In Our Time - Mathematics and Music

Episode Date: May 25, 2006

Melvyn Bragg and guests discuss the mathematical structures that lie within the heart of music. The seventeenth century philosopher Gottfried Leibniz wrote: 'Music is the pleasure the human mind experiences from counting without being aware that it is counting'. Mathematical structures have always provided the bare bones around which musicians compose music and have been vital to the very practical considerations of performance such as fingering and tempo. But there is a more complex area in the relationship between maths and music which is to do with the physics of sound: how pitch is determined by force or weight; how the complex arrangement of notes in relation to each other produces a scale; and how frequency determines the harmonics of sound. How were mathematical formulations used to create early music? Why do we in the West hear twelve notes in the octave when the Chinese hear fifty-three? What is the mathematical sequence that produces the so-called 'golden section'? And why was there a resurgence of the use of mathematics in composition in the twentieth century? With Marcus du Sautoy, Professor of Mathematics at the University of Oxford; Robin Wilson, Professor of Pure Mathematics at the Open University; Ruth Tatlow, Lecturer in Music Theory at the University of Stockholm.

This BBC podcast is supported by ads outside the UK. Thanks for downloading the In Our Time podcast. For more details about In Our Time and for our terms of use, please go to BBC.co.com.uk forward slash radio 4. I hope you enjoy the program. Hello, the 17th century philosopher Godfried Leibniz wrote, quote, music is the pleasure the human mind experiences from counting

without being aware that it is counting. Unquote. Mathematical structures have often provided the skeleton around which musician composed music and have been vital to the practical considerations of performance, such as fingering and tempo. But is there a more complex area in the relationship between maths and music, which is to do with the physics of sound, how pitch is determined by force or weight, how the complex arrangements of notes in relation to each other produces a scale, and how frequency determines the harmonics of sound?

How were mathematical formulations used to create early music? Why do we in the West hear 12 notes in the scale when the Chinese hear 53? What's the mathematical sequence that produces the so-called golden section? And why was the resurgence of the use of mathematics in composition in the 20th century? With me to discuss mathematics and music are Marcus Gisotoy, Professor of Mathematics at the University of Oxford. Ruth Tatlow, a lecturer in music theory at the University of Stockholm, and Robin Wilson, Professor of Pure Mass at the Open University,

and all of them are musicians. Markis Gisotai, how and why do you think music and maths are linked? I think the Leibniz quote is quite interesting, because it talks about counting, but I think that actually is a mistake about what mathematics is really about. I tend to find people think maths is about long division to lots of decimal places. And actually I try to describe a mathematician as a pattern searcher. And I think that's where the similarities with music really take off.

That music is a lot about structure and pattern and patterns developing over time. And I find mathematics shares a lot aesthetically in common with music. I mean, the way that I listen to music and the way that I read mathematical proofs are actually very similar. Can you disemble to? Yeah, I mean, a mathematical proof, you very often start with themes which you are quite happy with, perhaps an axiom or something like that. And then a proof as it develops interweaves the themes. You get new ideas coming in. You see things developing.

And it looks very much like the way a musical composition develops, where a themes are introduced for the first time, then start to start. to mutate in interesting ways within a sort of structure. And I think music, you know, sort of difference between music and noise, perhaps, it is that there is structure and pattern behind the way music develops. And they're both quite abstract worlds. And I think you belittle a piece of music if you translate it, for example, into a picture or into words.

And you need to spend time in that world to really appreciate the patterns in there. And it's the same with mathematics, I think. Do you think the appreciation and the ability in mathematics and music comes out of the same? Do we know anything about which part of the brain it comes out of? Yeah, I think there is evidence that the activity in the brain, when you're listening to a piece of music, is related to the same area which is happening with mathematics. And I think there's been research which says that somebody who learns a musical instrument

in their early teenage years, when the brain is developing very fast, actually helps to develop their mathematical abilities, as well. And I think it is because they're both quite about abstract structures, about patterns, and so both reinforce each other. In a maths department, you can always collect together an orchestra because there's invariably they're all musicians as well. Is there any sense in which over-severe mathematical application to music could constrain music composition? Well, I think that's true. People have this sense that mathematics is very deterministic and formulas and forcing things to happen. But again, I think

