Chaos Theory

with Chris Maslanka and Hattie Hayridge as Hell-O

(Opening Chord plays)

"One of the many rumours about Mozart is that he daubed his walls as a child with sums."

(Music continues)

"Maths is maths,and music is music,they inhabit their own worlds."

(Music continues)

MP3 58K

Chris Maslanka : Two angles on the connection between doing maths and writing music.Over the next half hour,I'll be hearing lots more and asking whether this really means that music is mathematical.A quest in which I'll be aided by the advanced computer android Hell zero.

Hattie Hayridge : Hell - O

Chris Maslanka : Er,er hello?

Hattie Hayridge : But you can call me Hell.

Chris Maslanka : Now at the simplest level,all music as a mathematical basis.As son as you go plink, plonk, plink,you can ask "How many notes are there? How do they reach my ear? How long do they last? What pitch are they?" all of which implies number,structure,proportion,the very stuff of maths.

Hattie Hayridge : The ancient Greeks studied maths and music as related disciplines,and Pythagoras said that whether two notes sounded okay together had to do with simple ratios,doubling the length of a string,gives you an octave.

(Two notes play and octave apart)

And one and a half times the length,gives you a fifth.

(Two notes play and fifth apart)

Chris Maslanka : Now,a proper walking encyclopaedia! Now according to this manual you have several domestic functions.Look,go and get me a cup of tea.

Hattie Hayridge : Oh,typical male carbon-based life form.You've got an IQ of 2001 and he uses me to make the tea!

Chris Maslanka : Oh and a cake too please.Right,let's start with a couple of really famous composers.What about JS Bach? Was he interested in maths?

Roderick Swanston : Bach was fascinated with proportions in music,and with the construction of music,and he spoke and wrote for an audience,who understood music in this way.But I think Bach has another aspect with numbers,and that is that Bach clearly thought of numbers often as symbolic.This kind of number symbolism appears in all sorts of quantities and qualities all over the place, particularly in religious music.He might for instance take the Christian symbol of the trinity,you know God,the father,God the son,God the holy ghost,or for instance when he came to write,while he was in Vymar,a series of small Chorale Preludes called the Auglobuchline,when he came to the one which is based on the ten commandments,the chorale Die Sinde Halgen Zengebucht,it has a tune which starts with five repeated notes: bom,bom,bom,bom,bom,and the accompaniment to that,is a much shorter version of that bom,bom,bom,bom,bom,which appears ten times.Now,given the fact that each time,especially at the beginning of the piece,he isolates them,one from another,you can't help thinking that Bach set for himself,the task of trying to make sure the little diminution of the main theme, would have been included ten times,and thus is a private reference to he fact that it is the Ten Commandments.

(Bach plays)

Roderick Swanston : But I think that Bach would have not wished to be a sort of cryptographer,in many,many cases,this was obvious and clear,and one has to remember spoke to an audience,whose approach to music,wasn't really the same as ours.It may have had elements that we have.People talked about "expressiveness" in the 18th century,and the knew the emotional power of music,but people recognised that music is not simply a succession of emotions,without there being constructed in some kind of way,related to numbers and proportions,there is no emotion,and there have been times when a composer has felt that the pleasure comes from the proportion of the way the piece is put together,and also if he can then incorporate something in a religious work,which as it were,tied up with some of the numerical symbols of the religion,then this was something that people would expect to see there.

Chris Maslanka : But what about the secular works?

Roderick Swanston : Well numerology in the secular works,is I think,less prevalent.But there are examples clearly,where Bach set himself numbers as a particular kind of challenge.For instance,in the Goldberg Variations, there are 30 variations,the third of every variation is a canon,the first canon is at the unison,the second canon is at a seconds distance,the third canon is at a thirds distance,till eventually he gets a canon at the ninth,which of course,as you're doing with your calculator here,gives you variation twenty-seven,and then the thirtieth variation,he has a quadlibet,which is a combination of four tunes but has nothing to do with that kind of canon at all.

(Bach plays)

Chris Maslanka : Professor Roderick Swanston of the Royal College of Music,who told us at the start of the programme about Mozart's mathematical grafitti.Mozart was into number symbolism too.

(Mozart plays)

He used three not to symbolise the holy trinity,like Bach.

(Mozart continues)

Mozart was a Freemason,and three was a symbolic number for their rituals.
(Mozart continues)

Mozart's opera,"The Magic Flute" is peppered with references like those three chords in the key of E flat.The key having,yes,that's right,three flats.But Mozart was a gambling man too,so I bet he was also interested in randomness and chance.

