Mathematics and Music

[knight66]

"Marcus du Sautoy is professor of mathematics at Oxford University and a fellow of New College he has recently been named as the next Simonyi Professor for the Public Understanding of Science." He is also very musical. I heard him being interviewed on radio and there was discussion around the connections between music and mathematics.

He goes way beyond obvious patterns and can feel prime numbers in music. It was mentioned that some Messiaen pieces are composed around prime numbers.

We have some mathematicans here. I was wondering if they could give their fix on where they see or feel the connections between the science and the art form. Is there something instinctual for some people in that they can feel the mathematical patterns in say Bach beyond the clear patterns.

What other composers seem to have been influenced by maths in their writing?

Mike

[greg]

If you can read French, Xenakis has the whole book of Formalized music on his website. But you need to be a math expert, too, if you want to be able to even understand it.

[DavidW]

It was mentioned that some Messiaen pieces are composed around prime numbers.

It's in the beats, and requires no understanding of higher mathematics--

This last element is especially important. Indeed, Messiaen referred to Turangalîla-symphonie as "a vast polyphony of rhythm." As he explained in The Technique of My Musical Language, a treatise explaining his compositional methods, Messiaen thought of rhythm independently of pitch. Rhythmic figures had for him as much integrity and interest as melodic shapes, and over the years he developed a stock of favored rhythmic figures that include not only Classical Greek patterns and Hindu tâlas, but also such original conceits as rhythmic palindromes and figures whose durations yield prime numbers (11, 17, or 23 beats, for example), which Messiaen regarded as having quasi-magical significance.

from http://www.carnegiehall.org/article/box_office/events/evt_8118_pn.html?selecteddate=02152008

[knight66]

David, Thank you for the link, it is useful to the extent that this week I bought the Turangalîla, a work I have not heard before. The programme notes on the link are much more detailed than the ones from my discs.

However, saying that 'it's the beats', is like telling me that maths is about numbers.

Thanks ' for the note of other composers who incorporate maths into their music.

I have a feeling we can now just watch this topic drop like a stone down the list.

Mike

[Sarastro]

Math can be applied to anything, but who wants to hear math when they listen to music, defeats the purpose me thinks.

Excellent point! *claps*

I major in mathematics but somehow do not feel justified to talk for math in music. My first calculus teacher (her husband is a conductor, btw!) told once that when mathematicians try produce some music using math, it is usually a very lousy kind of music. Schoenberg is a nice illustration to that opinion. The more composers think mathematically and analyze music, disintegrate it into pieces, the more unnatural it becomes.

I like Mozart's mathematical precision in music, though it has nothing to do with math - just classicism style. And fortunately I do not hear prime or imaginary numbers, transcendental functions or matrices in music, though such analogies are tempting. (and Queen's of the Night passage in the second aria can be represented as a sine or cosine function! )

One thing they share in common, I think, is that they both are imaginary and created by humans. Mathematics, as well as music, is very powerful, but only in people's minds. They both do not exist in nature, they are artificially created and very well fitted in the perception of the world around us. Mathematicians create rules of the game called "math", as well as composers make up scales for the game called "music". Music is the world of organized sounds and mathematics is the world of organized theories and symbols. So I think the reason some people try to realte them is just because they are part of our imagination and somehow exist only there.

[knight66]

I think you will find disagreement that maths does not appear in nature. My understanding is that it does, the movement of the planets is in part a mathematical expression. Maths is also claimed as being the basis of all science.

Mike

[knight66]

I disagree; maths is not invented, it is discovered. Tied to physics, it tells us about our world.

However, my hope was to prompt any members here to provide insight into the remarks in my initial posting. Perhaps another time when some posters happen by who can relate to that statement.

Mike

[71 dB]

no Sarastro  is correct, math is just another abstract language created by us, it's used in an attempt to explain things...but it hardly does any explaining, more describing in a very limited way, there is probably so much more to the larger picture, far beyond what our tiny brains are capable of understanding....

Sounds religious crap to me. The way that math explains things is not "very limited". Perhaps our tiny brains are too tiny to understand everything but that's a stupid excuse not to try our best.

[Sarastro]

the movement of the planets is in part a mathematical expression

But who has expressed this expression? Planets themselves? The Sun? A squirrel? Initially it is not a mathematical expression, it is just the things are. Mathematics is the way we try to describe it, and it works. Sometimes it does not work, and we seek other ways...

Physics, what I have to study as a math major as well, is so quite different... Even mathematics in physics and in pure math is different. For instance, in physics "infinity" is considered to be the approximate number of stars in the Universe and is placed nowhere. In mathematics "infinity" is NOT a number and is a mathematical idealization, as well as the differential and the concept of limits. There are closed and open forms, imaginary numbers that can not exist physically (square root of negative one) or the base of natural log.

[71 dB]

My first calculus teacher (her husband is a conductor, btw!) told once that when mathematicians try produce some music using math, it is usually a very lousy kind of music.

Of course it's lousy because they use WAY too simple (but somewhat deceptive) algorithms. Science does not know yet enough about human brain. Maybe 50 years from now it does (and computers are powerful enough to run brain simulations in realtime). Computer composer sounds funny to us as an concept but just imagine how much the world will change during the next 50 years! Computers and robots will be able to do almost anything so why not compose too!

We must always be careful about what we are actually doing with math and computers and make the right conclusions.

[knight66]

But who has expressed this expression? Planets themselves? The Sun? A squirrel? Initially it is not a mathematical expression, it is just the things are. Mathematics is the way we try to describe it, and it works. Sometimes it does not work, and we seek other ways...