that's a mistake about what mathematics is about. There's a lot of freedom within mathematics. You have your logical framework, but there's still a lot of movement to play with things inside there. And so we get a lot of surprises in mathematics, and that's the maths that I like, where suddenly a proof goes

off and you find itself in a completely different place. I think musicians quite like the sort of structures, as you said in your introduction, as a framework within which they then put a personal element to it. But you don't want to be too predictive such that your mind is running ahead of the actual music so you sort of know what's going to happen. That's boring. Robin Wilson, rhythm is possibly the most fundamental

feature of music. Would you say that was fundamentally, sorry to repeat to it, mathematical? I think it certainly is. It's certainly something that's been in music from the very, very beginning, whether it's African drums or clapping hands and everything. It was used to accompany dancing, for example. And dancing, of course, is rhythmical too. Eventually, of course, people started gathering notes into groups. You might have groups of twos, what we now call two in a bar, three's, three in a bar, as in the national anthem or waltz time, four in a bar.

Then, of course, as music developed, it got more complicated. You find pieces of music with five in a bar, like Dave Brubex take five, for example, or even more than that, six, seven. And so eventually, as time went on, and as music became more sophisticated, then rhythm became more and more complicated. And in fact, there are even some composers

who have actually used rhythm as the basis for some of their compositions. There are some musical compositions which don't have any pitch, but they're based on rhythm. Steve Reich invented some clapping games. We all know listening to Flamenco dancing, that there's a very clever cross-rhythms in the clapping that they use.

And Steve Reich formalized this in his clapping games, which are also used in mathematical education for children. There are other pieces, there's a famous piece called Geographical Fugue by Ernst Toch, which will start Trinidad and the big Mississippi in the town Honolulu and so on, and more and more voices come in. And it's entirely rhythmic piece.

So whereas rhythm probably goes back, well, it certainly goes back thousands of years, it's now been developed and it's now part of composition. I'd ask much the same question and ask Marcus you so took, because I'm intrigued now, I guess our listeners are too. The rhythm numbers, you mentioned lots of numbers, 2-2, walls and so on and so forth. Was that conscious? Did they say numbers, music, or did it just, as it were, grow out,

one grow out of the other, the way it developed the idea of rhythm? What's your take on this? It's probably an impossible question to answer, which case say so, we'll move on. Well, I think it depends on the particular. culture, obviously if rhythm is being used as an accompaniment for dance, then it certainly has to be sort of rhythmical and it has to sort of fit in with a sort of given sequence, a certain number of beats in the bars, as we now think of it, whereas a lot of early music, say, plain chant, for example, a rhythm played considerably less, I think, of a role.

Obviously it had to be sung in a certain sort of rhythm, but I think it was a much freer rhythm than, say, in dance music. Can we switch to musical pitch now, Robin. What are the mass political connections with musical pitch? Well, this is where I think we ought to mention the Greeks. The Greeks seem to be the first, in some sense, to link mathematics and music. In fact, to the Greeks, the four mathematical arts were arithmetic, geometry, and astronomy, but also music. Now, why music? Well, it's the Pythagoreans are always being said to believe that all is number.

They tried to quantify everything. And in music in particular, what they were quantifying was intervals. they described intervals, in other words, two notes sounded together. What's the difference between them? They described that in terms of small numbers. There's an old legend, which is almost certainly untrue, that Pythagoras discovered this link between mathematics and music when he passed a blacksmith's forge,

and he heard the sounds given off by the hammers. The pairs of hammers whose weights were in the ratio two to one gave sounds differing by an octave. Now an octave is playing it on the recorder. The first recorder ever heard in that time. It's a welcome to the recorder. And of course octave means eight, eight notes. So it's when he heard pairs of hammers whose weights were in the ratio three to two,

that gave sounds differing by a perfect fifth, which is five notes. And when he heard the hammers whose weights are in the ratio four to three, that was sounds differing by a fourth. Sorry. And so, hearing these hammers, he was able to link the weights of the hammers with the octaves, the fourth and the fifths.

And so you can describe these intervals in terms of ratios of numbers. The octave is 2 to 1. The fifth is 3 to 2 and the 4th is 4 to 3. And so here we are. We've got the intervals and they're described by ratios of numbers. And then you can combine them. If you want to combine a fifth and a fourth, well, they give an octave.