Roderick Swanston : Very interestingly in the 18th century,Mozart and CP Bach and Haydn and many others remained interested in the proportions and numbers of pieces,and so there's a little tiny sort of sub species that grew up,of music which was composed by the throw of dice [Ref:Protext Files: PT3:Music Machine.txt],they called it Ars Combinatoria.First of all before you started,as it were,the game board would have a fixed harmonic pattern and a fixed number of bars.So you might say for instance,"For the first part of my piece,I will have six bars,and it will have the following chords,one in each bar,and then I will have a series of notes,I might go C,D,E,F",and any combination of those four notes could be determined by the throw of dice.Then I would need to decide on the rhythm,and I could do this bar by bar.Now I can actually set up the combinations,which will actually yield a very nice piece.This was done for a variety of reasons.One of them,as a sort of private game,was to allow a dilettante to compose a piece of his own.It's very much like,you know,using your mini keyboard,where you only have to play the tune,and you've got a machine which gives you the bass and the rhythm and you can say,"I want to be Latin American".You know,you might well buy a book where you could,as it were write a different dance each evening,and what ever combination you used would yield a reasonable piece.All composers said,"Nothing of this ever replaces the same amount of time of inspiration".On the other hand,they didn't very snootily say,"This is of no interest at all".I mean,Haydn for instance,says it stimulated elementary students to use their imagination.They often saw combinations of notes,and ways of presenting things and variations of a particular little group of notes,which they wouldn't otherwise have thought of,which were thrown up by the dice.

(Dice are thrown)
"You seem to have a five"
(Harpsichord plays)

Chris Maslanka : Roderick Swanston revealing that automatic writing was around long before the surrealists.Hey! What are you doing with my ruler?

Hattie Hayridge : If I snap it at the right point.....
(Sound effect)

Chris Maslanka : Oi!

Hattie Hayridge : .....then the ratio of the long bit to the thing you started from....

Chris Maslanka : My ruler!

Hattie Hayridge : ....equals the ratio of the short bit to the long bit.It's called the Golden Section [Ref:[Media 1] Focus 2;[Science 1] Front 4;[Maths 4] "The Power of One":Protext Files:PT4:Analysis4.txt;SN2.txt].It's just under 5/8.

Chris Maslanka : I've always wanted two funny length rulers.

Hattie Hayridge : Course you have! It's supposed to be an aesthetically pleasing ratio for you humans,just right for windows,pictures,architecture.

Chris Maslanka : Oh,go bake a cake! I want to hear Emma Dillon talking about the Golden Section and medieval music.

(Medieval vocal chorus plays)

Emma Dillon : Well unfortunately we've got no proof,exterior to the music itself.In other words no medieval music lover sat down,and wrote admiringly about having herd a medieval composer engineer a fabulous game with the Golden Section.But having said that,scholars have looked at a vast array of music from the middle ages,and have discovered a number of interesting coincidences,musically speaking which occur at what we call the Golden Section of a piece of music.I'm personally slightly dubious about trying to go to a piece with a calculator and figure out where Golden Sections occur.It's very easy to cook the books,when it comes to maths and music and so this is giving the Golden Section a sort of bad name in musicology. I think it's much more interesting,in fact to start with a piece of music and then you explore how it works and look at where interesting moments of musical activity occur.

Chris Maslanka : But would musicians naturally have had mathematics as part of there.....? I mean would maths have been a related discipline in those days?

Emma Dillon : Oh absolutely.Medieval composers had a very diverse and eclectic training,across the arts so a composer would have been well versed in mathematics,arithmetic,and so on.

Chris Maslanka : So they weren't just doing this intuitively? There was an element of self consciousness in putting maths into the music?

Emma Dillon : Certainly.Perhaps where maths really does meet music would be in theology and philosophy, because the staple texts of the times,such as Plato,Augustine and Boethius,were all considering the nature of the universe,as a series of divine ratios and proportions,and then it's just a small step,into music,to argue that the ratios and proportions in musical structures could also reflect a similar kind of divine order.

(Medieval vocal chorus plays)

The motet by Ghiam du Fye,the famous piece written for the consecration of Florence cathedral,New Parisar un Flores,is built on a proportion 6 to 4 to 2 to 3.For a long time,scholars thought that the proportional relationships mirrored those of Florence cathedral.Unfortunately it's now been proven that the original theory is fatally flawed,owing to the fact that its inventor,simply counted out the spaces of the cathedral incorrectly.Du Fye's piece in no way matches the proportions of Florence cathedral.

