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One of my first attempts at sorting out some basic areas of music, caused fundamentally by my confusion over the apparently 'static' nature of pre-composed music, and yet the evidence that the compositional process itself can be so fragile and insecure. It was originally written and given while I was at Durham University and subsequently given in a revised form at the Royal Society Conference at APU in 1991.

The ideas behind this seminar are attempts to understand the nature of music, and its place within the hierarchy of human activity. For some time I have been dissatisfied with a purely historical view of musical change, which although important and necessary for certain purposes, can offer a misleading view of the passage of musical ideas through western art, and by transfer to the individual work of art itself, which is of particular importance to composers.

It seems that historically the musical work is understood as an inherently pre-existent object placed within an historical hierarchy of musical style. I do not wish to detract from this approach as such, as it is particularly logical and indeed necessary to provide a broad view of change, from which a more detailed study can be made if necessary. However, from a compositional point of view it has disadvantages. It produces the concept of the ‘inevitability’ of certain events or pieces, something which could not possibly have been evident at the time, even if it appears so with hindsight. It is this inevitability I would object to, and would prefer instead to look at the results in isolation.

A clue to the nature of perceived quality lies in the fact that for whatever type of musical work one cares to mention, there seems to be an audience of champions (however small), who find in it something of interest and quality. This would imply that there are fundamental ideas within what is called music that are universal to all work of perceived quality.

While this may not seem obvious at first, there remains the question of how one can feel the same sense of appreciation on listening to for instance, Josquin’s ‘Miserere Mei Deus’, Mozart’s ‘A major Piano Concerto’, or Birtwistle’s ‘Triumph of Time’, and yet using any single generally accepted method of analysis find it difficult or impossible to relate them all. Of course there are broad links: they all use (of course) combinations of pitches moving through time, and sometimes similar formal procedures. And yet from a composer’s-eye view the differences in the resulting language are all too apparent, and this is explicitly realised in the analyst’s attempts to make sense of the individual work.

And yet the effects of all these works artistically, I would content are almost precisely the same - they are analogous, isomorphic. They are all successful works of art, and as such generate those feelings of appreciation we all know no matter what our likes or dislikes may be. They have all been written by composers with trained minds, and they all exist in a finite typographical form on paper. From this high-level they do indeed appear similar, and yet as one moves closer divergencies of ‘language’ appear. Apparent rules of counterpoint, harmony and melody that apply to Josquin mean little to Mozart and less to Birtwistle. On a detailed level rhythm, harmony and melody seem to be entirely different in each. Is this a common language?

Today composers seem to strive for a common language. Does one exist, and if so what is it, and if not, is one necessary?

These are some of the questions for which I wanted answers. I felt sure a common language of some sort existed, but the problem was to break through the dialects and distortions that are called style, and to look for similarities on any level. These would be the clues.

Of course, music is not a language in the verbal sense. There is no etymological dictionary of music, nor could there be. The only etymology available is that of analysis based on the historical view of style, and as I shall explain, this is an inherently high-level notion, based on principles that are ‘invisible’ to the composer as he or she works. Music is a basic language in itself, it’s words are the play on sounds, not the meanings attributable to those sounds. The Historical Dictionary simply passes us back through derivatives more and more distant in time, while the musical experience remains alive and present. For this reason, as far as possible, the music must be held in isolation, just as it was for the composer who wrote it, as it was written.

We know music communicates, the question is not what, but how.

When I was originally formulating this seminar I was still thinking along historical lines. I found most perplexing the introduction of explicit ‘chance’ elements of one sort or another in twentieth century music. I knew they could be effective and could work, but how does one explain them with regard to the music of Mozart or Bach, where detailed definition of most parameters seems a prerequisite. But as I sought to answer this I found the necessity to move further and further back into this ‘detailed definition’ itself. Pure randomness is an impossibility,. but between this and pure choice (also impossible) there are an infinity of shades of grey. In the end I think I do to some extent answer these questions, but only in an oblique and isomorphic way, by disregarding style, and ignoring the historical method. Historically, choice and chance are irreconcilable, like any good paradox, but ontologically - investigating the musical work as an individual and isolated example of communication, it becomes more and more apparent that the two are not only reconcilable, but essentially interdependent.

