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I thought about posting this in the teaching section but you pianists have a lot more traffic and might have some excellent suggestions.

I'll be teaching a course called "Contemporary Mathematics" this summer with a few prescribed topics but a few weeks on the topic of my choice. Of course I choose mathematics in music.

Does anyone have any suggestions as to particular pieces I should cover? I want to highlight things that even people with zero background will be able to see. I will be looking for things to point out to the students like visual patterns in the sheet music, tempo and key changes, polyphony, golden sections, Fibonacci sequences, etc.

On my short list: A Musical Offering, Chopin Etude Op. 10. no. 1, and possibly Cum Sanctu Spiritu or Et Resurrexit from the B Minor Mass so far.

Thanks in advance.

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Interesting! And I think it might be hard for us to come up with specific pieces, except to say that just about anything will do. smile

Originally Posted by dsch
....golden sections, Fibonacci sequences....

I think most people don't know about those; I know about them only sort of by accident.

About "Fibonacci": Just a thought, and I don't know if this is even something you'd be able to use anyway -- heck, I don't even know if it's true. ha
But, I think I've noticed that when a piece has sections or movements with different "speeds," the different speeds are often related in a Fibonacci way.

About "golden section": I've seen various things about what this means. When I first came across it, the idea was a 10-to-7 ratio of two numbers -- such as dimensions of a room or space, and I actually used it when arranging areas of a room. But thereafter, what I've seen is that it's just the same as Fibonacci numbers, which isn't the same as 10:7 at all.

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There are few endeavors as futile as trying to apply mathematics to esthetics.

I would suggest looking into things which can be looked at mathematically. You could look at a score, and count the dimensions: Pitch, dynamics, tempo, rhythm, etc. You could look at temperaments. But trying to analyze pieces of music mathematically is not likely to get anyone very far.


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You're on the right track with The Musical Offering. There's a lot of math in Bach's music. Do a little research into the Goldberg Variations and you'll see what I mean. If I remember correctly, some of Debussy's works (I couldn't tell you which now, but La Mer comes to mind) are thought to be based on the golden ratio and one of Bartok's works (once again I don't remember which) is thought to be based on Fibonacci numbers.



"And if we look at the works of J.S. Bach — a benevolent god to which all musicians should offer a prayer to defend themselves against mediocrity... -Debussy

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Cool course. But can you be a little specific about what "contemporary" means here, since I imagine we're not talking about "areas of active research".

Personally, I have never been a big fan of so-called golden ratio for anything, but especially in music, where all examples I've seen are pretty contrived.

To me, the only truly mathematical aspect of music is about harmonics (closer to physics I suppose) and as a result how why western music have 12 notes in an octave. And corollaries like how one tunes a piano, violin, etc.

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Originally Posted by BDB
There are few endeavors as futile as trying to apply mathematics to esthetics.

I would suggest looking into things which can be looked at mathematically. You could look at a score, and count the dimensions: Pitch, dynamics, tempo, rhythm, etc. You could look at temperaments. But trying to analyze pieces of music mathematically is not likely to get anyone very far.


Music is completely intertwined with math, whether you believe it to be or not.



"And if we look at the works of J.S. Bach — a benevolent god to which all musicians should offer a prayer to defend themselves against mediocrity... -Debussy

"It's ok if you disagree with me. I can't force you to be right."

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Originally Posted by stores
Originally Posted by BDB
There are few endeavors as futile as trying to apply mathematics to esthetics.

I would suggest looking into things which can be looked at mathematically. You could look at a score, and count the dimensions: Pitch, dynamics, tempo, rhythm, etc. You could look at temperaments. But trying to analyze pieces of music mathematically is not likely to get anyone very far.


Music is completely intertwined with math, whether you believe it to be or not.


Yikes. Well, this thread is heating up. New topic here on pianoworld in my short tenure here!

Your statement is so general that it cannot be refuted--since math is, of course, more or less intertwined with everything. So to make this argument meaningful at all, you have to be very specific what aspect of music and what aspect of math you're talking about.

We might yet agree on this but first we need to straighten out what we're even discussing.

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I forgot, there's Schoenberg. And lots of modern modern music. Although in a way it's "cheating" to find mathematical aspects in music when the music was engineered mathematically to begin with.

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although yes this is cheating smile

Last edited by Adam Coleman; 04/25/11 10:28 PM.

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If it's an advanced mathematics course, you could have students explore the musical applications of Markov chains:

https://ccrma.stanford.edu/~jacobliu/254report/
http://peabody.sapp.org/class/dmp2/lab/markov1/

Also, there's much to be said about how Schenkerian Analysis relates to fractal geometry.

(And of course, Godel Escher Bach should be the required textbook!)


"If we continually try to force a child to do what he is afraid to do, he will become more timid, and will use his brains and energy, not to explore the unknown, but to find ways to avoid the pressures we put on him." (John Holt)

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Originally Posted by Kreisler
If it's an advanced mathematics course, you could have students explore the musical applications of Markov chains:

https://ccrma.stanford.edu/~jacobliu/254report/
http://peabody.sapp.org/class/dmp2/lab/markov1/

Also, there's much to be said about how Schenkerian Analysis relates to fractal geometry.

(And of course, Godel Escher Bach should be the required textbook!)


I like this post! We need a "Like" button.

So much great music is so tight structurally that to me "music engineering" is what I perceive. In fact these days I have trouble getting out of this mindset.

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Originally Posted by Lingyis
I forgot, there's Schoenberg. And lots of modern modern music. Although in a way it's "cheating" to find mathematical aspects in music when the music was engineered mathematically to begin with.


Messiaen's "Mode de valeurs et de tonalites" is an even better example, because every aspect of the music is serialized then and used in permutations. And the net effect of exploring all these possibilities is a piece that is maddeningly repetitive. Go figure.