Chris Maslanka : Oh no!

Emma Dillon : However (sniggers),what has recently come to light,is that in fact those proportions 6 to 4 to 2 to 3 perfectly mirror the proportions of another,but much more venerable temple,the temple of Solomon,described in the old testament book of kings [I think we're in Graham Hancock, connect up anything and everything territory now -LB],to be 60 cubits long,20 cubits wide,30 cubits high,and the house of prayer,equivalent to the knave of church,was 40 cubits long.
[By the same reckoning one could make the association with any object that has these proportions, there is no reason to assume a link,it is when the maths is inherently connected with the music in terms of pattern that the association makes sense -LB]
The temple of Solomon was a fabulous resource for medieval writers.It was constantly being used in sermons,and in wall paintings as kind of ideal temple.So I think here is a case when you can be very sure that these are the symbolic numbers with which du Fye was playing.

Chris Maslanka : So the composer is probably aware of the mathematics,but would the normal medieval listener,whatever that means,hear this technique,or would it just be a private thing for the composer to play with?

Emma Dillon : Well it's obviously and esoteric game,which the composer is indulging in.But medieval audiences for music were very diverse,and there is evidence,particularly in the 14th century that those who gathered to listen to this kind of music were incredibly educated and learned,and so these are games,which,if you like speak to a sort of secret society of listeners.But as for,you know your common man in the street,would they be able to hear these games? Well I think it's very unlikely. I'd like to make an analogy here with architecture.Art historians have shown us often hidden out of sight in these dark recesses of gothic cathedrals,we find more and more examples of exquisite carved works which are left there as a private act of devotion by anonymous stone masons of the middle ages.You have to get on ladders and climb up to find these things.But I think it works as a rather brilliant analogy to the kinds of games that we've been looking at in this music.

Chris Maslanka : Dr Emma Dillon,of Bristol University.

( "Jupiter" from Holst's "Planets" plays)

The planets have always fascinated people.But it wasn't until the early 17th century that German astronomer Johannes Kepler explained how they moved in terms of mathematical laws.Kepler stood at the cross roads between the old mystical ways of thinking and the modern scientific way ,and he even imagined the planets humming as they swept round the sun.The interval between a planets notes,depended on the ratio of it's closest and furthest distances from the sun.So for Kepler the divine ratios of the universe were literally expressed in music.There was just one problem,Uranus,Neptune and Pluto hadn't been discovered,and so he didn't know about them.Composer David Bedford [Mike Oldfield's string arranger -LB] does,and he's brought the concept of the Kepler chord bang up to date.

David Bedford : What I did was I got hold of the Royal Observatory at Greenwich and they did the calculations for me,pretending to be Kepler,if you see what I mean,and they gave me the intervals for those three planets,so I've actually got my nine interval chord.After being an astronomer for a bit,I turned into a composer,and I said, "Well,I'm going to use this chord".So I rearranged all the intervals into nice sounding chord,you know,things that I like as a composer [Doesn't that defeat the object?- LB].

Chris Maslanka : Whether it's good music or not? Whether it's what you want to make,is more important than the formula underneath it?

David Bedford : Yes,in fact I've just taken each planets interval and put it next to the next planet's interval but in a way that makes a good chord.

(Chord plays and adds notes as David calls the planet names)

Mercury,Jupiter,Saturn,Mars,Uranus,Earth,Pluto,Venus stroke Neptune.

Chris Maslanka : What d'you mean "Venus stroke Neptune"? (sniggers)

David Bedford : Well (sniggers),I mean that my keyboard won't do quarter tones at the moment and when I finally use the chord,which will be with a choir,then I'll be able to use that properly.

[In fact some of the modern instruments can cope with microtonal scaling,rather than use the chromatic conventional scale,lending themselves to this sort of work -LB]

The reason I like it,is that,is you just pick out say three notes at a time,you can have a major chord,a minor chord,a diminished chord,all sorts of different chords within it.Discordant ones,not so discordant ones.So it's a good base for musical material.

Chris Maslanka : And how do you use such a chord in making music?

David Bedford : Well,I have used it already,last summer,in August,in Dartington,the Dartington International Summer School of Music,I devised a project called the PRS Dartington eclipse project,and this chord formed the basis.The music stopped before the eclipse,because I knew nobody would want to listen to music.So it was a sort of welcome to the eclipse.

