Composing With Numbers

Professor Jonathon Cross

14/12/2009

The title of my talk, "Composing with Numbers", you might say, is perhaps such a truism that it is not even worth saying - composing with numbers is like breathing with oxygen.  To put it the other way around: breathing without oxygen is an impossibility, just as composing without numbers is an impossibility.  When I breathe, I take a breath to deliver my next sentence, I am not consciously aware of the fact that I am taking in oxygen, it is going into my blood and being carried round my body to keep my cells alive, and similarly, many composers are unaware of the mathematical, geometrical, numerical aspects of what they do.  Of course, at a very simple level, composers all need numbers, whether it is at the level of counting bars or beats, writing numbers to show metre, or indeed, finding a vocabulary to talk about intervals or chords or the geometry of forms.  So numbers are everywhere.  Therefore, composing with numbers is what composers do; it is what all composers have always done and will continue to do.  So, in a way, there is little point to my talk because maths is unavoidably in composition, but my subject today is, in particular, the music of the 20th Century, and, through a series of examples, what I want to try to show is how composers this century and last century, the 20th and 21st Centuries, have explicitly engaged with number in many different ways, in order to renew their art. 

But I would like to begin with a lesson from the past, because where would a talk on music and mathematics be without Johann Sebastian Bach?  I want to talk about a moment from the first canon from the Musical Offering, in Bach's own hand.  The first half of it is a theme that was given to Bach by the King of Prussia, and Bach elaborated this into an most extraordinary document, a series of puzzles of complex canons.  It begins with a relatively simple idea which makes the point very clearly.  It is a canon for two voices.  As players, you have to realise the puzzle, which is that one player begins at the beginning and plays through to the end, and the other player begins at the end but plays through to the beginning, and then they swap over - well, indeed, it is perpetual.  It is a cancricans' canon, because it goes backwards, but it could go round and round forever.

The idea of translation, of course, is central to the idea of canon, taking the same thing and moving it at a distance, but we get all sorts of wonderful reflections here as well.  We get reflection about the vertical axis, the idea of what a musician would call retrograding, so the same thing is going backwards and forwards, and we get, in places, reflection about the horizontal axis as well, at certain kinds of intervals.  We will encounter this a little later on, so one version where the intervals are rising, and another version where the same intervals are going in the opposite direction, as if seen or heard in some kind of mirror. 

As a musicologist, I need number in order to talk about this.  I would find it very hard to analyse music without number.  Even if Bach did not explicitly use numbers when he composed, he would have still understood it in something like these terms.  It is made up of a series of horizontal intervals, which I would call thirds and seconds and fourths, and so on.  There are also intervals that occur vertically, in the counterpoint, and you hear them when it is realised, as the two instruments coincide.  Again, one would need number in order to be able to deal with those, and the way in which counterpoint was studied was taught in those ways, the distinction between dissonance and consonance - how you treat the dissonant fourth is a vexed question passed down the centuries for composers.  But we would also need number to talk about the interesting play of durations, durational lengths, in this music, and of course we need number to talk about metre.  So it is a little mathematical or geometrical puzzle, as is all of the Musical Offering.

But what I have not yet really mentioned is that it is also a piece of music, and it works on an entirely musical level too.  The numbers, the geometry, is there for those who choose to see it, and it is an interesting question also whether you can hear those geometries.  We can make ourselves hear them, but do they work orally in the same way that they work visually?  But that is not necessary.  You do not need to know about those geometries in order to make sense of it musically, although, again, at an instinctive level, one recognises certain patterns of repetition and so on, which are intrinsically mathematical.

So we get the ancient adage: "Music is number made audible."  It is true at all sorts of levels, though it is also highly problematic, as I hope to come back to at the end of my talk.  Certainly, music being number made audible is a necessary, but, I would argue, it is not a sufficient condition for music to exist. 

