Continuums and saturated spaces

My parents took me to the square often. It was more like a park for me, because of it's size. Going through it involved pebbly curved paths. For some reason I keep that memory: me, walking with my dad, thinking exactly how many little stones would there be. Then, some other questions came like how big can a little stone be without being a regular stone, or how little can be before being just sand.

This memory came back to me trying to think about some ideas related to sound, of the composer Antoine Beuger. The text I will refer to is translated to English as Fundamental Decisions¹. I will just take some of its initial issues related to some concepts of my own.

Beuger starts his text with two categorical questions: the first one about the material of music ("what is it made of?") and the second one about its form ("how is it made?"). His answer to the first question is that the material of music is "the ubiquitous noise of the world; i.e.: everything that sounds". The second question is answered saying that music is a cutout, an extract of this "noise". Form is "the fragment that music cuts of this infinite diversity".

In relation to the first concept ("the ubiquitous noise of the world"), the key would be to think of an infinite, continuous and monotonous matter. The concept is of wholeness: it's not only everything that sounds, but all this is included in a set that brings it together: everything that sounds, simultaneously. This is easily referable to the idea of white noise, chaotic, messy. But there is a subtle but fundamental difference with this expression of sound that will give place to other ideas. White noise is a chaotic evolution of sound, an "unpredictable" evolution. What is chaotic there is the organization in time and spectrum of sound. But as the sound continuum proposed by Beuger includes everything that sounds, it also includes every possible evolution of sound. I.e.: every possible duration, every possible time. For our perception, one thing cannot have more than one duration. That's why Beuger will say that this object (the sound continuum) as a result of a speculation is mute, inexpressible.

Sound is only expressed as something that is extracted from there, as physical actions that reproduce a possible part of the whole. Beuger emphasizes that the qualities of sound are given by the cut that is made. That is equal to say that on the physical level, sound is form. The nature of the continuum is also given within the limits of sound shape: the last one is made of a lot of frequencies in its spectrum and if we consider its evolution it's also made of an infinity of different moments².

I see this continuum as a saturated set. In other context, making reference to this idea, Beuger will say that in the sound continuum there are no spaces³, there is no way to identify discrete elements. This is equal to silence. A presence so vast and complete that cannot be anything but nothing: so full that every space and duration is occupied, so much that there is no possible change, no possible movement. This concept is even more definitive if we think in a saturated topological space, where every element is related to the others at the same level and with the same hierarchy among all. Not only including a new element is impossible, but including a new relation between elements is also impossible. But then, deleting a relation is to make a cut. What I try to say is that composing is to think of elements in relation within a topological level⁴. And this is achieved by selecting part of the available relations in the sound whole, i.e. cutting it.

How is this different from what we were discussing? Well, in his text Beuger goes from the sound continuum directly to the form⁵ and does not talk about the procedure that this passage takes or better, he does not consider the procedure as a different level in this scheme. I'll say that as we have seen when sound is sounding is form, when sound is thought it is structure. With structure I refer again to the topological level, to the structure mounted as relations between elements. The difference here is that when we think in sounds as topological elements we are thinking them as points. That is to say that composition goes through a state of discrete organization (separate elements) that will not be translated intact to the sound reality: the fact that a sound is made of many elements makes it highly unlikely, if not ontologically impossible, that an identical sound exists. But this doesn't prevent us from thinking a sound appearing more than once, from the structural point of view, having the same relations with the rest of the elements. A topological form (written note) carried out (played note) in has infinite possibilities of being. "Pure differentiality of what exists. No need to worry about the differences. They do not need us to find their own. They do not need us to recognize them: they already are. They are what exists. Each sound is different. There is no repetition⁶."

Moreover, Beuger opposes to the idea of music as a construction of minimum units that build larger ones. It's clear that in a structural level this can be thought this way or not. For Beuger, however the composition is organized, the result is a cutout from a continuum that also embraces aesthetics, geographies, archaeologies and histories of sound and music. Similarly, I consider that the structural level is in the end analytical, and does not make music by its own, but its scaffolding, thought either before or after the piece realization. The saturated topological space would be identical to Beuger's sound continuum, but the first will now represent all paths and structural relations of aesthetics, geographies, etc.

Now I'm thinking you can reduce the category "little stones" to only one element if you can answer the questions of the first paragraph. Like a single little stone that includes in its properties "being all the little stones in the world existing in this moment". Let's put a number if we want it to be more spectacular. But I think I like it more the other way around: thinking that all the little stones are attracting each other, however immutable, beating on a dark backround that unites them.

¹ Beuger, A. Grundsätzliche Entscheidungen. 1997. Wandelweiser Editions. http://www.timescraper.de/_antoine-beuger/texts.html#Antoine_Beuger__

² Sinze the Fourier Theorem, any periodic waveform can be decomposed in many sine waves. See for example Roederer, J. Acústica y Psicoacústica de la Música. Melos (Ricordi Americana). Buenos Aires, 1997.