No, I don't see it this way. Of course the planets do not not move in a specific way in order to describe maths. But they move according to physical laws and those laws are mathmetical in part. Sorry, I stick by my remark that maths is there and is being discovered, not; an invented language or a mere descriptor.

Mike

[Sarastro]

just imagine how much the world will change during the next 50 years!

Yeah - global warming, air pollution, energy depletion, robot revolution, possible vanishing of the human race.

Actually, all the mathematicians I know who like classical music say that math people should not touch music, as well as computer people. Music is not just a set of meaningless sounds but consists of whole pieces that possess an idea. Putting it into calculations is killing anything live in it. And I don't think computers can ever replace the human soul and brain. The difference is that the computer has a finite set of operations, and it is not organic; human intelligence depends on the area of the gray matter of our brain, it may increase as well as decrease. Of course it can not increase infinitely, but even very small changes bring immense results. And the ability of living creatures is to learn from experience and feel other living beings. How can you imagine electronic music touch a living soul? There have always been tales about that, such as a tale about a mechanical and a live nightingale - for instance in Stravinsky's Le Rossignol (I'm being on topic ).

[Sarastro]

that maths is there

Well, gravity is there, not math. What you are saying is that Jupiter was always there, I mean, called Jupiter, and we just discovered its name in English. And this also implies that we did not invent English - we discovered it, it was always there, as well as other languages.

[knight66]

Well, gravity is there, not math. What you are saying is that Jupiter was always there, I mean, called Jupiter, and we just discovered its name in English. And this also implies that we did not invent English - we discovered it, it was always there, as well as other languages.

No that is not what I am saying.

[71 dB]

For instance, in physics "infinity" is considered to be the approximate number of stars in the Universe

You have misunderstood the idea. The estimated number of stars in the universe is 7*10^22. In many calculations we can use infinity instead of this very large number because the result is the same with "more-than-enough" decimals. That's the main difference between math and physics. In math we can (in most cases) use infinite precision while in physics we have to play with finite presicion because of physical measurements of finite presicion.

In mathematics "infinity" is NOT a number and is a mathematical idealization

However, math is able to set rules for infinity. In many cases infinity is not a problem at all. For example the infinity sum 1+1/2+1/4+1/8+...+1/2^n = 2, n E R -> infinity. 

There are closed and open forms, imaginary numbers that can not exist physically (square root of negative one) or the base of natural log.

Imaginary numbers does not need to exist physically to be mathematically meaningful.

[Sarastro]

No that is not what I am saying.

So what are you saying?
That tangent line to the graph, or integrals, or derivatives, or polar system of coordinates, or Fourier analysis, or infinite polynomials were always there? That right triangles filled the space? That linear transformation existed on the Earth long before?

You have misunderstood the idea.

No, I haven't. I am just in math camp. And I didn't get your point in showing a geometric series problem. What infinity? This geometric series converges absolutely and can be very easily explained on a piece of paper - geometrically.

[knight66]

Sarastro, I am not in interested in going there and wrestling about this. My interest is in the coinnection between music and maths, by all means have a discussion about what maths means divorced from music, but not with me.

Mike

[71 dB]

Actually, all the mathematicians I know who like classical music say that math people should not touch music, as well as computer people.

I agree. We need to know a lot more about human brain, but let's not deny the right to compose for the math/computer people, ok?

Music is not just a set of meaningless sounds but consists of whole pieces that possess an idea.

Someday computers will be able to come up with ideas.

Putting it into calculations is killing anything live in it.

What kind of calculations? The creative prosess of a composer is in fact about VERY complex calculations, so complex that we don't recognise them as calculations but as a creative prosess. There is no reasons why computers could not do that someday. It takes fuzzy logic, neural nets and other things to be invented but it's doable. We already can teach computers to do simple things like control a complex manufacturing process in a factory or recognice sounds/images. The commersical applications are increasing and so is the attraction to finance this kind of scientific research.   

And I don't think computers can ever replace the human soul and brain.

Soul does not exist. Computers should be build to complement human brain rather than replace.

The difference is that the computer has a finite set of operations, and it is not organic;

Infinite calculations can be do with finite set of operations. Simple operations are used to have simple algorithms which are used to build little more advanced algorithms which are... ...finally we have extremely sophisticated algorithms. If "organic operations" are needed then we simply make computers simulate organic structures.

human intelligence depends on the area of the gray matter of our brain, it may increase as well as decrease. Of course it can not increase infinitely, but even very small changes bring immense results. And the ability of living creatures is to learn from experience and feel other living beings. How can you imagine electronic music touch a living soul? There have always been tales about that, such as a tale about a mechanical and a live nightingale - for instance in Stravinsky's Le Rossignol (I'm being on topic ).

What do you mean by electronic music here? Is acoustically played (by human players) but computer composed music electronic?

Today computers can't compose like humans. Someday computers are intelligent "creatures" with self-consciousness. They learn new things by observing their environment. They feel sadness, joy, fear, etc. When computers are that capable (human), there is no reason why they could not compose like humans.

[71 dB]

No, I haven't. I am just in math camp. And I didn't get your point in showing a geometric series problem. What infinity? This geometric series converges absolutely and can be very easily explained on a piece of paper - geometrically.

The geometric series showed that infinity isn't a problem for math. Not a number? So what?

[DavidW]

However, saying that 'it's the beats', is like telling me that maths is about numbers.

Math is not about numbers actually.  In elementary mathematics courses that might be true, but abstract mathematics is much broader in scope.