I see mathematics in "music" as occupying three distinct branches that are loosely coupled. I call these the three Ms, the model, the method, and the music (for want of a better term beginning with M :)
The model posits declarative knowledge firmly grounded in physical reality. It's about real physical vibrating structures and signals that are "real" in the common usage of the word. It's largely a branch of fluid dynamics, or just general dynamics if you consider electronics as a production mechanism. This is good old school engineering math with a huge dollop of classical physics. Here we concern ourselves with such things as propagation, waves, superposition, turbulence, boundary conditions and reflection, diffraction and refraction...
The method is the imperative branch. As C Henry indicates, operator theory, or the mappings/transformations of spaces onto other spaces is the highest view of this. We concern ourselves with parametric equivalences, identities and so on. For the mostpart it's very theoretical and personally, quite beyond my mind. We deal with matrices a lot. But it's very important from a computational perspective, it's the "how to" part. For example, we know what a vibrating string is and what it does, but one may choose a number of methods to implement the model, scanned synthesis, FM, Karplus-Strong, modal/tensor and so on. Each method picks a salient feature of the model and helps us to realise it in an computationally expedient way.
The music is a whole different branch that links the two together. It says what those salient features are, at least to human beings made of wetware neural networks educated by learning, examples, and context. It is the psychoacoustic/cognitive aspect of signal mathematics. It basically says why a certain method fits a given model for *PRACTICAL* uses. To do so we leverage a lot of statistics and empirical studies. Those practical uses (for the production of sound) might depart a lot from the practical parts of another discipline (say building bridges that don't collapse).
Making music/sound involves all of these branches. As C Hubbard indicates, here is the gestalt. We design instruments/synthesisers using methods which in turn implement models which in turn correspond to internalised music (sound).
From a creative viewpoint a good understanding of each, explicit or implicitly
empowers the artist because the quicker one can move from an imagined (target) sound to its realisation as a signal in the real world.
This is of course all "philosophy", which may engage your interest. In my own work I try to avoid maths for it's own sake, at least in the studio it all happens at a subconscious level until something doesn't work and I have to fall back to the textbooks to figure out why. In the "conscious" process I "see" the signals
- geometry plays a big part. Most "practical" maths in the studio
is basic mental arithmetic, everything else is abstracted into black boxes with well known behaviour, it's the only way to manage complexity while working.
I hope that gives you some ideas for where to sail your boat.
Andy
ps I was flicking through "Mathematics and Music" - Assayag &c just this morning in Borders, it looked very good from a composers POV.
On Tue, 1 Aug 2006 14:23:12 -0500 "Charles Henry" czhenry@gmail.com wrote:
To the point: I'm hoping to find a topic that combines "computer music" with mathematics, or maybe more correctly; a mathematical topic on a fitting level, that has (some) relations to computer music (in a broad sense).
If you're interested in synthesis/DSP, I recommend to study operator theory and numerical analysis. All of your transforms come from analyzing differential/integral operators, their eigenfunctions/values, and inverses.
Music theory itself is pretty mathematical, but it's practically a pointless endeavor to try to apply mathematics to "explain" music. You just wind up finding structures (like the lattices with minor and major thirds along the axes) with practically no relevance to the real experience of music. However, I believe that studying dynamic systems (non-linear operator theory) has some promise for physiologically based theories of music. I've been thinking about doing a math masters as well....and those are the two topics that I consider most significant.
Chuck
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