I want to begin, by way of preface, with three very short examples of composers making numbers literally audible.  The first, you might perform for you yourself.  It is a little piece, for want of a better word, by an American composer who has been living in France for many years, Tom Johnson.  His piece entitled "Counting" comes from towards the end of the 20th Century.  Johnson calls himself a minimalist composer.  Indeed, he claims that he coined the term for that kind of music.  But the voice line of the music maintains a single pitch for each of the four singers, Soprano, Alto, Tenor and Bass. The words for the first five bars are as follows:

1 2 3 4 1 2 3   1 2 3 1 2 3   1 2 3 4   1 2 3   1 2 3 4

And it goes on like this - I am sure you get the idea.

In another version of this piece, Johnson had it in Russian, and it is very interesting to hear it performed in a language like that, in a language other than your own.  The structure is the same, whereby content equals form and form equals content, and there is nothing else.  The content and the numbers generate the form in a very obvious way.  But it becomes musical, through its repetitions, because there is a sound beyond the structure.  This is where the non-native language is important, because if I said it in English, we are so used to hearing those sounds and what those numbers represent, we hear them, as it were, as numbers, whereas doing it in a language like Russian, one focuses on the sonic quality of the words themselves and it makes a simple but obvious point about the kinds of repeating structures that one hears in music more generally.

The second example is from another minimal composer, Philip Glass - somebody else for whom number was important.  His opera "Einstein on the Beach", made in the 1970s, was the first of a trilogy of non-narrative and highly ritualistic pieces of multiple theatre produced for opera houses, but in no sense conventional operas.  Glass' music is very simple, highly repetitive and it uses very simple musical building blocks, arpeggios, which repeat over and over again, generating this sense of ritual or timelessness.  So Einstein on the Beach is, if you like, a meditation on the life and work of Einstein and what it stood for in the Twentieth Century.  At various points, we have so-called "knees" that punctuate the course of the work, where you will hear the choir singing numbers, literally singing numbers.  Glass says that originally these numbers were just there, if you like, as placeholders, until he found the appropriate text to put in, but he then realised later on that he did not need any text, that the numbers were what was needed.  Of course, Einstein's life was full of number, and in some senses, it is appropriate to the subject matter.  But I thought that it would be appropriate to mention this second example of another composer working literally with numbers.

What is of interest if you listen to this music by Philip Glass is that you have this progression, and of course, through repetition, it does not progress at all; it simply moves round and round in circles, just as the music does.  The music, at some level, is very familiar.  It is like what I would call a simple cadential progression that one would get at the end of a section of tonal music, but that does not go anywhere either.  It is as if Glass has put a huge magnifying glass on a moment in tonal music and that becomes the whole.  So the numbers, again here, take on a significance beyond the mere counting.

The final of these three introductory examples comes from John Cage, a little earlier, in 1939.  It is an example of what he calls square root form, which links back to the Tom Johnson example.  By this, he means that the micro structure is mapped onto the macro structure.  So each segment of the music lasts 16 bars, with little rhythmic motives which have a duration of four units, three, two, three, four, so it contracts and expands to make 16.  That then makes up one unit, and those units are also organised according to exactly the same ratios on the larger scale, so you can build up structures in that way, in a more complex way from the way that Johnson was doing it, but nonetheless, it is there.  The particular piece is called "construction in metal", presumably because all the sounds we hear are all metallic, of various kinds: metal bars being beaten, metallophones, vibraphones, glockenspiels etc.  Why is Cage dong this?  This is an interesting question.  In a way, by resorting to number and explicitly to number in this way, he is distancing himself from the European tradition.  He does not want to be dependent on old European forms and structures, but rather is trying to find a new kind of structure, appropriate to the new kind of music that he wants to write.  The end result, sonically, sounds like something entirely un-Western.  It is a very different kind of music, again, very timeless, static, concerned with rhythm rather than pitch.  If you listen to it, you will find that it is probably a rather alien sound world, but I think it makes the point very well about how it is something very different from this tradition that Cage was originally educated in.

