Thursday, January 26, 2006

Is there any merit or purpose in computing on discrete musical objects?

SVMs project the data from the original domain into some (possibly infinite) Hilbert space and then use a REAL-VALUED function f: X -> R for discrimination, so that if f(x) >= 0 then x is labeled positive class, and negative otherwise. f(x) is a linear discriminant in this projected Hilbert space (aka feature space), and thus can be expressed as an inner product in the feature space, so that f(x) = <> + b. Now skipping lots of details, in the original space, it can be shown that if your function is of the form f(x) = sum(i=l..m)a[m]y[m]k(x[m],x) + b and the kernel function k has certain properties you don't even need to know what is the mapping and it is guaranteed that it exists and that there's some inner-product space in that feature space. The codomain of this kernel function is either R (the reals) or C (complexes). (Be careful about the word "kernel" here. I still haven't figured out how the "kernels" in SVMs relate to "kernels" in algebra.)
So I thought: "Ok, if I can express musical objects as an inner-product, then I can let these SVMs compute directly on music". HOWEVER, I had a problem with the musical objects. Note that k: (X,X) -> R and k(x,x') = <> (phi is the projection from X to the Hilbert space). The inner product space is a vector space. A vector space is a module over a ring. In SVMs the ring is the field R, the reals. For this module, I can probably model my musical objects in the input domain X as an abelian group. My problem is the scalar multiplication of the module R x X -> X. How can I have the reals multiplying inherently discrete musical structures resulting in discrete musical structures and still maintain vector and scalar distributivity? For example, what does pi x {0,4,7} (a chord) mean?
Then I assumed that it wouldn't work this way. I began to think that perhaps I could form a free module over an ordered ring instead of a vector space, and create an inner-product using Milnor's definition of
inner product, which doesn't require the bilinear's codomain to be a field. That can take care of the domains in the SVM theory. But the theory only has R as the codomain. That's the assumption. I could try to see if I could enhance the SVM theory to allow Z as the codomain. Specifically, that amounts to taking a serious look at Mercer's theorem and finding alternative ways for the kernel function to fulfill certain properties assuming that the codomain is a commutative ring. But before going down that quest, I wonder if this is of any interest to the research community. Is it of interest to find ways to compute directly on musical objects? For instance, I remember how Elaine Chew projects the musical objects in that spiral in some continuous space, with some geometric calculations there. I remember David Temperley's center of gravity in the line of fifths, which again is non-discrete. I'm more interested in the discrete representations. I'll understand Mazzola after three reincarnations as a math PhD student, four as a musicologist and two as a philosopher (note: reincarnations somewhere in Europe, somewhere between France and Switzerland). But it seems that the foundation is so general that the musical objects may be represented by anything.
I see the nice discrete structures in the works from Mazzola, Andreatta, Forte and Noll. For me they are just there waiting for machines to compute directly on them by glueing their theories. But I still can't answer whether there's any merit, purpose or any benefit in doing that. I'm looking mainly for pragmatic answers to this question rather than philosophical answers (if possible). Comments are also welcomed.

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