And the reason you get that is if you multiply the fractions, the fifth is three over two, the fourth is four over three. And if you multiply three over two by four over three and simplify, you get two over one, you get the octave. So this shows how using these numbers that the Greeks liked using to represent a fifth and a fourth,

you could combine as a given octave. We'll move on from the Greeks, and we'll come back later to that, that notion. Ruth Tatler, can you explain for us how numbers can be used in a very practical manner by musicians, such with fingerings and to figure bass himself? Yes, on the most banal level, if a child is going to have his first violin or piano lesson, you'll find within minutes that the teacher is taking a pencil and putting numbers above notes. The child might have thought he was learning to read music, but very quickly he's asked to make a relationship

between a number, for example, the number one, and a physical action. So he's having to use use his first finger. If it's the piano, it's his thumb, and if it's a violin, it's his index finger. But the number is on the score, it's a practical tool, and it's making an association with a physical movement. And then the child might recognize something that looks like three quarters that appears after the treble clef and the key signature. But of course, it's not three quarters. It's three over four, which is a time that Robin has mentioned in Walsh, Walsh time. And so the has to learn to reinterpret what looked like something that he already knew, three quarters,

into something else. So there we have another practical use of numbers. And then when you get to grade five theory level, you meet this very lovely shorthand method of using numbers to indicate the harmonic content of a score, which is called figured bass. It was very common in the 17th and the first half of the 18th century. And it looks like this. You've got a baseline with a bass clef, and beneath it you get two figures or three figures. And it is literally an indication of the harmonic content. So, for example, if you have a C,

and underneath it you have the number two and the number four, it's telling you that you're going to play a chord which has got a D in and an F in, the second note above the C and the fourth note above the C. And people become very, very adept at reading these. a good continuo player can read at sight this as quickly as he can a full normal, normally musically notated score.

And also a very experienced person can read and hear that as well as he could read and hear a normal score. Can you tell us about the division into bars and how that came about and why that's important in music? Well, I think that's a question really about rhythm, The bar has been a unit in music for quite a long time. And the number of bars was a very useful way of counting the length of a movement. And that did become very important, in fact, the fact that there are numbers in the length of a movement and a work can be divided. And in Bach's time, composers did know exactly how many bars there were in a work. for example an orchestral score

because their apprentices had to copy those, for example, copy out the violin part. They had to know that there were, for example, 26 bars in this minuet. Because otherwise if they copied 25 bars, it wouldn't work when the whole group played together. So bars were a unit that were counted. And in older schools,

you do actually find very often at the end of a movement that somebody has notated how many bars how many bars there were. Marcus, you want to come in on that? Well, I think it's interesting how notation is, I mean, it really reflects this combination between mathematics and music that we're finding mathematics is often about finding a language and notation

to express quite abstract patterns and the way that we've found to express music, which, after all, isn't a pictorial thing, yet a piece of music is actually very visual when you're given a piece of score. And you can actually start to see the patterns in the music. You see a theme and then perhaps there's some geometrical game played on that theme. And so I think it's interesting how actually the way we've notated music is quite mathematical and geometric.

Well, also, you not only got to the fact that it was a unit, but it was an indication of duration. So we get numbers again used to make an equation between duration and what was physically written out in the number of bars. In 1619, Michael Prytoris and his syntagma musica gave this rule of thumb. He said 320 bars take half an hour. It's a very nice equation. 160 bars take a quarter of an hour

and 80 bars take half of a quarter of an hour. So we have this lovely, yes, you can work out 10.6. bars per minute. Interestingly, 100-something years later, Mitzler, one of Bach's admirers, founder of a scientific music society, said something similar and he updated it, except this time it was a bit faster, which is very interesting. He said, in winter, because he's talking about cantatas, in winter, 350 bars of varying time signatures, and that's interesting, it really is a broad rule of thumb, varying time signatures will take about 25 minutes, and in summer It can be 10 minutes longer, and so 35 minutes, so you get 490 bars.