This Cage piece is rather difficult to listen to if you are not used to hearing those kinds of things, but, rather like the Tom Johnson example, it throws the attention onto rhythm, onto these repeating patterns, and onto the very sounds that are being made as we try to grasp it and find something familiar.  But indeed, that is what we do when we listen even to Haydn.  We are evaluating it in terms of patterns of repetition and other such things that we recognise.  It has been there for so long that we do not think about it when we do it, but when we are confronted with something like this Cage piece, it forces us to listen to it afresh.

So, having gone through this brief introduction, I would like to approach the main focus of my lecture, which is music in the Twentieth Century.  Perhaps the subtitle should have been, "New Sensibilities, New Sounds, New Structures".  What were these new sensibilities, and how did composers respond and adapt to them?

Now, there are many ways of telling the story of Twentieth Century music.  There is not a simple line that runs through the Century.  This is never true of any history, but particularly of the Twentieth Century.  The path I have chosen to follow today is one essentially to do with the avant garde, that is, with those figures who have attempted to challenge norms, who have attempted to challenge tradition in the way that we saw that Cage doing.  Particularly fascinating is the way that many of these composers have turned to number as a way of pursuing and prosecuting this particular challenge. 

It is conventional to start the story of Twentieth Century music with Arnold Schoenberg.  It is by no means the only story of the Twentieth Century, but his own particular telling of that history has attracted a certain prominence.  He tells the story about the progressive abandonment of tonality, which he claims goes back to Bach, through Beethoven, and then we reach Wagner, where the tonal world is becoming more and more chromatic.  The bonds that are holding this tonal music together, this system that had controlled music for something like 250 years, was on the verge of collapse.  Eventually, there came a point, according to Schoenberg's story, around about the year 1907, where those bonds could no longer hold the music together.  The expressive needs of that music demanded that the bonds of tonality were broken, and so Schoenberg entered this murky world of atonality.  Through his discovery of the twelve-note method, he emancipated the dissonance; the rules that had governed music where dissonance had to resolve to consonance no longer held. 

These early years of the Twentieth Century were seen as years of crisis, the crisis of modernism; a crisis of faith, if you like, in what music was, what music could do, what it could say. So, in the years in the run-up to the First World War, we see this extraordinary break-up of the old order in certain kinds of music.  A work of Schoenberg's such as "Erwartung", is the most extraordinary piece, where you can hardly grasp what is going on.  It is a piece for theatre which consists of musical fragments with no tonal centres and even the text itself does not make a great deal of sense.  It is a bit like some kind of psychoanalytical nightmare.

What were composers to do with this?  After the First World War, things could never be the same again.  It had been an absolute tragedy for Europe.  People tried to pick up the pieces, start afresh, and you can see this happening in music in many interesting ways, trying to find new kinds of order to reflect the new age, a new post-War sensibility.  So, some composers, such as Stravinsky, looked back to the past, but distanced in a new way, through a kind of neoclassicism.  Other composers were asserting their national identity by looking to the properties of national music and absorbing that into their own music.  And, with rapid changes in audience and usage and technology, you see the rise of commercial music of all kinds, and the music responded to the demands of the new technology.

In Schoenberg's case, this notorious twelve-note method was one response to the "problem" of free atonality - how to give an order to seemingly chaotic or instinctively structured music, the idea of a kind of new beginning.  So Schoenberg turns to number, as seemingly something neutral, rather than something loaded.   "Nothing is added by my twelve-note method," Schoenberg said, "much is taken away."  It is a kind of composing with numbers, but what Schoenberg said is that you adopt the row but in every other respect, one composes as before.  So once the distance has been emancipated, how do we proceed?  Schoenberg was not the only one to come up with this system, but his was the one that received the most attention.  The twelve-note row is simply an ordering of the twelve notes of the chromatic scale, a particular ordering of those twelve notes, or perhaps, more importantly, the intervals between them.  The intervals that the composer chooses gives that row a very particular character.  Different composers did this in different ways.  This is the material then that the composer works with.