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It should be mentioned that music lends itself better to certain kinds of math than others. Simply looking for examples of arithmetic isn't very interesting. Discovering a Golden Mean here and there or a hidden Fibonacci sequence trivializes the connection.

Music lends itself much better to discussions of patterns and transformations.

Here's an example of a vague connection I find somewhat intriguing:

In mathematics, one often finds that two things that seem different on the surface are actually related. This happens in trigonometry (the relationship between a circle and a triangle, both of which work very neatly with trigonometric functions) and in calculus (the relationship between derivatives and integrals, which are two sides of the same coin.)

In music, I find that this is much like the relationship between chords and scales - especially in jazz music. To a jazz musician, a D minor 7th chord and a D dorian scale are the same. The only difference is how they're used. In a chord, you stack the notes by thirds:

D-F-A-C-E-G-B

In a scale, you line them up in order:

D-E-F-G-A-B-C

But the notes themselves are exactly the same.

A neophyte in jazz sees Dm7 and D dorian as two different things, but to the expert who's looked a little deeper, they're just two ways of thinking about the same thing.

Back to mathematics - beginning calculus students are often taught that a derivative is the slope of a curve at a point (yeah I know, actually the slope of the line tangent to the curve at that point, but I digress...), and they're taught that an integral is the area under a curve. But looking a little deeper, we see that the integrals and derivatives are the same thing, simply in opposite directions. (And some texts do teach integrals as "antiderivatives."

It's these kinds of patterns and ways of thinking about things that links mathematics and music in a meaningful way. Not counting measures and performing simple arithmetic.

In contemporary music, set theory starts to become more important, especially in the work of Lewin, Forte, and Babbitt. (And you can see aspects of this in everything from Schoenberg to Barber to Messiaen to Crumb.) Also, Neo-Riemannian analysis has much to say about applying some mathematical techniques to the analysis of tonal music, but that's likely beyond the scope of what the OP has in mind. (Understanding Neo-Riemannian analysis requires a basic knowledge of harmony and counterpoint.)


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Originally Posted by Kreisler
It should be mentioned that music lends itself better to certain kinds of math than others. Simply looking for examples of arithmetic isn't very interesting. Discovering a Golden Mean here and there or a hidden Fibonacci sequence trivializes the connection.....

I don't agree at all, Kreisler. I find the simple arithmetical things fascinating, and I was especially interested to notice what I said about the apparent Fibonacci nature of tempo relationships in a piece. One could also look at things like the relative lengths of sections of a piece, like "theme A" vs. "theme B," and exposition vs. development vs. recap, or movements of sonatas and symphonies -- and I'd guess that there would be some interesting arithmetical patterns.

As you said, it does depend on how advanced the math course is. But I would think the simple arithmetical aspects is the best place to start, for any level.

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It's a freshman-level course at a 4-year college but it covers many diverse topics. Subjects I am required to cover include everyday applications of logs, exponents, groups, graph theory, voting/apportionment, statistics, and number theory. The rest is up to me.

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Originally Posted by dsch
It's a freshman-level course at a 4-year college....

IMO then it's clear: Go with the simpler arithmetical things.

If you see that they're interested enough, then maybe after that you can get into more complicated things like what's been mentioned. But otherwise, I think you'll be losing and snoozing them, and they won't have any feel for why you're talking about what you're talking about.

Lingyis mentioned harmonics, and I think that would be something to think about including. I think it's a pretty striking thing to many people that tones with certain audible relationships have such neat ratios of frequencies. In fact, I'd go so far as to say that it would be a major omission NOT to include something about that in what you're thinking of doing.

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Maybe this as something interesting?
Enumeration and Construction in music theory
http://www.kfunigraz.ac.at/~fripert/papers/musik.pdf

I once saw another paper wich defined some transformations on melody lines/chords (inversions or transposition for example), and then talked about the group structure of these transformations, and there was also neat thing with a torus...
I can't for the life of me find it again though... that paper I linked kind of covers it, but the paper I'm thinking of was much cleaner and simpler.


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Originally Posted by dsch
It's a freshman-level course at a 4-year college but it covers many diverse topics. Subjects I am required to cover include everyday applications of logs, exponents, groups, graph theory, voting/apportionment, statistics, and number theory. The rest is up to me.


So pretty much everything in this thread is fair game.

My personal favorite is, again, why did the 12-tone system "win"? That's worth just about 1 lecture. The math is nothing more than arithmetic.

4 weeks 12 lectures so 11 more lectures to go! I'm curious what topics you end up deciding. 4 weeks of music related topics seems heavy though.

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Originally Posted by Lingyis
....My personal favorite is, again, why did the 12-tone system "win"? That's worth just about 1 lecture.....

Well, it did and it didn't.
Mostly did, I admit. smile

But lots of stuff is "pentatonic" -- most kids' jingles and lots of folk stuff.

Now if he can come up with some mathematical explanation of why the pentatonic scale is the same as the black notes on the piano.... smile

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Can I "like" the Godel Escher Bach suggestion as well. Your students will enjoy this book I bet laugh

Another more trivial example could be Elliot Carter's metric modulation. Mathematically it's nothing more than fractions but as a musician I found it an interesting possibility. Modulating the rhythm using simpler fractions (so that the listener can perceive the relationship between tempi) is an alluring compositional tool.

Originally Posted by kreisler
Also, there's much to be said about how Schenkerian Analysis relates to fractal geometry."
I really should get a book on Schenkerian Analysis - I think I would enjoy exploring. ?? fractal geometry - you're kidding right? It's pretty hard to imagine where this would connect with musical analysis - there aren't that many levels...
If Kreisler was teaching a course around here I'd go!


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