Chris Maslanka : But it must be fair to say,because it is to a certain extent,arbitrary,would it had mattered how you arrived at that chord,d'you think? Does it matter that it came from Kepler?

David Bedford : Well it matters to me......

Chris Maslanka : Yes.

David Bedford : ....because I know I've used the correct intervals for the planets.So it's got a meaning for me,because a lot of my music in the past has been based on astronomical or science fiction ideas,and that's going to be the next use of this chord.I've received a commission from the (indistinct) Festival Chorus,for what they're describing as their "2001 concert",where they're going to do music from the film "2001" [[Science 1] Behind 3] in 2001,and I'm basing my piece on a short story by Arthur C Clarke,where people have gone out into the solar system,and so this chord seems ideal for that.

(Chord plays)

In theory I could have created this chord myself,but I like the feeling that somebody once thought that this is the sound the planets make as they go round.

Chris Maslanka : David Bedford and his amazing Kepler chord,expressing the harmonious order of the solar system.

Hattie Hayridge : Think of a number.

Chris Maslanka : Mmmm,one.

Hattie Hayridge : And another number.

Chris Maslanka : Er,one again.

Hattie Hayridge : Oh,you're so predictable.So one and one is two,one and two is three,two and three is five,three and five is eight,and so on.It's called the Fibonacci sequence ,1,1,2,3,5,8,13,21,34,55,89.......

Chris Maslanka : Okay,okay,I get the picture.

Hattie Hayridge : Yeah,but the ratio of any pair of numbers in this series is close to the Golden Section,and the higher you go,the closer it gets.So 5/8 isn't bad,but 55/89 is even better.I could go on forever.

Chris Maslanka : I'm sure! But that won't bake any cakes.Ahhh silicon skivvy! Now notes and bars come as whole numbers,so it's useful to use whole number ratios like 8 to 13 or 55 to 89,if you want to get the Golden Section into your music.The pianist Roy Howard,has found it among the bar lines of Debussy.

Roy Howard : One will find in his Sixth Image for piano,three out of these six are proportioned on a start and end quietly and have their main climax,absolutely smack on the point of Golden Section.

Chris Maslanka : So that is a very gross architectural feature.

Roy Howard : Mmmm.

Chris Maslanka : Is there anything happening on a smaller scale in the form?

Roy Howard : Most definitely,and there has to be,because over a 5 minute piece,it's hard to make sense of a big architecture like that because it's not like a picture,which your eye scans in a tiny fraction of a second.So what happens is that there are smaller rhythms based on similar proportions building the piece up,marking the main turning points of the pieces.

(Debussy's "Reflects D'en Leau" plays)

In the first piano image,Reflects D'en Leau [[Science 1] Front 4] -Reflections in the water,this piece is a beautiful decorative surface.Half of the time you don't really know what key the music's in and that's Debussy's design,but there are certain anchoring points where he very clearly marks out a particular key,and the sequence of keys that is marked out is this,it could be Mozart.

(Roy plays chords)

Now I'm playing in a few seconds what happens over the course of 5 minutes.From the first of these (Roy punctuates his speech with chords) to the next one,you can measure in the score that there are 34 bars,one of the Fibonacci numbers.From there to the next of these chords,that follows after 21 bars,which is going down the Fibonacci sequence,by this time we're almost knowing what to expect. You measure another 13 bars further and you get to this chord,and then you measure 8 bars and that's exactly where the music finally settles on. So that tonal skeleton is 34 plus 21 plus 13 plus 8 bars.It's an extremely strong pattern.

Chris Maslanka : Do you think it's possible for people to keep that in their heads when they hear the piece of music? That they actually perceive that?

Roy Howard : Yes as long as you say "Perceive in an intuitive sense".

Chris Maslanka : Mmmm.

Roy Howard : I don't think the composers expected people to be counting the bars mentally,as they listened through a piece,but it's astonishing of course,what the sub conscious mind will do [Ref: Video N5: Music and the Mind] .In the case of Debussy,first of all he wanted his music to sound right without it being obvious why,and the other thing was of course,he certainly knew about Golden Sections and these numbers,because they were being written about in magazines he read,when he was young.We know he read these magazines,the articles are still there,and they were written by people, mathematicians,very interested in the arts,this was all the rage in Paris in the 1880s and 1890s,when he was learning his craft.