The first thing that the composer can do is to translate that, in order words, transpose the row onto all other eleven notes of the chromatic scale - that gives you twelve versions of the row to work with.  You can also turn it upside down, as people such as Bach had been doing long before then.  You can invert it, so, when an interval goes up by a third, in the prime version, in the inverted version, it goes down by a third, so you can invert the row and then you can transpose that onto the other eleven notes - that gives you 24 forms for the row.  You can then also retrograde it, so in the prime form you can go from the end to the beginning, and you transpose that, and similarly with the inverted form, retrograde that.  So you end up with 48 different forms of the row that are available to the composer to work with, a variety of material, but essentially all the same thing.  There is here a kind of variety within unity.  What Schoenberg said he was doing, that this guaranteed, as it were, a kind of unity and comprehensibility, which are very old values, but in a new context.

It is very interesting to see what happened after the Second World War after the model that Schoenberg set up.  The Second World War, in some respects, was viewed in Central Europe as even more of a cataclysm than the First World War, and many composers were writing using a German phrase after the Second World War, talking about "Nullstunde", Zero Hour, the idea, after the horrors of what had happened, of trying to begin again, and many artists were looking at ways of rejecting the past entirely.  The past had proved to be problematic.  The past was tainted and coloured.  Let us break off from that and see if we can find entirely new forms, structures, sounds, to express a new age. 

So we find a composer such as Karlheinz Stockhausen, a German and therefore deeply implicated in what had been happening, rejected the past entirely and started to explore one way through newly emerging possibilities of electronics, inventing entirely new sounds that had not been heard before.  Number was crucial to this of course, playing with sine-tone generators, looking a sine patterns on oscilloscopes, looking at the way in which these wave forms interacted with one another to produce new kinds of sounds, working with tape, measuring lengths of tape in order to combine sounds, and so on, in all kinds of interesting ways.

Another figure that I want to look at here, an exact contemporary, is a French composer, Pierre Boulez.  He talked about blowing up the opera houses, the idea of getting rid of tradition.  Indeed, it was only a couple of years ago that he, now in his eighties, was arrested by a rather vigilant Swiss policeman as he went over the Swiss border.  This was just after the events of 9/11 but it shows how  well-educated the Swiss police force must be, because they had obviously been reading this comment of Boulez about blowing up the opera houses, so they arrested him as a potential terrorist!  They thought he was coming into Switzerland to blow up the opera houses, so he spent a night in a prison cell.  Anyway, that is by the by.

Boulez also said, after the War, "All non-serial composers are useless."  Indeed, even Schoenberg was useless.  There was a famous essay he wrote in 1951 called 'schoenberg est mort!', 'schoenberg is dead!'.  In fact, Schoenberg had died that same year, but the essay was about how Schoenberg had sold out, as far as Boulez was concerned.  Schoenberg had invented this new structure, but in every other respect, as we have just seen, Schoenberg was writing old music - new wine in old bottles, if you like.  So what Boulez attempted to do was to serialise every parameter of the music. 

