Metaform and Metaforming



Throughout this talk I'll be presenting you with a series of quotations. Due to time constraints, I cannot read these out, but have provided you with the quotes separately, so that you may peruse them during my talk. Please read them as we proceed, as I won't have time to pause sufficiently for you to read the texts completely from the screen.

As a composer, one of my key interests has been the role of analytical methods in the understanding of the origin and development of the musical event.

These methods usually deal with one aspect or another of the music, but rarely with the entire phenomenon.

And yet the act of composition as a whole is substantially intuitive and may be based on what inspires us at any given moment - a sudden feeling, a slip of the pencil, a brief memory. Like an artist drawing, musical inspiration at its best can be sudden, immediate and capricious.

How, if at all, can we come to terms with this? Which approach is best to help us understand the musical event?



Why use the term metaform as opposed to metaphor?

In music, there may be an argument for a different term because of the difficulty many musicians have with the attribution of meaning in music.

Literature and language is steeped in metaphor (like that) - it is its most powerful tool. In music metaphor might be taken to imply programme music - which is generally considered less valuable than purely 'abstract' music.
[Steven Pinker].

In terms of this statement [SLIDE]: I find it significant that a language expert should so willingly accept Cooke's basic ideas, whereas, as Pinker himself points out, so many musicologists scoff.

Musicians prefer to talk about 'form' - an abstract collection of lower-levels elements that, together, form... a collection of higher-level elements..., and so on.

But why do we like these collections?

What is it about collections of differently leveled elements that makes us respond as we do to music?

I would imagine that very few academic musicians, including myself, would agree with Pinker's last clause. However, most of us would accept that music can and does communicate something - although whether that something is culture or rule based, or intrinsic, might be more debateable.

I'll be arguing that there is a meaning, and that this meaning is metaphorical. In musical terms, this means that there are many levels of form, each of which has a bearing on those in closest proximity. These might be termed metaforms if only to make it clear that these meanings are not necessarily linguistic or emotional. I'll argue that not only is metaphor the central meaning of music, but that metaphor is the reason that music exists, that we cannot avoid using it, and that its presence is fundamental to the inexplicability of music in analytical terms.

Metaphor does not need to be deep and fundamental - some of what we think of as the greatest and most profound music may have been the result of whimsy, caprice and accident. Such associations do not need to have any particular 'meaning' (in western music they usually don't), they just need to be there. (cf Britten yesterday)

To emphasise this, the following is another, frustrated quote from Pinker concerning a final, catch-all explanation for music...



Although sprinked with sarcasm, is there not equally an admission of failure in his search for the 'meaning' of music?

Could music not be all these things (and many more)? Pinker is using metaphor to imply a lack of understanding. Even if we know that experiencing music is caused by

"a resonance in the brain between neurons firing in synchrony with a soundwave, and a natural oscillation in the emotion circuits"

does this 'explain' or 'enhance' the perception of music in any more efficient way than a musicological account?



Increases in our scientific understanding of many areas of human activity has meant that, increasingly, knowledge of more and more of these areas are necessary for a complete understanding of any one in particular.

Work in many fields has seen a 'leaking' of ideas from one domain to another, and one of the most influential areas is computer technology. Richard Dawkins compares the rate of development in computer technology with the sudden acceleration in the evolution of the human brain over the last million years, and suggests, among other possible catalysts, that

the ability to see analogies, the ability to express meanings in terms of symbolic resemblances to other things, may have been the crucial software advance that propelled human brain evolution over the threshold into a co-evolutionary spiral


5 - Virtual Reality

The impact of computer technology on our idea of perception has been enormous, especially in scientific philosophy, as the computer reveals in practical terms how so many of the ideas we have taken for granted are in fact very complex.

?From artificial intelligence and virtual reality to pyschology and evolution, from speculation concerning what our technology may enable us to become, to what enabled us to begin the development of that technology six million years ago, the catalysts for such a growth of interest in analogy span the whole of human development?


