View Full Version : Mathematics and Music

Robert M
07-29-2005, 02:53 AM
Tonight, while following links for DX-7 presets, I stumbled across this page:


Once there, you will find a link to a book called Mathematics and Music, written by David J. Benson. The author is offering it online as a pdf file, and he says a hard-copy version will be published sometime in 2006.

My own math skills are woefully inadequate for grasping the full measure of this book, but I thought it was be something that other folks on the forum might be interested in (assuming this link hasn't previously been shared, of course).

Robert P
07-29-2005, 04:34 AM

That's a great link.


07-29-2005, 06:54 AM
So did you come across some good DX7 presets? I would be interested in those links also...

07-29-2005, 09:35 AM
wonderful stuff.

I'll let you know more about the section on symetries which is my area of expertise. It appears only a survey.

Mathematics is VERY helpful for me. But I found you need to make your own choices on what is actually useful. And how that fits into your other techniques.

The acoustic math stuff here is nice to look over

Robert M
07-29-2005, 05:27 PM
So did you come across some good DX7 presets? I would be interested in those links also...



Go down the page to the Patches section, and there are a few options to choose from.

Robert M
07-29-2005, 05:30 PM
Mathematics is VERY helpful for me. But I found you need to make your own choices on what is actually useful. And how that fits into your other techniques.

I tell myself if I had a better grasp of higher math, it would help with understanding music. I'll still read the book (as much as I can manage) for the basic ideas presented.

07-29-2005, 06:34 PM
Most of the math related to composition has been devoted to set theoretics. It's been worked to death. And it's easy to take it on face value, w/o thought that musical and cognitive understanding isn't necessarily based on being mathematically perfect. In fact, it may be the opposite, in that people are using generalizations when they listen to music not perfect details.

The mathematics for set theory is rather simple. Don't believe what you hear when people talk about how dogmatic and ~~~~like composers of the 60-70's were and those that use it are out of date blabla...Like anything...keep an open mind as gems are found where fools tend to tread. There are aspects that are truly astonishing and useful. And there are more specific parts that are useless and impractical. You can take parts of it in your own way. Finding books on these theories is very hard to do. And most of them are out of date or never get beyond an old school understanding. When you get into the advanced stuff, there is a lot of very powerful - some useful and some not so useful - materials. It just takes self discipline to read through it and filter out what was there for peer review and what is there that is practical from a composers perspective. University dissertations are another great place to get information. Dont believe everything you read. And dont discard everything you read.

chet reinhardt
08-04-2005, 03:12 AM
I've been meaning to post something here for a while but the topic has its quirks. I would conjecture that there are more types of mathematics than there are types of music. And the treatments of the different topics are subject to a great deal of variation. So here is a short list of books you might (or might not) want to look at.

Two good qualitative discussions of math and the pursuit of mathematics that have musical applications are: Chaos by James Glieck and Digital Mantras: The Languages of Abstract and Virtual Worlds by Steven Holtzman. Really quite nice.

I like the recreational mathematics of Smullyan and Martin Gardner. Lots of fun.
( see http://en.wikipedia.org/wiki/Raymond_Smullyan) .

Logic programming and automated reasoning might be of interest to some (Automated Reasoning: Introduction and Applications by Wos, Overbeek, Lusk, and Bolye or perhaps Logic for Problem Solving by Kowalski). I personally liked this stuff a lot.

Cliff Jones wrote an interesting book called Software Development: A Rigorous Approach which provides some interesting ideas on denotational semantics influenced methods that can also be used to specify complex systems such as compositions. Or you might find Object-oriented Modelling and Design to be more interesting or useful (Rumbaugh et al) or perhaps Object-oriented Analysis and Design by Booch. Some might say that this is not really math but there is a lot of math that is less clearly specified.

If this stuff is too elementary for you take a look at Topoi: The Categorial Analysis of Logic by Goldblatt. Categorical purists prefer Categories for the Working Mathematician by Maclaine but I like the fact that Goldblatt defines his notations and starts out at set theory. Category theory was constructed to provide an alternative to set theory for the foundation for mathematics.

There is of course a great deal more math of varying sorts and a great many ways of using it to assist in composition (or to using music as a way of understanding mathematics).

Best wishes


08-04-2005, 09:24 AM
I have read most of the books you post here. I would say nothing really demonstrates any practical use...including OOD or OOPLA..which isn't so much mathematical but "crisp" logical.