³ Saunders, J. Antoine Beuger. The Ashgate Research Companion to Experimental Music. pp. 231–241. Farnham: Ashgate, 2009. http://www.james-saunders.com/interview-with-antoine-beuger/

⁴ Topology is the branch of mathematics that studies the properties of figures regardless of their size or shape.

⁵ Surely this is for clarity on the concept that is intended to express and because his interest would rather be to talk about the matter and form in the procedures that apply to them. 

⁶ Beuger, A. op. cit.

Pre-form topology

There is a nice text of Cornellius Cardew about notation and interpretation where he says that what makes a score different of a drawing is the relation that is established there between the space (the paper) and the time (the form). That's why in the score there is a condition of readability, rules that set at least the relationship between the page and the form so that the score is no longer a drawing¹.

A posible orientation of notation is to establish with it the musical form. In the same text Cardew notes the following common idea: the composers works with sounds. But the things that are written in paper are not sounds. Earle Brown would say that too². There is a whole scheme of steps (a "production chain") that gives place to form. Anyway, in certain practice it is intended to determine form with notation.

Indetermination is other way. The same idea of something indeterminate in the score says something about the relationship between notation and sounds. If the score must be submitted to a process to get to sound, a process where information passes from one state to another (paper, player, action, sound), ¿what is the limit of the determinable? If the score is only part of the work, ¿how can it pretend to determinate it all?

So: form in music is what you hear, just as you see the shape of an object. Everything that sounds in the piece is form and the perception of this form gives us something like the identity of the piece (another concept shared between Cardew and Brown). The question now is ¿what is written?. Not only a series of steps or possibilites for interpretation. There is a pre-form, something that is there before time is put into play.

To understand this I use the concept of topological figure. A topological figure doesn't stop beeing it when you apply continuous transformation to it (for example, if you strech it, shrink it or twist it). A square or a rectangle are topologicaly equivalent, even through they're visibly different from each other. A topological figure becomes another when you aply to it a cut or when two or more points of the figure are joined. A square is topologicaly different form a cylinder (a square with two parallel sides joined).

The pre-form I mentioned earlier has topological characteristics. In this state, the piece has potentialy infinite forms, one for each possible interpretation. Doesn't matter how specifficaly written it is. There is an interview to Antoine Beuger where he mentions Leibniz to say that in nature (let's say, in facts) no two leaves are alike³. Same can be said about interpretations. The score is not the form because it has no place in the facts, no place in time. Only through interpretation a from can be extracted from it, by means of processes in which always something is lost and something (or everything, at last) is added. Two forms can come out from the same score and this is a central fact of indetermination.

¹ Cardew, C. Notation-interpretation. Tempo, Nueva Serie, No. 58 (verano de 1961), pp. 21-33
²  Brown, E. The notation and performance of new music. The Musical Quarterly Vol. 72, No. 2 (1986), pp. 180-201
³  Saunders, J. Antoine Beuger. The Ashgate Research Companion to Experimental Music. pp. 231–241. Farnham: Ashgate, 2009. Extraído de http://www.james-saunders.com/interview-with-antoine-beuger/

Form in fact and latent form

En español aquí

Some silences are made, of each one sitting there, in the ground, in some couch around. Some silences and only a long, dying breath from the air conditioner, oblivious to us all. The street outside and soon the passing busses, announcing themselves or stopping at trafic lights making the windows shake; thus the atention is sharpened. The musicians look at the ground. I'm with them, behind at left. With my first gesture they look at me. I breathe. With my second gesture the music begins. But there is nothing that excludes the busses, the air conditioning or the steps of latecomers, looking to settle into the small room. There is nothing in this music letting you release those silences without feeling that a catastrophe would occur.

If is there a question about how music "transforms" time, there should be a question about how music "transforms" space. Is form "habitable"?

I'm interested in course untied of discourse. When the elements of the narrative loose the tension that keeps them tied: you can't tell when "B comes after A" because is difficult to tell what A and B are, what are their differences. The atention "flotes", is no longer "directed"; the form elapses but it also sourround us.

Thinking the form here is to think in this unmeasurable force, in the fact or event simultaneously invoking every listener and going through each one individualy. It implies not only the work sounding but also the space and the actions in it, the memories and the spectations. Multiplicity leaking.

From this perspective, composing is not a direct operation on sound. After all you can only speculate about it: it's impossible to foresee every contingency. Even more if you deal with notation. As Earle Brown says: music writing is finite, sound syntax is infinite. On what the composer operates? Saying that the only thing that he does is speculate on what will happen, via more or less control in notation or in the arranging of materials is insufficient.

From here, thinking in composing becomes more abstract. What we can say up to this point is that composing is timeless; that form substantially passes and because of that the composition doesn't operate directly over it. Then, even if composing involves organizing elements, this organization isn't "in time". In any case the time is divided for the composer in a) another parameter (duration) and b) that on which the form "unfolds". It's interesting to stop at this point, because it may be the key to understand this issue. I'm thinking in a "folded form", a form without time. What remains: only the relations established between the elements of the piece. Notation is the codification of this structure. It remains to be seen what features it has.