6 - Mithen's Schematic

Cognitive psychology investigates the way the mind is structured. Is it a series of unrelated modules, each specialising in certain tasks - the number and specialisation of these modules growing as we develop individually, or does an element of 'leakage' occur between them, as illustrated by our use of analogy and metaphor? Some psychologists have gone so far as to say that there is a separate module of 'rerepresentation' which is has developed specifically because of the value of these cross domain associations.


7 - More Mithen

Steven Mithen, amongst others, argues that it is these very associations that caused the 'cultural explosion' in human development 60-30,000 years ago and that continues to dominate our development today - a rate of development that contrasts starkly with the many millions of years of 'fumbling in the dark' characterising the lives of our ancestors.

Ian Cross has taken this very much to heart and suggested that music, through the way in which it combines form, intellect, emotion, language and motor skills amongst others, is perhaps what makes us human.


8 - Hofstadter

Since first reading the book based around this network in 1986, I have been continually drawn back to it and its unique juxtapositions of artificial intelligence, computing, mathematics, music, graphic art, Zen, Language... There have also, since its publication, been a bewildering array of similarly cross or multi-disciplinary books....

And I've always been drawn to this figure - itself a bewildering array of connections and associations.

Each connection has a meaning, and yet, by crossing connections, one makes whole sets of new, apparently arbitrary ones.

Elaborate though it is, of course, our minds are very much more complex and rich in associations than any two-dimensional drawing.


9 - Detail

Music is often presented in one or more contexts according to specialism and personal preference.

It would appear that each of these approaches is necessary according to the task in hand.

I intend to outline a few of these approaches, (and this may be one of the problems). They represent very broad areas and I do not intend to cover them in an way other than schematically. Each 'node' in each network can be expanded into a network of its own - I have necessarily left out an indefinitely large number of nodes and each time I review this talk I think of others that ought to be included [especially after yesterday] - however, for reasons which will become clear in a few minutes, I cannot do this.

The accompanying texts are intended to be supportive, entertaining and, perhaps, revealing of certain associations.


10 - Musicology

Native ground for most of us. From here, one might comment that Beethoven's early period shows many signs of Haydn's influence. Beethoven himself showed significant musical development, his entire oeuvre having significant influence on, for instance, Brahms.

This view also allows those wonderfully intense discussions concerning Schenker, serialism and set theory, the usefulness or otherwise of studying Bach's Chorales, the tonal system, matrices, etc.

This view forms the basis of most degree courses called 'Music'.



For another example, we may take the first few lines of this Mozart Sonata and analyse any in accordance with various techniques. These techniques, hopefully, reveal aspects of the music's detail helping to make our experience of it more informed and enjoyable.

It was my own attempts to understand Mozart in accordance with 'traditional' systems of analysis that originally led me to doubt their universal validity - this excerpt represents what would usually be called the 'first subject material', as it is quite clearly more than 'a' first subject, but as a group of ideas more or less related (by contrast in terms of the passage I've labelled '4') and creating a sense of movement through this, association (2a - 2b, 4a - 4b), and, what? A sense of intuition?

What is any analysis but the imposition of one sort of metaphor on the music? As with any metaphor, can it reveal more than one particular perspective of the 'real thing'?


12 - Psychology

NB Sloboda's comments concern music with a significant performance aspect.

From another perspective, I have included six possible areas of interpretation from Steven Pinker:

Neurologists, clinical psychologists, developmental psychologists all have a perspective, and usually an entire vocabulary to complement it. I've had to ignore the lot, I'm afraid, and come up with this poor specimen. I apologise…



…and move hurriedly to a related and perhaps more picturesque idea... (you'll see it again later)


14 - Evolution and Genetics

In chapter 4 of his book, Unweaving the Rainbow, Richard Dawkins describes the methods by which it is expected that we hear music, which he calls 'barcodes in the air'.



This seems to provide a good explanation of music, or is it a metaphor?

Note that I am not claiming that birds 'make' music:

"That whales and birds link song sequences together is also not evidence of versatility. The most mindless behaviours are often linked, the completion of one calling forth the next...Indeed, the more complex and 'purposeful' the behaviour is, the further it may be from intelligent behaviour, simply because natural selection has evolved a surefire way of accomplishing it, with little left to chance."

