forming a grid, and subdivide each line going up from the fundamental into two, three, four units and so on, like overtones, but in the field of rhythm, until the shape was completely filled. So I was thinking in terms of shapes, of musical masses, and I could also make negative shapes, windows in these masses of sound.

In 1956 I realized the electronic work GESANG DER JONGLINGE 'Song of the Youths' in the studio for electronic music of West German Radio in Cologne. In this work it is possible to hear the realization of statistical processes. For example, I would give my three collaborators each a sheet of paper with a curve drawn on it, all of them to be executed in 20 seconds, to make a certain sound-event. And I would say to the first collaborator, this time start the pulse generator at 4 pulses per second, follow the curves of the drawing and end up with 16 pulses per second. To the second, who was working the potentiometer controlling the loudness level, I would say, let us take the dynamic range as being 40 decibels, from this maximum to this minimum, following the curve. To the third assistant who was in charge of the electronic filter, which lets through only a narrow band of frequencies from the signal, I would say, start at 3000 cycles per second, follow this curve for 20 seconds, and finish up at 400 cycles per second. We had a large stop-clock, we would start it, then on a count of three we would do the curves, and do them again several times until we agreed it was all right by everybody. After that, we would make seven more layers of 20 seconds, each one a little different, according to my definitions, and superim-pose them all.

Naturally I can't say exactly at which moment a pulse will occur: all I can indicate is a general tendency during the curve. And the same is true for the dynamics and the filter. But if we superimpose a number of curves which share an overall characteristic tendency, then it leads to a certain result which is a mass: a mass moreover with a very distinct shape and a very precise tendency compared to another mass. This method of composition of musical microtextures by statistical methods has become very important in music. All the different applications of chance and random techniques in music are nothing more than derivations of it.

After finishing my music studies in the early fifties, I started again at the University of Bonn, studying communications science and phonetics. Simple analyses of noise sounds led us automatically to statistical wave structures. And the structures I found in individual sounds, like consonants, I expanded into a larger time-frame, deriving entire musical sections behaving in the same way as a single noise. I think that the most important innovations in musical form come about from building on the relationships of the three time regions: form, which is everything that happens between, say, eight seconds and half an hour; rhythm and metre, which is everything that happens between one-sixteenth of a second and eight seconds; and melody, which is everything that is organized between one-sixteenth and one-fourthousandth of a second, between 16 and 4000 cycles per second. It is almost technically possible to stretch a single sound lasting one second, to a length of half an hour, so that you have an overall form which has the characteristic structure of the original sound. On the other hand, if you are able to compress an entire Beethoven symphony into half a second, then you have a new sound, and its inner microstructure has been composed by Beethoven. Naturally it has a very particular quality compared to the sound resulting from the compression of another Beethoven symphony. Not to mention a Schoenberg symphony, because there are many more aperiodicities in Schoenberg: that would be more of a noise, whereas the Beethoven would be a vowel, because it is more periodic in its structure.

What I have said about point composition I would also say about composition in groups and in particular masses. We need to know better how to determine tendencies in masses of notes with greater precision. We have very little experience in this field; no research has been done in how a composer might continue in this field. I haven't done any systematic research, and research would be necessary from my point of view because in other fields I am working with very precise measurements. I simply have no precise knowledge of the behaviour of certain shapes in certain contexts; I just do it intuitively. And I don't deny that mistakes often occur: when I try to combine points with certain masses, and the points are completely masked, or when I think there is a difference between certain textures in a mass, and they don't come out in practice. I make mistakes in speed, where I think there are still differences recognizable at high speeds, and when it comes to the performance they are not, and I have to make changes in rehearsals. Finally, when I have to talk about these things to other composers, younger composers let's say, who want to work with me, then I find my language is not developed enough to speak about these things: I have to use mainly words from statistics and other fields.

We come now to the second set of terms, the terms determinate, variable and statistical. Points, groups, masses - all can be composed in any one of these ways. In most of my works I have composed points in a determinate way, or groups, or masses. What does that mean? It means that one can hear very clearly the intervals which make the proportions, the durations of the individual points, the shapes of the individual groups and masses. I don't really need to give examples of determinate structures of points, groups and masses because all of my scores, up to 1955, are completely notated in all characteristics in terms of regular scales and measurements. But a method of composing variable structures was a new phenomenon: what does it mean? I will give one example, and then leave it to you to invent more from your own imagination.

I composed a work ZEITMASZE 'Time Measures' for five woodwinds, in 1955, at the same time as I composed GESANG DER JUNGLINGE. Five woodwinds: oboe, flute, cor anglais, clarinet, bassoon - is a traditional ensemble, but in this work you hear determinate structures alternating and mixed very clearly with variable struc-tures. For example, there is a point where all five instruments start together. They have been playing chords. Everything organized in the vertical, such as chords, is clearly determinate because you would never find notes falling by accident together in a series of chords. It's an obvious sign of determination; even more so if there is a progression in a sequence of chords, from simple to more complicated and back, or as in classical harmony, from consonant to dissonant and back to consonant again. In this example, however, the starting-point is a point of departure where every instrument starts playing at a different speed.