That way you get 14 bars per minute, so in the 150 years it's sped up. Can we, Robin Wilson, can we take on, you gave us brilliant traditions on your recorder. Can we take on after the Greeks? How did the, what happened to the development of musical intervals after the Greeks? It was found eventually that the Pythagorean system, which sounded very, very good, is actually flawed. If you look on a piano, if you take seven octaves, you go C to C to C, right the way out of the piano, that, of course, you're doubling the frequency each time. So if you take seven octaves, you've actually gone two times, two times, two times two,

you've actually multiplied by 128. But also on a piano, if you go up in fifths, if you go C to G to D to A and so on, right the way up the piano, you find that on the piano seven octaves equals 12 fifths. Now, seven octaves, as I said, is two to the seven, which is 128. But if you take 12-fifths, you're multiplying together one and a half, three over two, 12 times. That doesn't give you 128, it gives you just a little bit over 129. So the Pythagorean system is basically flawed. Also, as music developed, say around 14, 1500s and so on, thirds and six were coming in more and more.

There's the major third. Now, the Pythagorean, if you actually work out what the ratio is, it comes to the horrible ratio of 81 over 64, which isn't at all easy to deal with. But if you lower that slightly to 80 over 64 and cancel, you get 5 over 4. 5 over 4, that's nice small numbers again. And so with the 3rds and 6th coming in,

the major thirds, the major 6th, had ratios like 5 over 4, 5 over 3 and so on. These are much nicer to deal with. But then they found that that was also flawed. Supposing you want to go from one note to the next C to D, if you use the Pythagorean system, I won't go to the detail. Well, it's essentially you can go via G. C up to G and down to D.

If you work it out, you're dividing three halves by four thirds. Don't worry about the detail, but you get nine over eight. But if you go a different way, you go up to F, which is four thirds, and then down to D, that's a much. of third, that's six-fifths. That gives you 10 over 9. The point is, not the details, but you can actually get C to D. If you calculate it one way, you get one answer, nine over eight. If you calculate it another way, you get 10 over nine. How can you have two different fractions representing the same interval? The question is, how do we smooth over those

difficulties? Can you take that up? How do we smooth it over? Well, I was going to pick up a point. I mean, the amazing thing is, why do we have these 12 notes on a piano, the semi-tones. And why have we divided an octave up into 12 notes? What's so special about 12 as opposed to 10? I mean, we've got 10 fingers. Why didn't we divide the octave up into 10? And it is essentially mathematics, which explains, it's a curious, I mean, we had a lot of numbers there, but in a sense, it's a curious coincidence of numbers, which has led us to the 12-notes semi-tone scale. And it's the fact that if you divide, you've doubled a note to get to an octave. So if you have 440 hertz or something is an A and you double it an octave 880 hertz,

well the semitones, okay, here's a little bit more maths for you, but you take, to get 12 notes. I'm blinking in the face of this as short. I know, okay, well, this is, you actually, the semitones are divided by the 12th root of 2. Now, okay, there's a curious coincidence of numbers where if you raise the 12th root of 2 to the power 5, you suddenly get very close to 3 over 2, which is this perfect 5th. So, okay, don't worry about the maths in some sense, but it is a curious coincidence. We need to worry about that. The point of the programme is that we're all trying to worry about the maths.

I mean, we're worrying away all over the United Kingdom matter about the maths. Okay, so, exactly. So there's something about the 12th root of 2, which is very close to being this very nice number, 3 over 2. If you raise the 12th root of 2 to the power 5, you get very close to a rational number. and it's this curious coincidence of numbers which are very close together so in a sense you can't hear the difference which has meant that we've very naturally divided the 12 note

the octave up into 12 notes and other cultures you mentioned in your introduction the Chinese went for 53 why have they gone to 53 from 12 to 53 because that's another place where you get a very curious coincidence of numbers Ruth can you tell us how the scale of equal temperament tried to sort these numbers out

Well, we had to solve the problem of this Pythagorean comma, which is the difference when you go up your seven octaves of fifths and your octaves. And it became very important around the 1700s when people really did start to explore composing in different keys and not just composing in a few like 12 keys. Suddenly, Bach, for example, he did his famous 48. He wrote 24 preludes and fugues 1722. And he played them all at one sitting.

We know that. Therefore, every single key had to sound in tune. So suddenly there was a reason why this was more than theory, why people had to find these different temperaments so that it was relatively in tune and not just sounding a sweet chord of D minor. We had to find all the... And people were experimenting the whole time.