Some of the theories of sets as described by Cone, Morris, Starr, Forte, Rahn, Babbitt would be a good place to start, and probably end too. I'd say there are only a few practical abstracts that are there.


The problem with mathematical approaches is that is not necessarily how the brain "generalizes" when it understands musical abstraction. Fuzzy based systems and expert systems gear toward this world, but fall short. Ambiguity is a very nice thing when it comes to music...which is why mathematics, with all its discreteness is an impedance mismatch.

Lerdahl is worth reading....although difficult


chet reinhardt
08-04-2005, 11:26 AM

To each his or her own.

I''ve made extensive use of scale-invariant self-similarity with respect to the large scale structure of compositions and the Glieck book is certainly sufficient to be the catalyst for that realization.

And even if one skips the equations and just reads the text Goldblatt provides a way of freeing the mind to see relations between structures (considered quite abstractly) that I think has helped me see analogies between quite disparate compositions that I've found useful. And generally I find it more fruitful to think in terms of categories and morphisms between them than in terms of sets.

It is true that I never got around to constructing the system for generating compositions from compact specifications that I once envisioned. But the hard part of that for me was reading and writing midi-files. If I had something that could translate between Prolog clauses and midi I might brush the Cantor dust off some of my old files and see what I can do with the idea.

But generally I think that for a great many people what Herb Simon called the laws of qualitative structure (the ontology of the domain and some general ideas about what sort of things one can do with these ontological objects) is quite sufficient to spark new approaches.

It is true that the approach reflected in my references veers towards the logical and the qualitative rather than in the numerical quantitative direction but I find it to be the more fruitful direction for me. And of course one important aspect of foundational studies is that one can be translated into the other.

As I said before to each his own.

By the way, I've been following your latest set of posts with great interest. You are making a great contribution to the discussion. Keep up the good work.

Best wishes


08-04-2005, 11:34 AM

I, myself, have gone down a few tooooo many roads with the theoretical. A good dose of Cage or Feldman would probably do us both a bit of good, at least oin the weekends. Although, it's still extremely hard for me to write in the NOWness....without my interval vectors, and my interval cycles and and....

chet reinhardt
08-04-2005, 08:04 PM

As you say, it is all about balance and what works for the individual composer in the here and now.

Best wishes


08-06-2005, 01:57 PM
Hi everyone,

This topic is of interest to me for an uncommon reason. I have two undergrad degrees, one in mathematics, and one in music. While I clearly see the application of mathematics to explaining how a string vibrates, or analyzing various forms of scales, to my mind the union is quite different. Since we are mostly musicians here, I'll touch on math for just a second. Part of math is a way to model reality, as best we can know it. Math (as we know it) was born in the pursuit of very practical problems, though now the problems modern mathematicians work on do not lend themselves to immediate application in the "real world". In my humble opinion, at its heart math is not like "math" as most people think of it. Math is much more like a child's game, which is tons of fun, while you try this and try that in pursuit of solving a puzzle. A few similarities with music to me include:

It has its own notation
It is commonly done alone (think of practicing!)
It can be used to convey a very wide range of ideas
Some of the best kids can out perform some of the best adults
It requires a certain abstract, broad, wide open view of the world
It is eminently fun, but the general public does not see it that way
It requires a great deal of creativity, literally the more the better

I can go on and on, but you get the point. While a little advanced calculus never hurt anyone, you don't need it to appreciate that math and music are very much related. Your well developed talents in music would transfer quite well to math (though not without practice) and I suspect the trip would be much more rewarding than others might believe.

08-06-2005, 02:44 PM

Gee...And here I thought music was related to drawing and painting...


hmmm I was wrong...
I guess these are mathematical terms though riiiiiight??? :p

10-12-2005, 10:12 PM
check out Joseph Schillinger: http://www.schillingersystem.com/

chet reinhardt
10-20-2005, 12:02 AM

I downloaded and read Eric's paper on Schillinger and read some of the posts at the website. Reminds me in some ways of constructive mathematics and of Bourbaki reconstructing mathematics using set theory.

I don't know yet how far I'll be pursuing it but it sparked some interesting speculations and I constructed a logical conjecture with respect to the Schillinger Determinant which amused me.

Thanks for the reference.