Chris Maslanka : Bartok used Fibonacci numbers too,for example in the music for strings, percussion and celeste.There's even a xylophone rhythm which goes 1,2,3,5,8,5,3,2,1 (Xylophone plays),and Roy Howard's got a rather more subtle example from the first movement of the piece.

Roy Howard : If you measure the length of the piece in the score,the main climax is centred after 55 bars,but the piece doesn't end after 89 bars,it ends after 88,so one looks back to see why this might be, and then you discover that the real quieter turning point of the piece tonally comes after 44 bars.So Bartok's playing two sequences off against each other.There are also things after 33 bars and 77 bars,so he's going symmetrically,with multiples of 11 bars or multiples of 22 bars sometimes, that's one aspect of the music,and then the overall structure is following a large Golden Section shaped wave,but he's letting the Golden Section proportions give or take a little bit on the large scale to accommodate the symmetrical sequence.

Chris Maslanka : It's nice to know composers aren't above fiddling with the figures a bit,when they have other musical points to make.That was Roy Howard.

(Orchestral piece plays)

Now we've heard how proportion and whole numbers fit into music,but if music is really the playground of maths,where are the modern concepts,such as erm.......?

Hattie Hayridge : Chaos,chaos,your flat is full of chaos.

Chris Maslanka : Mmmm,that's an interesting theory.

Hattie Hayridge : Yes chaos theory,it's the maths which describe situations where a tiny change at the start of a process can cause a huge change later,whether a butterfly flaps its wings in Brazil or not determines whether there is a hurricane here in Bradford.

(Sound of falling books)

Chris Maslanka : Bloomin' Brazilian butterflies.Before you pick all those books up,remind me about Fractals.

Hattie Hayridge : A site that looks the same on the large scale as on the small scale.Have you been to Baker Street tube station?

Chris Maslanka : Not lately why?

Hattie Hayridge : Well,there's this picture of Sherlock Holmes,as you get closer you see it's made up of lot's of little Sherlock Holmes's,all the same as the original.

Chris Maslanka : Yes?

Hattie Hayridge : And as you zoom in one of those,you start to see it's also made up little Sherlock Holmes's ...'s,'s,'s,'s,'s,and so on.That's Fractal.

Chris Maslanka : Oh,I can't see anyone building that into a piece of music.

Hattie Hayridge : Actually,Bach and Beethoven without knowing did it.(sings)Debussy,Nillson and Sibelius did it....

Chris Maslanka : Okay,spare me the ditty! I need someone who's alive and still does it,such as the composer Peter Paul Nash.

Peter Paul Nash : It's to do with an extreme fluidity,a kind of making the music like life.The first composer that I know of who came up with these ideas,formalising the musical term,was early as 1959,which is before the word Fractal was coined,and before Chaos Theory was formalised.

(Piece plays)

That was the Danish composer Per Nergal,with early versions of what he later called his "Infinity Series",and if I could name just four principles,in which,to me it seems,these principles are applied both in music and in maths at the same time,First of these is infinity,this is the notion that a piece of music using certain particular principles can last forever [Ref: Video A4 RI Lecture;A19 Horizon].

Chris Maslanka : You mean as a series which never really repeats,but which carries on forever?

Peter Paul Nash : Never repeats but could carry on forever.If you wished it to (sniggers),interesting.

Chris Maslanka : It would make concerts very difficult.

Peter Paul Nash : Far more interesting is to do pieces of music which have the implication that there is an infinite before....

Chris Maslanka : Mmmm.

Peter Paul Nash : ...and an infinite after.Just like nowadays Fractal mathematics are used in say making pictures go down telephone wires,by saying this is a little bit of the picture,from the little bit you can read the total [Ref: Video BB17:OU "The True Geometry of Nature";AB1:Equinox "Chaos"].This is a way of writing music which says "From what you hear,you can imagine a far bigger universe than the one I'm actually giving you".The second principle,I would call it weaving organic shapes [Ref: Video A5;H1;K3 {Bill Latham};Media 1: *.jpg],a kind of fluctuating feel to the lines of shapes you're using.This might be in melodic shapes up and down,both Sibelius and Nillson,early in the century referred to "organic melody,organic rhythm" sort of flowing.The third principle,which is directly connected,is what in Fractal mathematics is called,"hierarchy",in other words,what happens on a small scale is replicated in one degree or another,on a larger scale,it's also called "self similarity".In the pre Fractal days of my composing,form was always difficult,its, "Oh my goodness,how do I make a form out of these bits?".But there's all of a sudden,a page in my composing, turned,which meant that it was just as much fun,making a whole form as it was making all the different bits of it. I could think one moment about how the form was going,the next about certain details,also about the colour of it,also about the emotional content of it,because of this ability of the small to reflect the large.What's interesting about this way of composing is the way the memory behaves,because the whole point about this is that the ear does not necessarily consciously pick up what it will remember hearing later.In fact it has heard them,and they can be brought back in a way that can be presented to the memory.This is a way of using the listeners memory,so that the listener becomes in a sense,much more of an active participator in the musical process.