William H Calvin, How Brains Think, 1996

In the final paragraph he is clearly suggesting that certain practiced, developed or genetically given birdsongs may have the effect of a physical drug. Could this not be a description of the effects of some music on some people? Fascinating though the subject is, I must hurry on…



There is relatively little work in this area. One of the best books concerning art and evolution, quoted earlier, by Mithen, does not include any mention of music!



This little account of the point of view of a Venda chief seems to point to many uses of music often not considered in music courses:

  • the use of music as an 'official' tool of the state
  • the use of music as a status symbol at social functions such as weddings, funerals, etc.,
  • the financial element of music - money is power
  • status and tradition - music included for its traditional rather than musical value?



In contrast to the earlier 'musicological' view, we could argue that Beethoven's earlier work developed due to the political and economic status of his patrons. It could be argued that the size and nature of Beethoven's ninth symphony owes much to the Royal Philharmonic Society of London.

This does not explain the detail of the music, but does explain why the music exists.



To me, one of the most intricate and fascinating areas - it is especially interesting to see how many physicists and mathematicians like to use music as an example of... something.

However, we have to be especially careful in relating levels here. William H Calvin:

"Quantum Mechanics is probably essential to consciousness in about the same way as crystals were once essential to radios, or spark plugs are still essential to traffic jams. Necessary, but not sufficient. Interesting in its own right, but a subject related only distantly to our mental lives."

(Physics, especially, is in a tricky position. As the carnivore of the scientific world, it has over the last century, devoured one discipline after another and some are concerned that some of its representatives are now after psychology, too, on the basis that if the brain is a physical object and the mind is a phenomenon arising principally from the brain, then thoughts must be physical and therefore, in principle, subject to the laws of physics. This is a powerful argument that has provided the fuel for significant portions of the Artificial Intelligence debate. With Roger Penrose, there are now factions within factions, the latter stalwartly claiming that the experience of consciousness itself points to difficulties with our current understanding of physics itself. )

As the above quotation shows, however, psychologists and neurologists - those dealing with one or other level of emergent phenomena are fighting back hard.



Mathematics, computing, AI, virtual reality, quantum mechanics are missing...



Clearly, this area overlaps quite significantly with the evolutionary and genetic view described earlier. Here, I've chosen to elide ideas concerning the ecology of music - the idea that the sound of a given music is tied to the availability of material resources, with the more musicological one that music is heavily conditioned by the motor skills required to perform it. This is also, of course, of growing interest in the view of cognitive psychologists, as demonstrated by John Sloboda earlier on.

I find the idea that the physical environment plays a significant role in determining what music should be is both charming and easily forgotten in western tradition where instruments are so.



This simplistic combination of simple networks is static and constrained to two dimensions - our thoughts are neither.

Composers, psychologists, mathematicians, physicists, performers and others may experiment with combinations of more than one approach.

So, a purely scientific view of the perception of sound may be influenced by evolutionary or ecological arguments.

Composers of traditionally notated music may be influenced by musics from other cultures, the sounds of electroacoustics, or the traits of an individual performer.

An archeologist may develop an interest in the psychology of those of our ancestors he or she is studying.

While accepting its limitations, this graphic presents, to me, a satisfying if necessarily incomplete picture - I like the way in which, mirroring Hofstadter's earlier network, moving short 'metaphorical' distances across the diagram, can result in 'moving vast distances' between apparently unrelated areas.

However, it does not tell us anything in detail about any single activity or area. I would suggest that this is not because of the quality of the diagram, but its inherent physical weakness as a two-dimensional representation of a multi-dimensional set of mental domains, each of which have complexities of their own, and between each of which, given sufficient knowledge and imagination, exist further and equally intricate sets of associations. Each of us possesses such a network. None of these networks are the same, although most are similar, or at least be structured in the same way, their precise contents will vary. The way in which we interact with others, in more or less related fields and with more or less related interests will depend not only on the contents of these networks but on our very ability to relate our own networks to those of others.