Instead of heading for the first score that comes to mind I intend to take a brief journey around some ideas that may offer clues as to what fundamentals lie behind our notions of communication and information, and I ask for patience when I discuss areas that may at first seem rather remote from what could be seen as the linqua franca of music. They may offer what to some may be rather radical ideas about the communicative properties of any medium, and indeed about the nature of communication itself.

To begin with I shall look at the general idea of the system.

Next (Systems)


I Systems

I use this term broadly. Everything exists within a system, and all systems lie within, or make part of larger systems. For example, I could take the system that we are presently in. I am talking within a specific framework we call a seminar, which can be said to be a system of its own. It has its own rules and conventions, which are generally accepted. However, it can easily be seen that this is also part of a number of other systems, and that it itself is made up from the interlocking of many systems. Thus, on a geographical level, this seminar is taking place in a certain location. This location is part of a greater location, which is part of a greater location called England, part of Europe, part of the world, and so on all the way to ‘universe’ I presume.

Alternatively, I could move another way, and once again beginning with the seminar, take my position in it, and investigate myself as my own physical system, taking into account my voice, my vocal chords, the brain urging muscles to work through the nervous system, the neurons in the brain, the molecules which make up the neurons, the cells, the atoms . . . It can be seen how complex this can become, and yet how ordinary the results can be. The levels of hierarchy change with the ‘position’ of the observer, and to some extent each level is isomorphic with any other. The only physical boundaries, which become mental boundaries, are the unimaginably huge universe, and the unimaginably small quark or lepton, both concepts which in physics are not fully understood, and veiled quite literally in uncertainty.

Figure 1 is a list of four such inter-related systems. It can be seen that they all from part of the overall system that forms the ‘universe’ as we know it and as seen from our position, and each spinning away from this overall idea, moving through many isomorphic levels, before ‘coming to rest’ again amongst the quarks. The levels I have chosen to list in the ‘natural’ lists are relatively arbitrary, they are in fact, probably infinite in number. Amongst the four lists are two ‘natural’ systems: that for natural existence, and that for a member of that overall system, an ant-colony. The ant-colony’s system is a breakdown of its social behaviour, which is extremely complex, and my reasons for choosing this example will become clear. Another system listed is that of the computer, a man-made mechanical system. While this is much simpler than the ant colony, it displays remarkably similar features in the way in which it ‘chunks’ together bits of information, to make further atoms with which to build. There is also, and more controversially, an attempt at an hierarchy of music. Is this man-made or natural? Once again, as a system, it lies within the overall, and while the article of composition is ‘man-made’, it is obvious that the system within which it lies is natural, and that the materials it uses are composed of predominantly natural elements. Furthermore, it will become clear that there are remarkable parallels to be found between human biological systems and the ants’ social system.

There is one point to which I would like to draw your attention regarding the lists. It may seem obvious, but it becomes important. The idea may be termed ‘leakage’ and involves the links between the various levels within each system. It is fairly obvious that the closer together two levels are, the greater the relationship and the more continuous the flow from one to the other, but as the levels grow further apart, the relationship grows less and less intimate, until it could be said that the relationships are beyond effect. For instance, just as the individual neurons in my brain have no direct effect on my thought processes, I myself have no effect on the cosmological processes that lie behind the universe. In the same way, a sonata in music may be made up of notes, but any one individual note need not effect the whole, either when written badly by the composer, or played as an error by the performer. However, the whole sonata does ultimately rest on the individual notes as objects.

I intend to briefly expand on two systems of the list: those of the computer and the ant colony. I hope those who are conversant in either of these fields will accept my apologies.



I(i)Computer Systems

On the lowest level of constituents that make up the computer as an object are the atoms that make the hard-wiring and the silicon chips. These are ready-made for man. However, the lowest operating man-made level is that called machine language. The hardware consists of several partitioned sectors called memory, the central processing unit or CPU, and several input/output devices. The memory is divided into distinct physical pieces called ‘words’, for instance 65,536, or 216 in any one computer. These words can be divided into ‘bits’, and there are, for example, 36 bits per word. Physically a bit is a magnetic switch that is either on or off. Words can be used as either data or instruction, or both. Usually the first part of a word is interpretable as the name of the instruction type which is to be carried out, and the rest of the word makes up a pointer, which points to another word or group of words in the memory, but way of an address. Every word has an address to locate it and in addition, an instruction may point at itself, that is, instruct itself to execute a change on itself.