We have Verkmeister 1, Verkmeister 3, Kienberger 3. We have all these different temperaments. And Bach obviously had his own temperament. But the problem was, how did you actually tune the harpsichord because even though theoretically you could say we know the mathematical answer, we divide our octave into 12 equal semitones, you don't tune a harpsichord in semitones. You don't hear it.

You aim for the fifth. You then tune in a third. You go for a fourth. You try and get these pure things. So you had to make compromises the whole time. And that's what the temperament was about. But it was actually motivated by a practical thing. The people were becoming more harmonically adventurous

and composing in these different keys. Robin? Yes, I think it's important to realize that the problem arose with keyboard instruments because, of course, if you're playing a violin or you're playing a flute or something, you can adjust your pressure, you can adjust your fingers and play in whatever key you like. It's only when you have instruments which are pre-tuned like the keyboard instruments, that you have to get this sorted out. Another way of thinking of equal temperament is because of all these problems about keyboards

that might sound fine if you play in the key of F, but might sound horrible if you play in F sharp. This is why you have to have equal temperament. The equal temperament is essentially a compromise. There's a compromise which means that you can actually play in any key, any key you like, and it'll all sound the same. But if you make that compromise, then as Marcus said, you get very, very close to our favourite intervals.

I mean, the Pythagorean 5th, as we've seen as 3 over 2 or 1.5, whereas an equal temperament is very close. 1.498. That's very, very good. The fourth, Pythagorean's 4 over 3, 1.333, equal temperament 1.335. Very good. But if you take the major third, that's 5 over 4, that's 1.25, an equal temperament, like a piano, you get 1.26. In other words, if you tune a keyboard to equal temperament, which we always do now,

well, generally, not except when you're playing, say, Baroque music. if you tune a piano, then it is automatically out of tune. Would you say that Mark was a numerological composer, Mark is this, I tell you? Well, I wouldn't have said, I think there is numerology in there, but I think he was somebody who loved patterns. I mean, he's always picked out a sort of mathematician's favourite composer because there seems to be so much structure in his work. I mean, for example, something like the Goldberg Variations,

I think it's a perfect example. It's an exercise in symmetry, because each, variation. I mean, variation is about doing something to a theme to make it slightly different. And mathematics provides Bach with great ways to do that, for example, reflecting a theme or moving the notes up a little bit. So as you go through the variations, you find the theme being moved up one step, two steps, three steps, four steps. And so there's a huge amount of structure. Also, the rhythm, he plays, if you look at the rhythm structure in the Goldberg variations, he's picked out every single combination of rhythms that was possible.

So he's obviously very aware of mathematics behind that in order to make sure that he'd covered every single possibility in his variations. Is it a stupid question to ask, where is melody in all this? Oh, well, melody is very important as well. I mean, and melody as well is, I mean, the interesting thing is getting Bach was an expert in getting melodies which when you shift them, say, in time, will fit over each other, and sound fantastic together. Ruth, would you talk about a little more about Bach?

Yeah, well, Bach and numerology, numbers with significance. I mean, they should never really go together, but unfortunately this wonderful theologian and musicologist Friedrich Sment did put the two together in 1947. Well, numbers have always had significance right across the board. But here with Bach's three had great significance for many reasons, but the religious reason being one of them. Well, yes, but he took it further, and he introduced a number alphabet,

and he started to say that Bach used a number equivalent, a number alphabet, A equals 1 to Z equals 24, you get 24 letters in the alphabet. If you take I and J as 9 and U and V is 20, so you get 24 letters. And this way, it explained for Sment, the many 41s and 14s that he had recurring. So 41 is J.S. Bach, J is 9, S is 18,

and Bach is 14. And he started, he posited, it as a theory, but unfortunately many people took it literally. But one of the most outrageous things possibly, he said and this did get very with religious significance unfortunately because he was a theologian

and he was very interested in Bach's faith, which is wonderful, but he didn't need to use numbers to prove it. He took the number 84 that appears at the end of the patron omnipotentem in the B minor mass. At the end of the movement of 84 bars there is the number 84.