So the oboe, for instance, is playing a given number of notes as fast as possible, the bassoon is playing as slow as possible, in an unrelated tempo; the English horn is playing an accelerando from as slow as possible to as fast as possible. (By 'as slow as possible' I mean for a woodwind player as long as possible to play a group of notes in a single breath. Actually that will vary with the size of the player's lungs, but that's not what I mean by variable: the timescale is clearly determined for any one player.) Another player starts as fast as possible and slows down in turn until he reaches a speed four times as slow. Then, after a longer time which is a time value within which everybody can finish playing their written notes, they wait for each other and come together again in the same metronomic tempo. This is easy to hear because the groups follow one another in a very precise manner, in sequence, or playing in chords, or in clear beats of three or four.

Sometimes one instrument is out of time with the others; there are sections where all five have individual tempos. There is a continuum between complete determination and extreme variability. And when we listen, we can feel when the music is very determinate because we know exactly where we are: on a certain beat, in a certain sequence of timbres, in a certain rhythm, but when we are in a region of high variability, the music is floating.

I said the word statistical is new in the context of music. Let me give an example from the seminar in communica-tion sciences and phonetics which I studied with Professor Meyer-Eppler. This was a teacher who had come from physics and phonetics. In phonetics he was analyzing the different sounds of language, in communications science he was engaged in studying statistics, because he wanted to know more precisely what all the different noises were, and analyzing the wave structure of noises and con-sonants in language led him to use statistical methods of description and analysis. He would give us exercises demonstrating the principles of the Markoff series; in one we were given cut-outs of individual letters from newspaper articles, and we had to put them in sequence by a chance operation and see what sort of a text came out. Then we would repeat the operation with individual syllables, and then with combinations of two syllables, and so forth, each time trying to discover the degree of redundancy, as we called it, of the resulting texts.

Statistical means that you can permutate or change the order of events without it really making any difference, whereas if I were to change the order of the words and syllables I have just spoken, then there would be no direction or determination any more to what I am saying: it would just be an irregular distribution of phonemes. My composition ZYKLUS 'Cycle' is worked out according to different degrees of indeterminacy, with different degrees of statistical behaviour affecting certain groups of elements, groups or masses. That means the resulting music is constantly fluctuating between determinate and statistical behaviour. In ZEITMASZE, which is based on lines, it is possible to follow individual lines without difficulty because each is played by a different instrument and all you have to do is follow the colour of the instrument. In a statistical composition, however, the field becomes very much wider and it is not possible to follow precise lines.

In ZYKLUS I have worked with nine degrees of statistical distribution. I'm not saying that you are supposed to identify these nine degrees when you hear the music, nevertheless the music that results from such a method has very particular characteristics compared to other music, which is important in the context of this discussion. Statistical methods are introduced into musical composition in terms of bands and band-widths. By band I mean that every aspect is considered as occupying a position between a minimum and a maximum value: in pitch, a highest and a lowest pitch; in rhythm, a shortest and a longest duration; in timbre it may be between dark and bright. What is in between such limits is called a band and the band has a certain width. When the width of the band is zero, then we have a highly determinate situation: there is no choice. At the other extreme, when the band extends over the whole range of possibilities and I can choose, for example, any pitch, then the band-width is maximum. So between extreme determinacy and extreme relativity, indeterminacy, in a given composition, there is an entire range of degrees composed in terms of different band-widths. And some composers have become pretty specialized in statistical composition since these methods were introduced.

It happens every once in a while, in music as in other fields, that you find people specializing in one new aspect of musical forming, and becoming famous because they just specialize. A composer like Ligeti specialized for years in microstructures, the detailed composition of textures; or Xenakis, who has concentrated on stochastic distribu-tions; or Penderecki, who was the cluster specialist for a long time. In another context we can see Feldman as being the specialist in music that is as slow, and as soft, as possible. Every once in a while music produces its specialists, people who go very deeply into their narrow specializations, and vary them all the time. This is something we take for granted in painting, more than in music. Everyone has his so-called personal style. By which is meant that he has narrowed down his field of activity so completely that it only takes a fragment of a work for you to say, ah, that's so and so.

And we can really say that universalists are becoming very rare in all fields, all the sciences. I tell my own students, if you want to become famous just take a magnifying glass and put it to one of my scores, and what you see there, just multiply that for five years. For example, if you see snare drums, then you start composing around twenty pieces only for snare drums. Snare drums of all different sizes: for fifty snare drums, for twenty, for thirty - snare drums on the roof, snare drums in the basement, big snare drums and very tiny snare drums, snare drums amplified and intermodulated. Then he will be the snare drum specialist, he will be known in Japan, he will be famous everywhere.