The CPU has a special pointer which stores the address of the next word to be interpreted as an instruction. The CPU first copies this and then executes it at the appropriate time, for instance an instruction to PRINT or JUMP.

In machine language, the types of operation constitute a finite repertoire which cannot be extended, and all programmes, no matter how long or complex, must ultimately be comprised of these basic machine instructions.

The next step up from machine language is a fairly gentle one. The language here is called the Assembly language, and represents machine language on a one-to-one basis. Essentially all it does is to ‘chunk’ the individual machine language instructions, so that instead of writing the sequences of bits that comprise the instruction that makes addition, the user may simply use the word ‘ADD’, which will automatically give the machine level the address of the relevant machine bits.

This level is in fact a programme itself (called the assembler), which is written in the machine language it is to ‘translate’. So in fact when a computer assembles, it is running on two levels concurrently. This compares somewhat to the way in which dot notation in music works. Rather than write the acoustical and temporal information directly into a score (eg 440Hz for 1.0456 seconds), this information is chunked into a dot based notation. Of course, this chunking means the loss of a certain freedom, (of for a computer - information), but it aids facility on a higher level of acoustical composition. Of course, when a dot composition is played, just as in assembler, the two ‘programmes’ - the written piece and the actual sound - do indeed run concurrently. Figure 2 gives an example of these two languages in comparison. On the left is the machine language, given here in Octal, and on the right its translation in assembly.

‘Above’ the assembly language is the compiler and/or interpreter. Compilers basically use algorithms, which are groups of commonly used instructions, and larger chunks called subroutines and procedures. As the compiler’s algorithms comprise assembler versions of instructions, the compiler itself cannot ‘see’ the low level machine language that it is implicitly relying on.

Once again, a similar relationship between computer operations and musical notation can be made, for once the musician has learned the ‘shorthand’ called dot notation, he then learns how to manipulate that as a system, rather than thinking on the acoustic level, which is to some extent ‘invisible’ or irrelevant to him, although as sound the music implicitly relies on the stability and regularity of these physical laws in order to function properly.




When seen from our point of view, ant colonies seem quite well-defined units. Not only that, but we can see in the trails the ants make when collecting food or materials for next building a level of organisation which quite belies the apparent lack of intelligence or awareness shown by the individual ant. This point is not flippant because there must be some way for the individual ant to become part of an apparently effective team, and yet there seems little way for this to come about, if none of the individuals seem to be aware of their overall purpose, which is the preservation of the colony. In this sense they are unable to step outside of their own ‘ant-consciousness’ and see the overall colony.

While ants are often observed to roam about as if at random, it is obvious that at some point there must be a statistical increase of information within a group in close proximity, which gradually builds up until a trail is formed. A trail can in this way be seen as a package of information which is formed by the ants themselves, even though they as individuals may be unaware of this information. However, there must be a certain communication - for instance, the awareness that they exist within a specific team of ants. As the trail, and thus the ‘information’ content grows, so does the communicative power of the whole.

However, the colony does not consist simply of a certain number of identical individuals. The overall population is divided into ‘castes’ - basically the female queen, the male workers, and the larger soldier ants. Moreover, within castes the ants develop specialisations which may vary according the age and caste. At any one moment and in any given area of the colony, ants of all specialisations may be present, although the density will vary according to the ‘information’ contained within the area. Indeed, to all intents and purposes, the distribution of caste and function is the equivalent of ‘information’ - the two states of being are working concurrently.

As a colony evolves over a period of time, this distribution becomes finely balanced, and allows an ever-greater expansion of the whole. The success of the whole, indeed, depends on the effectiveness of this distribution. As a living ‘object’, it cannot remain rigid, but must ebb and flow as the environment in which the colony exists changes. The movements of individuals within this system allow it a certain flexibility, and if some disturbance effects the colony it will react as one, but through the movements of its myriad individuals. Just as a human brain reacting to stimuli can be said to be taking in information, so the reaction of a colony to an external disturbance can be said to be effectively the equivalent of knowledge.