And he said, this is Bach deliberately putting it because it stands for J.S. Bach, which is 41, and Credo, which is 43, C is 3, R is 17, and so on, it adds up to 43. So you get 84. And the problem is people liked this, and they started to follow it. And any number they could count in Bach, they started to interpret, which shows you far more about the interpreter than it does about Bach himself. But, I mean, the upshot of this is actually, recent scholarship tells us that the number 84 was written by

C. P.E. Bach later, while he was copying the score. Do you want to comment on that? Certainly the idea of numerology does come in all over the place in music, and we've heard a lot about in Bark. A lot of people have gone into things like numerology and things like Mozart's the Magic Flute. I mean, the number three there is a crucial number. It's linked with the Masonic ideas that occur in the Magic Flute. If you look at the

if you look at the overshaw, it starts off with three chords, and the three chords occur later on throughout the opera on a number of occasions. The key signature is E flat, which is three flats. The goodies in the opera,

like Pimina Tamino and so on, they all have three syllables in their name and so on. So there's numerology. I don't know how far one can push it, but certainly there are many instances where numerology does come in in music. Other

Northern Bark and certainly right up to the 20th century. Yeah, I think it's quite curious how people like signing their names in their music. I mean, Begg is an example in the 20th century. His number was 23 for some reason. Perhaps like David Beckham chose 23 to play in for Real Madrid. Well, Begg played a number 23 in his music. I wonder when we're going to get football in. Sorry.

Next program is football and music. Football, music and maths. But Beig would use 23 bar sequences, for example, in the lyric suite, as a way of saying his presence in that music, and his lover had the number 10. And so he would use these ideas. And I think that's the nice thing why musicians love structure. I mean, it's a bit like poetry.

You know, poetry is very artificial in a sense, to bound yourself within a particular framework to get rhyme and rhythm and things like that. But it pushes you in interesting directions. Non's fret not. Right. Never mind. Okay, sorry.

But I think that's why musicians like structure because it pushes them in interesting directions. They set up a framework, and then they let their imagination wild within that framework. Ruth, do you want to come in and finish off on that before we move on? Right, Marcus, can you tell us about the contribution of the French mathematician, Jean-Baptiste Fourier, to the understanding of physics. Yeah, the physics of music and sound. Let's make it really hard now.

Let's make it. Well, this is, Fourier's discoveries are the reason why you can hear my voice now. I'm not sitting in your living room or in your car or wherever you're listening to us, but you're able to hear the sound of my voice. How can you do that? Fourier discovered that just as water can be broken down into its atomic components, sound has basic building blocks. So every sound is built out of a very basic sound.

It's actually the sound of a tuning fork. So I've got a tuning fork here, so you can hear the building blocks of our voice. voices and sounds. So that's a 440 hertz, it's an A. Actually, if you draw it graphically on an oscilloscope, it's a sine wave. So we're talking about something the equivalent of prime numbers in mathematics or molecules in chemistry. The tuning fork is that equivalent.

The building is it better do it again because I didn't get that significant until now. Here is the hydrogen and oxygen. Why is it just a second? Well done. Not just a thing. Tell us, before you do it, because we're all now a gog on the edge of our seat. Will you score the penalty or will you not? But tell it, tell it why the tuning port?

What is it about that little thing you have in your hand that's so important? Well, this is what Fourier discovered, that the sound that this makes is the purest... Sorry? What is you made of? This is made of metal. Yeah. So, and the metal is vibrating together, and I'm using the table now as a soundboard for hearing it. And this actually produces somehow the perfect wave.

If you were going to draw a sound wave, which you thought was the purest sound wave, it would probably be the sine wave. And the point is, if you can probably have a picture of what a sound wave. sound wave looks like music on your amplifier or something, you see this thing dancing up and down. Fourier understood that you could build very complicated sounds out of playing tuning forks of different frequencies together. So for example, the sound of a violin, I can kid you into hearing the sound of a violin by playing tuning forks of different frequencies together. And just like water's made out of two hydrogens and on oxygen, well, the sound of a violin is reproduced by the sounds of these tuning forks of different frequencies. So they build up a

sound, which, and that's why this is so important, this particular sound, and why it's so pure, that's why we use it as a thing to tune things to, because it's such a pure sound, not cluttered by other notes. And so even the sound of our voices is being broken down by a computer into the frequencies of the sine waves. This is transmitted across the wires, and your loudspeaker is then told to recreate these sound waves, sine waves altogether, vibrating altogether, suddenly you're hearing the sound of my voice. Absolutely magic. Robin, the simple thing. The simple form of mathematical structure is a musical canon or a round. Can you explain how that works and why it's so important? Yes. There are many rounds. A round is essentially a succession of voices coming in singing the same thing. So I might sing London's burning, London's burning. And then when I go on to the next line, then Marcus will come in with London's burning. And then when I come into the third line, Marcus is singing the second line and then Ruth is singing London's burning again.