23 - Jabberwocky

For the second part of this talk I would like to, in Hofstadter's words, 'jump out of the system':

[Hofstadter's collection of translations of Jabberwocky.] It's fun, but it also exemplifies the problems involved in translating from one language to another. He provides another example…


24 - Translation

He then goes on to give three versions of the translation:

  • "S. Place"
  • "Stoliarny Place"
  • "Carpenter's Lane"



In a similar way, I am offering two versions of what might be called 'jumping out of the system'.

Here's the first...

Any ideas as to what this might be?

  • a wave editor's interpretation of the file of this presentation... (Cool Edit Pro)
  • actually, there is a cheat here - because I wanted to arrange this earlier - in actual fact, this is a wave editor's interpretation of the file of this presentation as it was last Saturday. Does this matter?
  • there's a second cheat, too - what you've just heard has, as many of you will have been able to tell, some reverberation added and the first part of the sound has been treated with a filter - why? to make it more interesting (a matter of taste, or converging/diverging networks?)... Here's the original.

Now listen to one of these...

or try them together

They were produced with a piece of software that I wrote last year for to investigate links between image and sound, and are a number of interpretations of the same image. Many of you might have guessed by now what this image is...


26 - The Network


In other words, thery are 'audio translations' of the graphic data of the image of the network itself.



The quality of the music just created is up to me and you. How important is the way in which they're produced?

As a composer, one of the 'metaphors' I commonly use is the imagining of music visually or graphically. In other words, in attempting to notate a thought or sound or gesture, I might well use graphic or visual shapes (or physical gestures) to help, so this (draw) might represent a scale (or should that be a glissando?) I often might use such shapes as these (zigzag) to represent the way in which the music approaches a climax...

One of the incentives for developing the previous software was in order to investigate such visual or graphic metaphors.

Needless to say there are very profound difficulties in the implementation of this apparently obvious metaphor, more profound, I might add, or at least of a different nature to Hofstadter's difficulties with "Carpenter's Lane". This is mainly due to the differences in 'meaning' between language and music...

What does it mean? You tell me, but the materials it produces are, as I have said, quite suitable as electroacoustic material. Although the results are obtained slightly unconventionally, they are controllable.

How does it work?


It 'scans' the image, taking, two the Cartesian coordinates and an RGB value. This information is then 'translated' into a score and orchestra compatible for use with the CSound programme. As the sounds are of some complexity, I'm using CSound to generate a wav file which may then be played by any audio editor capable of interpreting wav files. Note my use of terms such as 'translation' and 'interpretation'.

Some of the many problems:

  • Which graphic information should map onto which sonic information - there are no rules.
  • At best, the graphic information here can offer only three parameters - X,Y and RGB value. Sound has many more.
  • Just as I explained with regard to the 'semantic network' - the 'explanation' in this case is limited by the physical dimensions imposed on the technology (as well as the time I have available for making points such as these!)

What do these tell us about music?

In my view, they tell us a great deal about what it is not.

It is not a graphic!

Does this reduce the value of thinking of music graphically?

Not if we are careful and use graphic thoughts and ideas for our own use - recall Nicholas Cook:


We must be careful not to assume that any metaphor is intrinsic to the idea of music.

While there is a clear and demonstrable link between the sounds and the words (even the letters!), there is clearly no meaningful link in linguistic terms. The typography is important. (See Hofstadter)

The link between the sounds used above and the words of the network is itself a metaphorical one. The graphic is in one respect the metaform of the sound. However, the leap from one to the other is itself a part of the process - I am jumping out of the system. In taking the results of that leap - a nascent, unformed musical idea, I can then analyse that idea in isolation from its direct cause - the pixel formations of the typographic details of the network I have just described. Attempts to find a real, meaningful link between the two cause a problem - because the link is clearly and demonstrably there, and yet it is only meaningful typographically.



My next example is another metaphor (or metaform).

It involves a link between music and mathematics, or rather, in this case, mathematics and number theory.

In this quotation, as in his books 'TENC' and 'SoM' Roger Penrose, the mathematical physicist, uses the Mandelbrot Set as an example of what he suggests may prove the Platonic nature of mathematics - in his opinion, built into our minds through the offices of nature itself. (Galileo: "the book of nature is written in mathematical symbols").