The movements of knowledge through a colony can be thought of in terms of the smaller teams which gather food. If the amount of food is great enough, so the effectiveness of the communication between groups will be increased, and once the supply begins to dwindle, so the teams begin to split up, and return to their multitude of ‘bewildered’ individuals. In just the same way, any ‘information’ which redistributes ants of differing castes throughout a particular area, whether it be food, building materials, or an external disturbance, can be seen as a stimulus which forces the whole to ‘think’ through its individuals. In the list in Figure 1, a difference is made between the terms ’team’, and ‘signal’. A ‘signal’ here represents a team that is actively carrying a piece of knowledge through the colony. Once a signal is formed, it is informed by the caste information that it already contains ‘where’ it is to move to distribute the new information. The whole, here, is working on several levels at once. On one level, there is the overall reaction of the whole to a certain stimulus. On a second, there is the reaction of the ‘individual’ team or signal to itself; that is, to the communication that arises because of the stimulus, and on a third, there is the reaction or non-reaction of the individual ant, who may also be aware of an ant behind and an ant in front, without the slightest notion or concern about where it is actually going.

From the human point of view it can be quite easily seen that is simply a number of different way sof viewing a single piece of information, rather as in the computer model a function could be described in machine, assembly or compiler languages. And yet our common sense reacts against the notion that any piece of information could be carried by the random movement of individual, entirely ignorant ants. It contradicts our idea of information as being purposeful or goal oriented, for the ant carries no such sense, or rather, the notion of it is not within its grasp. The colony as a whole has a sense of meaning and purpose, but although the relationship between the whole and the individual can be traced through interim levels, the overall meaning vanishes passed a certain point.

Once a signal or group of signals reaches a sufficiently ‘high’ or complex level, it may be termed a ‘symbol’, as in Figure 1. These are essentially ‘sub-systems’ of the whole, and are composed of lower level systems which have been called signals. They only exist in an active state, when in reaction to an external stimulus, rather in the way that the brain actively reacts to a passive symbol such as a word or a note or a phrase. To sum up:

The meaning which one attributes to any passive symbol, for example a word, derives from the meaning which is carried by corresponding active symbols in the brain. Passive symbols only have meaning when related to active symbols

In music, of course, these active and passive symbols have even greater consequence. Are the active symbols the sound of the passive symbols lying on the page, interpreted by the performer, or are the sounds themselves passive symbols that must be interpreted anew by the listener? If the isomorphism implied by the many similarities between the ant colony and the brain can be carried through at all into music, the relationship between the note, the figure, the phrase and the while becomes fascinating and a little fearfully complex.




I have tried to show in the description of the above two systems not only the complex way in which the levels contribute to one another, but also the manner in which as the levels are climbed, the ‘leakage’ between one and another several ‘powers’ away seems to dry up almost completely. These are the two themes that recur repeatedly in almost any system such as these. Even in the most basic of systems, that of successorship, 1 2 3 4 5 seems quite easy to appreciate, but if one were to ask the connection between 358 and 4,928,492, the answer, although one knows it, does not appear quite so logical. all of these systems are recursive; that is, they have elements and relationships that recur, and in which not only the elements may be altered in the recurrence, but the relationships themselves, while still maintaining the links between pairs and levels. (While giving this seminar, it transpired that according to some people, the definition I give to ‘recursive’ and ‘recursion’ here are not ‘mathematically’ correct. However, in this text the word should be interpreted using the above definition.)

Does the same idea apply to the ant colony? Or even to a human brain? Personally I believe it does, although the complexity involved in extracting the detail is quite hideous to consider. As has been said about the compiler language in relation to the machine language, and of the ‘sense of purpose’ of the ant colony with regard to the individual ant, the lowest levels are ‘invisible’, within the systems, as the higher levels are inconceivable from the lowest, and we are only able to see them at all because of our own relationship to these systems. As we have seen, we ourselves are part of a system, and although we have the gift of self-awareness, in that we are able to examine ourselves, it seems perfectly obvious that the level at which we are presently thinking has little if anything to do with detailed neuronic activity in the brain. it is simply not possible to consult an ant about what its colony is up to (language difficulties apart). And yet the paradox is plain. With our own logical minds we can see the link: how can an ant colony exist without ants, or a brain without neurons, or a piece of music without notes, or the individually defined sounds?

This seems to be the nub of the question. It is a familiar paradox, for of course paradoxes are essentially recursive. At the heart of each recursive system is a paradox, and both can be followed quite happily until we hit an infinity.