So, he takes a twelve-note row, and produces these two matrices.  These are grids wherein the twelve-notes of the first row are re-arranged in two different ways, so that there is a block of twelve rows of twelve notes.  The two matrices are the prime form and the retrograde form ('P' Matrix) and the inverted form and the retrograde inverted form ('I' Matrix).  In order to do this he converts the notes into numbers from one to twelve.  So every time you see number one, it refers to E Flat, every time you see number two, it refers to D Natural, and so on.  So we have these two matrices that he can then work with.  So, if you read the numbers, it will give you various forms of the pitches of the row.  But, once you have turned these into numbers, you can do all sorts of other exciting things with them.  The numbers can also become durational values.  So, if you take your basic value, as he does in the start of this piece, the demi-semi quaver, a very small note value, that is number one; number two is two demi-semi quavers, or a semi-quaver, and so on, all the way through to number twelve, which is twelve demi-semi quavers or a dotted crochet.  So you can control duration, durational values as well from that initial row, but that too is not enough for Boulez.  He comes up with other rows.  There is a dynamics row, from pppp to ffff.  Therefore, notionally, we have dynamics, and, similarly, a mode of attack row, though he can only come up with ten of them rather than twelve - normal, and then we have accents and staccatos and legatos and all sorts of other things.  So even form then becomes generated by the initial row.  Certain things are left to the composer - register - but it is an attempt to try and rationalise, to control from the row every single parameter derived from the initial row using numbers, which is an extraordinary idea.

This music of Boulez was an experiment, although the end result is music that one might almost say sounds quite random, despite the complete control.  If you listen to it alongside the works of John Cage, produced at exactly the same time in America using chance procedures, it is quite hard sometimes to tell the difference between them, so the circle meets at the bottom.

Another interesting composer was Conlon Nancarrow, an American who spent most of his life working in Mexico City.  He devoted his entire life to writing for the pianola, as we would call it, or player piano, as it is known in America.  This meant that he composed onto piano rolls, so that he could use his ruler and his hole punch to make music exactly as he wanted it to, where you did not have the intervention of humans who could not perform the music exactly as he wanted it.  This allowed him to imagine music of extraordinary proportions and complexities that no human could play, but that a machine could.  Nowadays of course, with computers, these things are very easy to do, but at the time, the player piano was the way forward.

His fifth study for player piano is a wonderful place to start to try and grasp the complexity of his pieces.  It starts with a relatively sparse musical texture which you can hear very clearly defined little musical motives, which are introduced one by one, and then are layered one on top of the other.  The distances between the appearance of these motives are carefully calculated to get closer and closer.  They are all moving at different speeds, until we end up with the most extraordinary complexity, where things cannot get denser or closer, and, at that point, the composer has no choice but to just stop, because there was nowhere else to go.  But everything was very carefully calculated.  Each line has a different, very complex, time signature, to give a sense of how these things are all moving together.  It is also canonical, just like Bach, very carefully calculated.  You can see this very clearly in the score for this music, which was, interestingly, of no use, since no human can perform it, but for the likes of me to study.

It is also very interesting to study the original hand-written score for compositions by Birtwistle, a man who composes very much with numbers even though you do not obviously hear it in the music itself.  In a melancholic work such as 'Night's Black Bird', you do not hear these numbers at all, but this is how the composer works.  The numbers, for him, do all sorts of what he would call low level work - choosing pitches, ordering pitches, ordering durations, groups of notes, even deciding instrumentation at certain points, deciding on formal proportions and so on.  There was a time when he even worked with sheets of random numbers.  So the harmonic world is very clearly controlled, but the moment to moment is left to numbers to decide.  It frees the space in the composer's mind for far more important decisions, to do with expressive intent, theatrical effect, large scale design, and so on.  What he does with these numbers is generate a harmonic world that is framed, is consistent, yet unpredictable moment to moment, very much in tune with the world as he sees it.  One might say, given this piece, it is a kind of melancholia of technology; number for him represents, as it were, a kind of technology.