What we often mean when we refer to the 'mathematical' view of music is usually in reality that branch of mathematics called number theory dealing with the patterns existing between series of numbers. (Serialism and Set Theory, and so on.)

I was introduced to number theory through Hofstadter's book, and the greatest impression, for both musical and aesthetic reasons was made by his exposition of 'wondrous numbers'.


30 - Wondrous Numbers

Wondrous numbers demonstrate, in Hofstadter's book, the inherent difficulties of proof. The rules that define whether a number is wondrous or not are very simple. Take an arbitrary number...

As you can see, this produces a sequence of numbers, rising and falling in accordance with whether the result of one of the calculations is an odd or an even number.

A 'wondrous' number never returns to one.

The task for number theorists is to prove whether a number is wondrous or not without using the algorithm itself.

Why should anyone want to 'translate' varieties of this algorithm into 'musical' terms? I don't know - maybe there's a psychological reason somewhere, but here is a brief example.

Here's another one, The Mixmaster algorithm.... Choose Harpsichord, delay 80, all other parameters on, then do it!

Friendly numbers, social numbers, other numbers…

Choosing any of these seems arbitrary and obscure. Applying additional parameters makes them even more so...

What have these to do with music? What have they to do with number theory? What, if any, is the link between the numbers, the function, and the 'music'?



Demonstrate psy as a simple idea... MY SOFTWARE!

Are we hearing a feature of number theory, in other words, the taking of a number and a demonstration of its wondrousness or otherwise... or are we hearing musical relationships?

[What does the wondrous algorithm say about serialism - is it related at all?

Is this not serialism? Is this not mathematical?]


32 - Cheats and Mistakes

You may well be able to guess some of the problems now...

These are, incidentally, specific and detailed and involve difficulties of the 'translation' from one medium to another.

In other words, these are the 'cheats' and the 'mistakes'.



The Copenhagen Interpretation was really an attempt at overcoming the inherently fixed nature of so much electroacoustic music. If 'performed' properly, (I recommend four synthesisers controlled by four versions of the software), each time the piece is 'played' it will be the same but different, much as, I hope, a piece of notated, interpreted music is different.

The wondrous algorithm plays a small but fairly significant role during a stretch of around four minutes in a piece that lasts about sixteen minutes. Here is a demonstration of a part of this small part:


There is a difficulty here that involves the cheats, mistakes and other anomalies - are they significant? For that matter, when the programme is running, it is impossible to tell exactly what will happen, and to compound the problem, whether the data presented on screen is an accurate measure of the 'reality' existing in the synthesiser, or an accurate measure of the algorithm at any particular time.

Here we have a good example of a musical metaphor. From the perspective of a programmer it is vitally important that the programme mirrors as perfectly as possible the 'reality' it is intended to control. If this does not happen then the best that can happen is that anomalies appear and that the programme ceases to behave predictably, the worst that the programme will simply stop.

As a musician I can understand this - to a point. There is clearly little purpose in writing a programme that is so unstable that it hardly ever runs, however wonderful the results. At the same time, there is no point in having a programme that runs perfectly but produces dull tedious rubbish. Ideally, of course, you want a programme that runs perfectly and produces wonderful material, but in order to get to this point, if you ever can, you have to develop many less than perfect systems on the way.

In other words, you have to learn.

As a rather nice side effect, analogy, metaphor, metaform or whatever, it also links rather well with the idea of Wondrous Numbers. As you heard, I'm using wondrous numbers as a series of patterns, 'translated' into musical pitches. These are contrasted with other formulae which create different patterns. I have cheated in this programme, but limiting my random input value to the range 2-127. We know that the input of any one of these numbers will create a finite pattern, always resulting in a similar 'cadential' pattern of notes, something like 16, 8, 4, 2, 1 (which results in the 'circular' pattern you hear). It is likely, but unprovable, that not limiting the input would produce a similar set of patterns - they would all 'cadence' eventually. However, it is not possible to prove this and there is a chance that the programme would hit upon one of the few (or the only) natural number that would spiral forever.