III Self-reference

There are examples of recursion which have slightly less viciousness to them however (although with enough level-weaving, I think ultimately the same conclusion would arise). Indeed, if everything we did was ‘reduced’ to recursive paradoxes we would never get anywhere. (Of course here I am talking at cross purposes again: I think anything of quality can be seen from a level on which these paradoxes exist, but we have the potential to quickly shift levels before any attack of ‘paradox vertigo’ sets in). This is what could be termed self-reference, as a broad term that could also include analogy, isomorphism, and self-reproduction. They are all forms of partial recursion, in that the analogy or isomorphism is either not perfect, or controlled by practicality so that not too many infinities arise together. It is, in a sense, playing on the edge of the infinite and like standing on the top of a high cliff, mountain or building can, if controlled, afford marvellous results.

While science despises infinity, for it exposes holes in any purely logical view of the world (as we have seen above), artists, I would contend, play with it. Not consciously, maybe, but the effects of recursion can be seen in the oeuvre of any artist, and indeed throughout a whole period of history. These recursive elements create style, whether on a local or global level. Extreme examples of self-reference are easy to find, especially in the twentieth century. In fine art, artists like Magritte, Dali, and de Chirico make explicit use of ‘trademarks’, in Magritte’s case, tubas, pipes, applies, and pictures within pictures (an obvious partial recursion) are all commonplace. In Dali’s work there is a profusion of ‘supports’ supporting collapsing figures, ants and melting clocks. In spite of their surrealism, these figures are equivalent to the ‘pet’ ideas evident throughout many artist’s work, such as Breughel’s peasants, or Van Gogh’s fields and flowers. A more profound example of self-reference within an oeuvre are the self-portraits of Rembrandt. While obviously not intended to come together as a single work, they form a cycle that is as revealing as any ‘group’ of similar works in music: one could mention Mozart’s Piano Concerti or Beethoven’s Sonatas or String Quartets.

In literature self-reference is even more evident. Shakespeare gives many examples, for instance the Duke of Gloucester’s sub-plot in Lear, or the play within a play in Hamlet. Here the device is quite explicitly used to enrich and expand upon the major or ‘real’ action of the plays, which of course can be extended on level further into ‘real life’ itself. Swift’s ‘Gulliver’s Travels’ is an example of a book entirely written ‘within’ itself, in the sense that Swift ‘himself’ ‘wrote’ none of it, but took on the character of another, in order to comment more freely, (but also symbolically) on his society. Here the invention of a character is obviously used to allow more room for the author to think, so he can remain, like all satirists, on a different level entirely from the action. Vladimir Nabokov, more recently, in his book ‘Pale Fire’, created the work from a poem by a fictional author, enclosed within a large editorial preface, and an even larger section of editorial notes as a ‘supplement’, all written by a fictional editor. In this way Nabakov seals himself off from the book on a number of levels via the poem, then the edition, and finally through the complete book, much in the way that Magritte plays with the idea of painting a painting of a scene, situated before the scene itself, which is also a painting. In both cases the artist or author desires to comment on the way in which the information contained within reality is approached.

all the above examples explicitly or implicitly use the idea of a ‘frame’ within a frame. Obviously, in most painting this has to be so, as all paintings are finite. In the same way, any work of art is automatically framed by its duration, no matter how that may be distorted, or however abstract this duration may be. In a strange way, it is this very ‘framedness’ of art that gives it its potential for commenting on the infinite, through the use of recursion and self-reference in varying forms. As in Hamlet, the play within the play gives insight into the ‘play’ itself, which in turn gives insight into the real world.

Although inherently more abstract, music makes equal use of frames within the ‘automatic’ frame that surrounds a sound with silence. These can take many forms, whether explicitly in the slow introduction, prelude or coda, or more implicitly by the use of formalizable structures such as sonata or Da Capo, where the frame is ‘built in’ to the picture. And once one starts to look in depth at any of these frames, the definitions begin to grow more hazy, and the infinity of recursion begins to creep up on you.

The opening bars of Beethoven’s fifth symphony seem a perfect example of this idea. They represent an introduction, that is a ‘frame’ and a ‘signal’ for the entire movement. But within a few seconds one becomes aware that this ‘frame’ is itself not merely part of the whole, but gets very close to being the sole material of the whole. Obviously the comparisons with other art forms become difficult here, because we are looking at the functions of the basic blocks of communication, and while the ideas may be analogous, the functions are radically different.

