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.

## Thursday, January 26, 2006

## Saturday, January 21, 2006

### What is music? - wikipedia answers that (or does it?)

From Wikipedia, on the definition of music:

"[there are] those that seek a platonic or quasi-platonic ideal of music which is not rooted in specifically physical or mental terms, but in a higher truth. "

[...]

"The platonic ideal of music is currently the least fashionable in the philosophy of criticism and music, because it is crowded on one side by the physical view - what is the metasubstance of music made of, if not sound? - and on the other hand by the constructed view of music - how can one tell the difference between any metanarrative of music and one which is merely intersubjective? However, its appeal, finding unexpected mathematical relationships in music, and finding analogies between music and physics, for example string theory, means that this view continues to find adherents, including such critics and performers as Charles Rosen and Edward Rothstein."

"[there are] those that seek a platonic or quasi-platonic ideal of music which is not rooted in specifically physical or mental terms, but in a higher truth. "

[...]

"The platonic ideal of music is currently the least fashionable in the philosophy of criticism and music, because it is crowded on one side by the physical view - what is the metasubstance of music made of, if not sound? - and on the other hand by the constructed view of music - how can one tell the difference between any metanarrative of music and one which is merely intersubjective? However, its appeal, finding unexpected mathematical relationships in music, and finding analogies between music and physics, for example string theory, means that this view continues to find adherents, including such critics and performers as Charles Rosen and Edward Rothstein."

## Friday, January 20, 2006

### Music and Mathematics course

Here's the Syllabus of an Music and Mathematics course given by John Rahn. Kool stuff. Very kool stuff. All these algebraic structures are there waiting for us to compute directly on them.

http://faculty.washington.edu/jrahn/5752004.htm

http://faculty.washington.edu/jrahn/5752004.htm

## Thursday, January 19, 2006

### The research community doesn't understand the customer

Folks in the Machine Learning 'research community' could think a bit more about their 'customers'.

It's a generic problem in academic research. In private companies you learn fast that the customer is king. In the 'academic community', though, they are more interested in their own problems and tend to ostracize the applications and users.

For example, Machine Learning helps users classify/rank their data. It should help the users understand their problem domain. There are two areas that I don't see evolving in Machine Learning.

First in feature selection, their ultimate goal is to optimize the set of attributes for classification. But in doing so, the process can reveal a lot about the problem domain itself. I can't find much interpretation in the literature of how to utilize the result of feature selection to interpret the original problem domain. For instance, the literature uses a lot of Information Theory, but it doesn't show how much information the attributes convey

The second symptom of the problem is again the assumption that you are always dealing with the Euclidean space. Mercer's theory in SVM assumes a vector space. They definitely should work with better abstract algebra and category theory to redefine their theory, so that the machines can manipulate other objects

It's a generic problem in academic research. In private companies you learn fast that the customer is king. In the 'academic community', though, they are more interested in their own problems and tend to ostracize the applications and users.

For example, Machine Learning helps users classify/rank their data. It should help the users understand their problem domain. There are two areas that I don't see evolving in Machine Learning.

First in feature selection, their ultimate goal is to optimize the set of attributes for classification. But in doing so, the process can reveal a lot about the problem domain itself. I can't find much interpretation in the literature of how to utilize the result of feature selection to interpret the original problem domain. For instance, the literature uses a lot of Information Theory, but it doesn't show how much information the attributes convey

**conditionally**on the class.The second symptom of the problem is again the assumption that you are always dealing with the Euclidean space. Mercer's theory in SVM assumes a vector space. They definitely should work with better abstract algebra and category theory to redefine their theory, so that the machines can manipulate other objects

**directly**.## Tuesday, January 10, 2006

### Rio de Janeiro - Elza Soares

Rio de Janeiro, gosto de você

Rio de Janeiro, gosto de você

Rio, Rio de Janeiro, cortado por montanhas

Mares desespero cortado por favela bala fuzila fuzileiro suicida dominantes das alturas guerrilheiros capitais

Guerrilheiros capitais

Formado o aparteid social que provoca o vazio preenchido pela droga

Da sociedade que explora o consumo ali do cidadão

Rio de Janeiro, gosto de você

Rio de Janeiro, gosto de você

Rio, Rio de Janeiro, cortado por montanhas

Mares de maneiro

Rio, Rio de Janeiro, cortado por montanhas

Mares de maneiro

Metropolitano brasileiro que mandou

Gasta dinheiro carioca de origem

Trabalha no sinal mas não metralha

Trabalha no sinal mas não metralha

Rio de Janeiro, gosto de você

Rio de Janeiro, gosto de você

Premiado pelo mundo simpatia tá na cara do turista enlouquecido na beleza guanabara

Premiado pelo mundo simpatia tá na cara do turista enlouquecido na beleza guanabara