Much speculation in number theory concerns whether such anomalies (such as the recently resolved Fermat's Last Theorem) are solvable paradoxes, or symptoms of something deeper.

Musically, while we may be fascinated by such ideas (I am), the important thing is the 'sound' of things.

In using a particular idea from number theory, I am, by definition, not using such an idea in its mathematical sense, but using a feature of it - its ability to pattern numbers. And yet this is a crucial part of the function.

If there is a confusion of levels here, I believe it is a direct result of our inability to resolve issues raised by the application of the metaphor in the first place. It is the 'frisson' of these difficulties that creates at least a part of our enjoyment of music.


34 Conclusions

Is it possible, as I have suggested, that the very basis of our musicality is that precisely because music cannot 'mean' anything, we have little choice but to make it mean all sorts of things if we are to appreciate it at all? In the above example from The Copenhagen Interpretation, we are quite capable of hearing, not just the patterning itself, but we can (if we know about it and feel a 'network response' to the idea) also appreciate the use of number theory in its creation. We can also choose to ignore these associations and respond on a more physical or emotional level.

Is it not only possible but rather likely that one of the major reasons for the existence of music is that it is an attempt to mate, no matter how elaborate and arcane it has become in some circles? If that is the case then it must be that there is something in the expression of music that reveals evolutionary advantages over those who do not have it. What puzzles some evolutionists like Steven Pinker is that music does not seem to reveal a particular, built-in, evolutionary advantage, unlike, for instance, being tall, which is an indicator of health.

There are other theories concerning the development of both music and language, and about the potential role of language or some other factor in the sudden increase in our brain size some 100,000 years ago. I like William Calvin's theory that it was our ancestors' ability to throw stones accurately - an immensely complex act involving the development of many cognitive and physical domains working together. This would have given the accurate thrower a significant evolutionary advantage - they'd have had more to eat, and so on.

It has to be said, though, that I like this theory as much due to its 'obscurity' in comparison to others, than due to any particular knowledge I have. In other words, it appeals to me for aesthetic reasons.


35 - Coda

Dawkins, in his description of birdsong provides an advantage music might give the musician - as a 'drug', although not one ingested physically. Ironically, Dawkins gives this example in defence of the scientific method of understanding, as opposed to the poetic view as exemplified in Keats' Lamia.

Hanslick would not have been happy with this conclusion. Any human listener responding to music in the way Keats responds to the nightingale 'may prevent the development of that strength of will and power of intellect which man is capable of'.

He felt that music listened to as a drug loosened 'the feet and or the heart just as wine loosens the tongue'.

Note the plethora of metaphors.

It may comment on my personality that I find it rather pleasant to consider that it may be possible that the very response Hanslick finds so suspicious may be symptomatic of the 'cognitive fluidity' that enabled modern Homo Sapiens to become what he or she is.

When I looked through this talk, I had the strong feeling that it would be interpreted that the first part is an apology for the second - that it attempts to justify the methods used in the composition.

I did not intend to do so. It is one of the fundamental points of this talk that there is no way of really understanding the details of a musical event - merely different ways of seeing them. Ultimately music can only be experienced - although that experience can be enhanced through the development of associations, if the listener has the appropriate network to begin with.

We may find it difficult to appreciate the metaphorical nature of music because we find it difficult to think without metaphor. As Steven Mithen has suggested, I feel that metaphor in the sense of 'many domains working together' is built into our thought processes for strictly evolutionary reasons - it is what made us what we are.

One of the effects of this is not that we experience music as, in Steven Pinker's memorable phrase, auditory cheesecake, but as complex series of interelated networks of ideas and associations. The nature of these networks, each of which is personal to ourselves although clearly partially defined by nature and nurture, determines how we respond to specific types of music or specific musical ideas or perspectives. It may well be that, as Ian Cross suggests, there is something special about music simply because unlike virtually every other form of expression except, maybe, mathematics, it cannot be appreciated in any way other than metaphorically.


Richard Hoadley, April 2000