One way to think of a topos is as some kind of fancier category of sets.
One way to make sets fancier is to consider sheaves of sets on some topological space; this is mentioned in Zhen Lin's answer. One can think of a sheaf of sets as a set which varies and twists over the topological space, but it seems that this could be a bit painful for you to think about with the background you're coming from.
Another way to makes sets fancier is to put actions on them. So:
Let $G$ be a finite group, and consider the category of all finite $G$-sets,
i.e. all finite sets equipped with an action of the group $G$. (Morphisms are maps between sets that are compatible with the $G$-action on source and target.)
This is an example of a topos, which is pretty small in your non-technical sense.
The subobject classifier is the two element set $\Omega$, with trivial $G$-action, and with
one of the two points distinguished. If $X$ is a $G$-set, and $Y$ a $G$-invariant subset, then we have the morphism $\chi_Y: X \to \Omega$ which maps
all of $Y$ to the distinguished point in $\Omega$, and all of $X\setminus Y$ to the other point. This morphism "classifies" the subset $Y$. (More precisely, $Y$ is the preimage of the distinguished point of $\Omega$ under $\chi_Y$.)
If we take $G$ to be the trivial group, then we just recover the topos of finite sets.
Let's instead take $G$ to be the cyclic group of order two, say $G = \langle 1,\tau\rangle,$ with $\tau$ of order two. Then to give a finite $G$-set is just to give a finite set $X$ equipped with an involution (i.e. permutation of order two) $\tau$.
Now in addition to the two-element set $\Omega$ with trivial $G$-action, which (once you designate one of its points as being the distinguished one) is the subobject classifier, you could think about the two-element set $\Omega'$ equipped with the non-trivial involution, which switches the two points.
We have already seen that $Hom_G(X,\Omega)$ (I write $Hom_G$ for maps preserving the $G$-action) is equal to the collection of $G$-invariant subsets of $X$.
What about $Hom_G(X,\Omega')$? You could try to compute this (of course it will depend on the particular $G$-set $X$). It's not particularly exciting, but it might help you get a feeling for the difference between the subobject classifier and some other objects, such as $\Omega'$.
There are many mathematical aspects of music, but there also many non-mathematical aspects that are inherently cultural.
As an example of a "math aspect", take a look at harmonics: we like hearing sounds that produce the "same" frequencies, and these are just integer multiples of the basic frequency that is being played. Another example would be that of equal temperament, which creates a semi-optimal distance between notes, such that the possible harmonies are maximized.
That being said, the actual number of tones in an octave as well as the choice of scale are completely culture dependent - Western ears are used to 12 tone octaves and certain scales but not others, in what seems to be a rather arbitrary choice. So I think it's fair to say that music is definitely not "just math" - it's very strongly tied to the culture we were brought up in rather than to some mathematical formula.
Best Answer
the theory has been applied in my composition software presto for atari (google it, it is still available for PC emulation), and for the universal software rubato for composition, analysis, and performance. These software were also used to compose music, see mazzola's homepage www.encyclospace.org, and go to CV there. Best, Guerino Mazzola