However, to move back from the complexities of Beethoven, we can see how, especially in the twentieth century, ideas of explicit self-reference have gained much ground. This goes beyond typicalities of style, of idiosyncrasies, and can involve the use of ‘borrowed’ material from a composer’s previous work, as happens for instance in Birtwistle’s Triumph of Time. In Berio’s Sinfonia, the borrowed material ranges from Bach to Berio himself, and includes a passage on the piano to be ‘taken from a work played in the same concert’. And in the same way that Swift ‘distanced’ himself from Gulliver, the whole idea of neo-classicism within the twentieth century can be seen as a similar frame with which a composer may distance himself from his work. This is not to say that the communication is any less effective because of this, as Swift’s comments on his society are no less acerbic because he comments ‘around’ Gulliver.

In this sense Stravinsky is no more Pergolesi in Pulcinella that Swift ‘is’ Gulliver. All that really seems to happen is that the ‘invented hierarchy’ within which any work of art exists, is taken to a level one stage further than ‘normal’. The end result, and its resultant communication, are not necessarily effected at all, for the basic level of all these communications is the infinity of recursion.



IV Mozart

Having found certain ideas which seem to recur at many levels and throughout many art-forms, I would briefly like to discuss how they could possibly be seen in a piece of music. For this purpose I have chosen the exposition of the first movement of Mozart’s F major Piano Sonata, K332. I choose only the exposition for the sake of time and detail, and Mozart because his music at once interests me for its natural lyricism and flow. It also has a way of defying rigid methods of analysis, for while a formal sonata form exposition can be seen quite clearly, with the second subject entering in bar 41, quite clearly in the dominant, a cursory glance at the detail of the music shows how hopelessly inadequate such a brief description is. My initial question was quite straightforward: where is the music? For instance, is the so-called first subject the melody of bars 1-4? Does it include the second phrase, bars 5-8, and the third, bars 9-12? And what about the idea in bars 12-16, repeated in 16-20? Or the accompanying figure of the opening bars?

One might put all these ideas together and call them ‘first subject group’, but merely giving a label does not describe or explain the internal structuring of the phrases, anymore than the formal description ‘sonata form’ describes a first subject. We may decide that there is no first subject as such, but a group of ideas which in succession communicate unity yet with expectation. The continually reiterated perfect cadences into F major (bars 4-5, 6-7, 11-12, 16, 19-20, 20-21, 21-22) certainly establish key, but they also serve the purpose of articulating the profusion of melodic material in which the passage abounds. These ideas are really quite distinct, although when isolated incomplete.

Here I feel there is a problem, for I am not quite sure how far one can go in describing, in a reductionist manner, the inter-relations between these ideas, without going into fairly obvious detail. One could mention the reflection of the f-a-c-a in the bass of bar one in the melody above it, but the nature of Mozart’s piano writing could explain this miniature example of self-reference. Does the articulation of the tonic chord warrant this attention, or are we looking at the obvious to help us describe things we do not really understand? From a reductionist point of view, the relationship certainly exists on the paper, but holistically, can it be that relevant? Of course, practically speaking, there is little else the left hand could do in that register without interfering with the right.

There are also the temptingly Schenkerian descending scales in bars 10 and 12-14, also outlined in the phrase of bars 5-7. Indeed, these descending scales occur everywhere explicitly, and the whole exposition could be described as a graceful series of rises and falls. Are these relationships ‘deliberate’, or sub-consciously self-referential? And how can we be sure, if we choose to analyse in reductionist depth, that we are not slipping levels and looking at something not unique in this piece of Mozart, or indeed in Mozart at all, but at something that lies ‘hard wired’ into the human brain, and that these relationships are the external manifestations of? If there is even the slightest possibility that this is the case, and if one can follow the clues that lie within other systems such as the ant colony and its symbolic equivalent, the brain, could we speculate that a great many of the lesser relationships that exist, are simply ‘invisible’ to us. That although physically present within the notes these lower levels, and possibly an infinite quantity more, are lying beyond our analytical powers. Here the passive symbol idea has relevance. The passive symbols that represent Mozart’s thought, when played, activate the active symbols within our own brains, which are placed there by nature and refined by understanding.

Ultimately, we may reduce the notes on the page to one, two or three note figures, but by this state the high-level has gone and as musicians, we would be attempting the equivalent of trying to work out the atomic structure of silicon by inspecting a computer’s Compiler programme.