Chris Maslanka : And what was the fourth principle?

Peter Paul Nash : The fourth principle is that very small gear changes can have a huge effect (sniggers),you just throw a little switch,you just do something and the course changes.I find that very small irregularities,can throw the system in a completely different way than you thought that it might do.

Chris Maslanka : So this is very reminiscent of that thing in Chaos Theory....

Peter Paul Nash : Yes,the butterfly flaps its wings one side of the Earth,and there's an earthquake on the other [An earthquake??? -LB] (laughs).Yes but as well as that,you hear the psychological strength of these changes easily.You know,you can feel the difference that it's making upon the development of the piece,so that then other changes can build on top of it.

(Piece plays)

It's a sort of organic form making and melody making all in one.

Chris Maslanka : So how fundamental is the mathematical aspect? Is it just that you've found another organising principle?

Peter Paul Nash : It is certainly an organising principle.I think there should be a cautionary note injected here,which is that maths is maths and music is music,they do inhabit their own worlds [I don't entirely agree -LB].But nevertheless they have many many connections,and always have done.From my point of view,it's the music that comes first.At school I was terrible at maths [Typical! -LB] (laughs),and when I've read about Chaos Theory and Fractals and so on,usually in popular science books,rather than saying,"Ooh that's a good idea,I think I'll have that",it's more a question of feeling "Goodness me! I've been doing that all along".

[Which only goes to show that musicians are doing maths instinctively on an intuitive rather than mental cognitive basis -LB]

Chris Maslanka : But isn't all this self consciousness about rules? Restricting the directness of music?

Peter Paul Nash : There isn't a conflict.In fact I think that my music has become ever so much more accessible,more emotionally immediate than it was before.The mathematical precepts behind it,in fact aid the listenability of it enormously.The important thing to understand is that music crucially depends on structure,just like mathematics is the study of structure.

Chris Maslanka : Yes,you couldn't manage without it.

Peter Paul Nash : And so what I'm trying to do is to make structure that is memorable,and useable and also which can develop,which can change.This is a friendly way of using numbers.I'm basically saying,"Look,structure is good,there's things we can do with it",and it doesn't have to war with the emotional and dramatic and exciting sides of music. [I agree -LB]

(Vocal piece plays)

Chris Maslanka : Composer,Peter Paul Nash.clearly the world is a structured place,and our brains have evolved to appreciate structure,so no wonder we like it in our art and music.Maths is just a description of that structure.But if structure is so fundamental to music,could maths and music be an expression of the same skill? [I think so -LB]

Hattie Hayridge : If they were,brain scans would show similar things going on when you do maths and when do music.Actually the Research Imaging Centre of the University of Texas,at San Antonio,(exclaims) are looking into it right now.But their results aren't out yet.

[There is a piece of the brain close to the ear that is thought to hold mathematical ability -LB]

Chris Maslanka : Sounds like a whole new programme.

Hattie Hayridge : It's hard to pin down,because music uses so much of the brain.In fact the whole thing lights up,like....well,like your little face when you see the beautiful cake I've baked you.

Chris Maslanka : Oh Wow! It looks good and by golly it.....uuuugh (coughs) it's awful!

Hattie Hayridge : But I made it mathematically,with ingredients in the proportions of 6 to 4 to 2 to 3!

Chris Maslanka : That doesn't make it cakey! You silly thing with colossal IQ.It's not just proportions,it's the right proportions,just like music.Maths can be useful,but we mustn't let the tail wag the dog,must we? [I think this misses the point -LB]

Hattie Hayridge : Don't look at me,I'm not programmed to answer aesthetic questions.You'll have to ask someone who is.Like Roy Howard.

(Piano plays)

Roy Howard : The mathematics are just a measure of what sounds best.It's what sounds right in the end.It's one's sheer musical taste. [I don't think it is -LB]

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