So around is, well, it's what it says, it's around. You go round and round and round. You sing a song, but the different voices come in at different times. Now, and there's a rather cute way to hear around, actually, because anyone who's listening on a digital radio and also a sort of conventional radio will get around going because the digital radio always produces the sound a little bit later. So you hear our voices sort of in canon, so you can try that now. That's a canon.

A round is where you go round and round and round. Frerejaka is another one. You go over and over again. So developing this is the idea of a canon, which is what Marcus was saying, which is where you have one voice. When I say voice, it might be an instrument, playing something,

and then successive voices copy the first one, but a little bit later. But they don't go around back to the beginning. So around, they're not very interesting. Cannons then developed, for example, the famous Packlebel canon, Now that's got a mathematical structure Because underneath you've got a thing called a ground base

Which is a certain line of notes in the base Which go on right they repeat right the way through the whole piece And on top of that You've then got the cannon Where the first violin for example comes in And then a few bars later The next one comes and plays the same thing And then the next one and so on

And there are a lot of cannons around the place One of the most amazing ones Is actually a very early one The 13th century canon called Summa is a Komenin, where you've got six lines, the two bass lines, if you like, the men's parts, they have a two-part canon. They sing the same line after each other over and over again

right the way through the whole thing. And then on the top, the four lines, they have a four-part cannon, one coming in, then another, then another, going right the way through the piece. It's a very, very intricate piece of music, and is as early as the 13th century. Ruth Biff, I'd like to move to the 20th century but I think there should be something said about Mozart's use of numbers.

One of the things that everyone knows about Mozart is that he was told when he was young he could be a brilliant mathematician as well as a brilliant musician. And so we're onto somebody of whom it was admitted at the time and that these two talents ran in tandem. So he's quite a good case study. Yes, I did a little experiment last month. I had to give a talk on marriage of figaro.

And as some of us know, the marriage of Figuero opens with numbers. Figuero is standing there measuring the room and he says the words 5, 10, 20, 30, 36, 43, which is very interesting because the opening words are numbers. The question is, did Mozart allow this to affect his structure? More interesting is that the original play, Beaumarche's play, from which Deponte, the librettis,

took these numbers, the Beaumarche original, were not these same numbers. So we know that DePonte changed it. Beaumache had his figaro opened the play by saying 19 times 26. But then we have DePonte having the sequence of numbers, which seems to be totally random, but it adds up to 144. So I thought, well, let's just see. What I found was very exciting.

The very next duet has exactly 144 bars. Not only does the very next duet have 144 bars, but the first 576 bars, which ends at the recitative between Suzanne and Marcellina is in those 576 bars, we have the complete plot exposed. Now, 576 is four times 144. So Mozart has played with this. He's taken this number and had it, as Marcus had said earlier, as a little structure, a little game or some kind of restriction,

and made a beautiful one to three proportion in those first. What do you make of that, I used that in the trial. I mean, I was intrigued by it. What do you make of that? Well, I think, again, it comes back to this idea of poetry, that you think you're self-consciously doing this. Yes, I do. I mean, I have a dialogue with a composer from New Zealand, Dorothy Carr,

and we spent a lot of time talking about math. I'm interested, I think what I wanted to be was a composer, and I couldn't be, as I became a mathematician. So I love talking to her to find out about how maths is using music. But she talks about the way she uses something like the Fibonacci sequence, which is about growth and things, the way nature grows through numbers. And she loves using that just as as a structure,

which then she is creative within. And I think too much freedom, you just end up with something which isn't interesting. But I think it's that nice balance between having enough structure that something's kind of meaningful and not too much such that it doesn't bound you. And mathematics does that at its highest level.