While the infinities and paradoxes here may be hard to work with, we know that music as an art is a finite temporal experience, and that whatever their inner complexities, musical phrases progress through time towards a certain point. On this high level we once again see the sense of purpose returning, and the information on the page takes on a more meaningful aspect.

Just as in the ant colony, we can switch levels from the meaningless wanderings of the individual ant, and see the whole reacting to some external force. In other words, cause and effect are to some extent returned to us. Even here, levels are quite clearly visible, and the music in Figure 3 gives my interpretation of some of them. Here we look on each note and phrase as a temporal event, and it is travelling through these notes and their relationships that gives us information. This corresponds to the idea of hearing music anew each time it is played, for even if one knows the eventual outcome, the importance lies in the travelling towards it. The music can be seen as travelling through a series of levels, much as one would read a list in Figure 1. As was mentioned earlier, the concept of leakage applies, and the levels one passes through in the Mozart (that is the variety of phrases), themselves take on relationships with each other, although if you ‘jump out of’ the temporal system and compare the end of the exposition with the beginning, you can see how far the music has moved from the opening bars, while still remaining within the stylistic system of the whole.

In this case, the numbers above various phrases represent one interpretation of the ‘levels’ passed through, and while some are obviously linked internally, for instance 2a and 2b, or 4a and 4b of the ‘first subject’, others have more complex relationships. In Mozart’s case all are articulated by cadences on this level, thus the move from 1 to 2 in bars 4 and 5. here I think phrase two has definitely moved from the level of phrase one, just as the more definite cadence in bars 11-12 represents the greater jump into phrase four.

In this manner, one can move through the entire piece, rather like travelling in a musical multi-directional lift, moving between phrases with jumps that are more or less extreme, while the linking material arises on a local level, and as one passes through more and more levels, the leakage grows less and less between those more remote. Here memory becomes of utmost importance, and our ability to contain information will at least partially account for our appreciation of the whole structure.



V Conclusions

Looking at Figure 1, I think the isomorphic links that lie between the lists can be said to be something more than coincidence. Most branches of modern sciences and arts have to some extent accepted the idea of the relative; that is the way in which a single piece of information is directly effected by the way in which it is observed. This is an unsettling idea, because it challenges our notions of the absolute or the pre-defined, but it is also responsible for natural phenomena of the most remarkable beauty. Primarily, and this is probably its most profound result, it states that no view is ever complete in itself, and that the more complete one attempts to make one’s view, the more thoughtful and logical, the open it is to recursions and infinities. An analogous way of describing this is by imagining a human as a flat two-dimensional figure on a piece of paper. Because of its two-dimensionality, it is impossible that it will ever appreciate the three-dimensional world in which we observe it. all the trompe d’oeuil tricks, all the logical arguments will never enable it to experience the third dimension, and will indeed only emphasise its two-dimensionality. One can cut the paper, fold it, roll it up, but the vision of the 2-d being will remain limited by definition. The third dimension can certainly be imagined, speculated upon, formulated even, but never fully experienced.

The arts, from a reductionist viewpoint cannot escape from this. Any single work can be defined in a simple typographical or reproductive way, in a book, a score, a painting, a photograph, or a record, but using the simple and natural tools of symbol and level, the human imagination can be enlarged and enriched. Music is by far the most abstract of the arts, for it not only combines the dynamism of temporal movement with the complex abstraction of created symbols, but in both of these dimensions it is capable of symbolising those very symbols, and so on. as a natural art it is hardly surprising that it should be the case that music mirrors the processes by which the human exists.

Neither does our own limited three-dimensional ‘vision’ preclude the existence of levels of understanding or meaning above our own. By the argument given above with regard to the two-dimensional being, our own 3-d limitations could quite possibly blind us to other levels of meaning we can suspect and intuitively feel, but can have no rational explanation for.

If a lingua franca of music exists at all, it exists here. It does not seem a terribly helpful conclusion, for the majority of the processing seems to go on at a subconscious level, with the results being ‘invisible’ to all reductionist intents and purposes. Above all, however, it provides a method of linking music of any type to natural processes of communication and form. As was mentioned at the beginning, it also answers the determinacy/indeterminacy paradox, not by justifying chance methods historically in relation to more defined music, but by jumping levels and seeing them as aspects of the same idea. When compared to the infinite recursions that lie at the heart of any work of quality, the mild indeterminacy of regulated chance is tame.