This is not so far removed from another composer, Olivier Messiaen, who is another composer fascinated by number.  In the pages of his posthumously published treatise in seven volumes, which cover most of his creative life, we see this man fascinated by number, by geometry - non-retrogradable rhythms, so palindromic forms of durations and playing with meter, additive rhythmic processes, taking an idea and then adding something and adding something to build up larger structures.  In terms of pitch, what he called modes of limited transposition, fascinated by the properties of particular modes, scales, if you like, when transposed, that could only operate in certain ways, where you have repeating intervals, which we would use numbers to describe.  The whole tone scale, absolutely symmetrical.  The octatonic scale, alternating tones and semi-tones, another symmetrical scale that is limited in terms of its transpositions.  He was fascinated by prime numbers, also interested in number symbolism, a deeply devout man, Roman Catholic throughout his life, expressed his faith through his music in all sorts of fascinating ways, and number plays its part.  One, the number of the divine unity; three, the number of the holy trinity; seven, the number of creation, made up of three and four, male and female, giving us perfection, and so on.  These things are actually composed into Messiaen's music, in the same way that he used all kinds of other materials, found materials if you like - plain chant, Indian ragas, birdsong - these things are all in Messiaen's music.

A particular example from Messiaen is a process, a fascination with symmetry, that he calls interversion - it is a way of generating rhythmic durational material.  To take a very simple example, we can begin with a rhythmic sequence: semi-quaver, quaver, dotted quaver - so we are adding a semi-quaver each time - which are then numbered one, two, three.  An interversion is then where you go into the middle and then move out to the extremities, so one, two, three, and then you take the middle number, two, and then one and then three, to produce a different pattern, which ends up as quaver, semi-quaver, dotted quaver.  You renumber it one, two, three, and then you go through the whole process again.  When you have only got three elements, of course, you very quickly come back to the beginning, but the more elements you have, the more complex it becomes.  For instance, with a variation with four elements, the same process can be seen to be at work, starting with the numbers one, two, three, four.  To go from the middle gives us three, two, four, one.  Renumber it one, two, three, four, go through the whole process again, and, very quickly, you are generating all kinds of variants, if you like, on the original form.

When you have up to twelve, the process produces a lot of material which will take a long time for the sequence to work out.  You have to listen to it on a kind of celestial scale - which is precisely what Messiaen is doing.  There is a very late piece he wrote called Des Canyons aux Etoiles, From the Canyons to the Stars - that is the kind of scale that Messiaen is working on.  But you also hear, on a smaller scale, what he calls little 'personage rhythmic', rhythmical personalities, three little ideas: one that stays the same, which has a very clear durational identity; another which contracts and then expands; and a third one which expands and then contracts. You hear these three ideas all moving in relation to one another, embedded in musical structures, and so on.  Number, then, helps Messiaen express the magnitude of the universe, the eternal sameness of God: number remains the same, God remains the same, and that is why it is so important to him.

Eliot Carter, who celebrated his 101st birthday last Friday, is also another composer fascinated by number, by symmetries, by palindromes, by inversions, using prime numbers in various ways, working with pitch class sets, turning pitches into numbers, durational systems, rhythmic ratios.  It is Eliot Carter who coined the phrase 'metric modulation', which is actually something of a misnomer because it is more tempo modulation, calculating accelerandi and deccelerandi very carefully.

We might start in two-four, regular two-four, at a metronome mark of crotchet equals 96, and he wants the music to speed up.  So, first of all, he turns the quavers into triplet quavers, and then he turns them semi-quavers, then into quintuplet semi-quavers, and then he re-notates that.  The music does not change speed, but he turns it from quintuplets in two-four to groups of five in 10-16, and he makes the point here that now the crotchet has become the crotchet with an extra semi-quaver on, still at 96.  So we move through all the way there, and then, at the last minute, then he changes the metrical marking again, back into 3-4, staying the same, but now we have speeded up so the crotchet is now 120, whereas it was 96 at the start, this is a very carefully controlled modulation of tempo in that way.

The last example I want to give is the work of a former Gresham Professor of Music, Iannis Xenakis, a Greek composer, who was fascinated by drama, architecture, philosophy and mathematics.  He studied initially as an engineer, which is how he encountered the famous modernist architect, Le Corbusier, and they collaborated together on the famous Philips Pavilion at the 1958 World Exposition in the World Fair in Brussels.