I love, I love, I love, I love you

I love, I love, I love, I love you

Rio de Janeiro, gosto de você

Rio de Janeiro, gosto de você

Rio de Janeiro, gosto de você

Rio, Rio de Janeiro, cortado por montanhas

Mares desespero cortado por favela bala fuzila fuzileiro suicida dominantes das alturas guerrilheiros capitais

Guerrilheiros capitais

Formado o aparteid social que provoca o vazio preenchido pela droga

Da sociedade que explora o consumo ali do cidadão

Rio de Janeiro, gosto de você

Rio de Janeiro, gosto de você

Rio, Rio de Janeiro, cortado por montanhas

Mares de maneiro

Rio, Rio de Janeiro, cortado por montanhas

Mares de maneiro

Metropolitano brasileiro que mandou

Gasta dinheiro carioca de origem

Trabalha no sinal mas não metralha

Trabalha no sinal mas não metralha

Rio de Janeiro, gosto de você

Rio de Janeiro, gosto de você

Premiado pelo mundo simpatia tá na cara do turista enlouquecido na beleza guanabara

Premiado pelo mundo simpatia tá na cara do turista enlouquecido na beleza guanabara

I love, I love, I love, I love you

I love, I love, I love, I love you

Rio de Janeiro, gosto de você

Rio de Janeiro, gosto de você

## Monday, January 09, 2006

## Saturday, January 07, 2006

### on kernels with integers as codomain and not a field, using Milnor's bilinear function

The problem of generalizing Kernels as functions with integers and not reals as codomain is in Mercer's theorem. The problem is that the theorem begins assuming that the Kernel function has the reals and complexes as codomain. Using Milnor's definition of a bilinear function that becomes an inner product space, and the non-degenerative restriction, I can construct a Kernel-like function that works with a free module instead of a vector space. But then, how can I be certain that it will have all the properties expected of a Kernel function for SVMs?

## Wednesday, January 04, 2006

### The Blank Sheet Desperation, The Student Nightmare

Among the worst despairs someone can have, one is called The Blank Sheet Desperation. It's the state of despair one faces in front of a blank sheet. It's because the person has all the ideas and has to start writing. The person has to have all the main ideas in order to begin. And how to encode correctly the ideas? Think of all the models that are isomorphic to the ones in your head.

This fear is similar to The Student Nightmare. The Student Nightmare is a dream like this: the student is in the classroom and doesn't remember there was a test. And then he/she gets The Test, and he/she knows no answer. It's a terrible nightmare. I read somewhere that it is The Student Nightmare because it happens to all students and there's no other nightmare that compares to it.

This fear is similar to The Student Nightmare. The Student Nightmare is a dream like this: the student is in the classroom and doesn't remember there was a test. And then he/she gets The Test, and he/she knows no answer. It's a terrible nightmare. I read somewhere that it is The Student Nightmare because it happens to all students and there's no other nightmare that compares to it.

### Céu, sol, sul

Eu quero andar nas coxilhas sentindo as flexilhas das ervas do chão

Ter os pés roseteado de campo ficar mais trigueiros como o sol de verão

Fazer versos cantando as belezas dessa natureza sem par

E mostrar pra quem quizer um lugar pra viver sem chorar

É o meu Rio Grande do Sul céu, sol, sul terra e cor

Onde tudo que se planta cresce o que mais floresce é o amor

É o meu Rio Grande do Sul céu, sol, sul terra e cor

Onde tudo que se planta cresce o que mais floresce é o amor

Eu quero me banhar nas fontes e olhar horizontes com Deus

E sentir que as cantigas nativas continuam vivas para os filhos meus

Pelos campos florindo e crianças sorrindo, felizes a cantar

E mostrar pra quem quizer um lugar pra viver sem chorar

Ter os pés roseteado de campo ficar mais trigueiros como o sol de verão

Fazer versos cantando as belezas dessa natureza sem par

E mostrar pra quem quizer um lugar pra viver sem chorar

É o meu Rio Grande do Sul céu, sol, sul terra e cor

Onde tudo que se planta cresce o que mais floresce é o amor

É o meu Rio Grande do Sul céu, sol, sul terra e cor

Onde tudo que se planta cresce o que mais floresce é o amor

Eu quero me banhar nas fontes e olhar horizontes com Deus

E sentir que as cantigas nativas continuam vivas para os filhos meus

Pelos campos florindo e crianças sorrindo, felizes a cantar

E mostrar pra quem quizer um lugar pra viver sem chorar

## Monday, January 02, 2006

### Schenkerian Analysis

It would be nice to have my system to compare the results of genre classification between one piece and one of its Schenkerian reductions. But I can't find any MIDI file or any encoding that has the Schenkerian version. If you find one, let me know!

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