I just say, I've kind of got, you've got so, you're so interesting, all that stuff, that I haven't got much time left for. We have to touch on the 20th century. So what do you think the most important development was in the 20th century, Robin, and if we can go on about that a bit? I think it was the complete decomposition of keys.

In the broad period, you had music in a definite key. But more and more, say, on the white notes of the piano, but more and more people started using black notes of the piano, they tried to get more chromatic, we say, and that certainly developed a lot in the 19th century. But as the 19th century went on, you've got more and more pieces of music which were far more chromatic, and this led eventually in the 20th century to the complete disappearance of tonality in some composers, and in particular to Schoenberg and the 12-tone row.

The natural development of this idea was instead of regarding particular keys as important, to regard every single one of the 12 notes in the octave as equally important, and then to compose on the basis of that. So what a composer would have done, he'd have, I mean, the number of ways of ordering your 12 notes in an octave, there are many, many millions of ways of doing it. So a composer will actually choose an ordering, and they might choose them for certain reasons.

And then the idea is that they will compose the music by introducing these 12 notes in that order, and everything will be depending on that particular order. the mathematical structure is actually choosing the order to start with and then the composer as a composer as a musician as opposed to a mathematician then takes over and that's where their creative business comes in. Do you want to refer to Messia in this regard?

Yes, yes. Yeah, I mean Messia is my favourite composer because he uses so much mathematical structure and I think it's partly why 20th century musicians came more to these mathematical structures is because of throwing away tonalities so they needed other sort of things to hang on to rather than say a nice

major scale. And so Messia, for example, uses my favourite numbers, the prime numbers in the quartet for the end of time. And he uses, he wants to create a sense of timelessness, the quartet for the end of time. And he uses the fact that prime numbers somehow can keep rhythms and tunes very separate. So quartet for end of time has a 17 note sequence

against a 29 note sequence, which play like two cannons. But they interlock in detail. different ways each time they repeat themselves. And so it creates in the piece. You know, he's used a mathematical device in order to do something which is to create the sense that you're never quite hearing the same thing in the same way each time it's, uh, the new, the voice comes in again. And so, you know, Messiaen just love the idea of symmetry and another composer, one of my favorites is Zanakis. Zanakis is sort of the modern day Bach. Bach used very simple symmetries

to do variations in the Goldberg variations. But Zanarchus uses the full range of mathematics. radical symmetries. The symmetries of the cube get used in a piece called Nomus Alpha. Not that I think you could hear the cube if you heard that piece of music, but that's hiding behind there. Ruth, can you tell us briefly about the golden section and why that applies now? Well, since we're talking about the 20th century, it's probably easiest to take it up on that. The golden section can be expressed with the Fibonacci sequence, which is an additive sequence 1-1-2-358, 13, 21, 34, etc. And a line is considered to be cut at the the golden section when three of these numbers come together.

So, for example, if you have a piece of music that's 55 bars long, and at bar 34, you have, at the climax, for example, it could be said to represent this perfect proportion. And there was a fashion for looking for these golden numbers and golden sections in the 19th century. And its aesthetic qualities have been explored very much in the late 19th and early and in the early 20th century. And they are naturally occurring as well. Coalhant-Stokhausen uses it in microphony too,

very deliberately with the number of seconds in a piece and trying to create this very beautiful aesthetic feature. Is finally his mathematics pushing on further and further into its influence on us. I was about to say interference. What a terrible way. Interimps and our music. Well, I think more and more composers

are exploring the possible structures that we've discovered, which may have come out of natural phenomenon like the Fibonacci sequence, but I think we have a whole range of very interesting structures, which musicians are very keen to see whether they can, as Mitzler, the Bach student said, well, music is about sounding mathematics, and a lot of composers are interested in about ways of sounding new structures. Very briefly, Robin.

Yeah, this is right. I mean, composers have used ideas like magic squirt, whereas fractals, all sorts of other things, but the important thing is they use the mathematics to give them the material out of which they work, and then they come along as composers and do the compositions. Thank you very much, Mark Tusseroy, Ruth Tatler, and Robin Wilson. Plenty to think about in that.

And next week's programme will be on the anatomical and cultural history of the heart. Thanks for listening. We hope you've enjoyed this Radio 4 podcast. You can find hundreds of other programmes about history, science and philosophy at BBC.com.com.uk forward slash radio 4.