Xenakis talks about a basic structure of conoids and hyperbolic paraboloids, curved surfaces made from straight lines.  It is no longer there, but it was an extraordinary space, and, incidentally, had a very early piece of electronic music written for inside the space, the 'Poeme Electronique' by Edgard Varese, played from something like 400 speakers inside this space, and people moved around it and so on.

Xenakis said, "I discovered on coming into contact with Le Corbusier that the problem of architecture, as he formulated it, was the same as I encountered in music."  So, for Xenakis, architecture represented, if you like, the idea of proportion in space, and music represented the idea of proportion in time.   This too is not a new idea.  It has a very long pedigree, going right back.  If you think of a work that Dufay wrote for the consecration of Florence Cathedral, Motet Nuper Rosarum Flores, the proportions of that motet are the same as the proportions of Brunelleschi's dome in Florence Cathedral, so the space and the time as it were, the geometries of the two match one another.  It is the idea of number geometry made audible in an extraordinary kind of way.

The last sound example I want to mention, is his first acknowledged work, "Metastasis Transformations."  In the score you will see the transfer, if you like, from architecture to musical structure, curved surfaces, curved musical surfaces, in this case, generated from straight lines.  In this case, they are glissandi on the stringed instruments. You know, a stringed instrument can fill all the gaps in a way that a piano cannot.  So, again, the idea of a projection of architecture into music, and now what Xenakis is interested in is not tunes, melodies, but rather using these shapes to construct these new sound masses, these blocks of sound, exploring new kinds of timbre, so that we are not aware of the individual sounds, but we are aware of the whole that these individual voices make up.  He later became fascinated in what he talked about as stochastic music - he played with game theory, set theory, Fibonacci numbers, all sorts of things, to generate his music.  All of this mathematics ends up being music.

I have given you all sorts of different examples, a number of different ways in which composers in the Twentieth Century, and in the 21stCentury, have worked with number.  It is clear that many composers have been obsessed with number, but as I said at the start, this much is nothing new.  All composers engage with number and geometry in various and different ways.  But what I have tried to show here is how certain avant garde figures have engaged with mathematics at various levels, for various reasons: number as an engine of the new, if you like, as a way of pushing the boundaries, pushing music in different directions, as a means of engaging with the technology of the modern world, as they saw it; number used as a means of avoiding direct connection with the immediate past, to distance themselves from those old forms and structures and modes of expression which were seen to be outdated and, as I said earlier, tainted in some way; number also used for its symbolic power, as something that was seen in various ways as either being neutral or eternal, as we saw in Messiaen; and number as a reflection of the concerns of the modern world, the complexity of the modern world.  Art does many things, but one of the things it does is hold up a mirror to the world; it shows us the way the world is, or at least it shows us the way in which artists see the world.

Nevertheless, for the most part, the links with the past seep through, revealing that structure has always been a concern of composers.  Xenakis talks about ?the effort to make art while geometrising, that is, by giving it a reasoned support less perishable than the impulse of the moment, and hence more serious, more worthy of the fierce fight which the human intelligence wages in all other domains."  That said, many composers, even in the Twentieth Century, have not explicitly or conspicuously, or perhaps even consciously, worked with number, but they cannot help it.  Despite themselves, they still do.  While music may well be "number made audible", as I mentioned earlier, in the end, I would argue it is more than mere number. 

The music of the Twentieth Century does what music has always done: it explores the sensibilities of its age in sound, holding a mirror to the world, but also, more than that, it is not just a record, it also constructs a reality for us, in order that we can make sense of the world, just as mathematics does, as a way of ordering the world, of making sense of the world out there.  Music, of course, expresses our intellectual curiosity.  Music is as good at doing that as anything else is, but it does other things too.  It expresses our passions, and our feelings, our joys, and our sorrows.  In short, music, like number, explores, in sound, what it means to be human.

©Jonathon Cross, Gresham College 2010