Using tvals to represent primitive category systems

The porcupine tval <15 2| maps porcupine intervals onto 15-EDO. And
the meantone val <12 7| maps meantone intervals onto 12-EDO.

If we assume that your garden variety "meantone interval" is more than
just a combination of 5/4 and 81/80 and so on, but ends up gaining all
sorts of non-ratio based additional perceptual attributes from scales
and acculturation and such - e.g. that it has a "function" of some
kind and is an interval "category" - then the above is quite
significant, isn't it?

I suggest this is a great way to start modeling categories which fits
into Gene's homomorphism paradigm. So if temperaments are
homomorphisms from A -> B, A -> C, A -> D, etc, then elements in B*,
C*, D*, etc are the evolved version of vals. Rather than mapping JI
onto a generator chain, tvals let you map a schema for a "regular
category system" onto a generator chain, which is a tval. This
property seems to make them a really useful generalization of vals.
Note also that every tval itself then has a map back to a JI val as
well.

Of course, this doesn't require any assumptions about "how categories
actually work", only that they exist, and that the temperament defines
a mapping between them and ratios. One way to think of this approach
is to think of JI as some sort of "perfectly intoned category system"
which has exactly one category per ratio, and then to think of tvals
as different category systems which have more than one ratio per
category. I think it's an exciting prospect to start looking how
different tvals relate to one another and how tvals in differed
tempered spaces relate to one another. Perhaps we can reason through
some things that a 12-EDO listener ought to expect when moving to
19-EDO, or what a meantone listener should expect when moving to
porcupine.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> The porcupine tval <15 2| maps porcupine intervals onto 15-EDO. And
> the meantone val <12 7| maps meantone intervals onto 12-EDO.
>
> If we assume that your garden variety "meantone interval" is more than
> just a combination of 5/4 and 81/80 and so on, but ends up gaining all
> sorts of non-ratio based additional perceptual attributes from scales
> and acculturation and such - e.g. that it has a "function" of some
> kind and is an interval "category" - then the above is quite
> significant, isn't it?
>
> I suggest this is a great way to start modeling categories which fits
> into Gene's homomorphism paradigm. So if temperaments are
> homomorphisms from A -> B, A -> C, A -> D, etc, then elements in B*,
> C*, D*, etc are the evolved version of vals. Rather than mapping JI
> onto a generator chain, tvals let you map a schema for a "regular
> category system" onto a generator chain, which is a tval. This
> property seems to make them a really useful generalization of vals.
> Note also that every tval itself then has a map back to a JI val as
> well.
>
> Of course, this doesn't require any assumptions about "how categories
> actually work", only that they exist, and that the temperament defines
> a mapping between them and ratios. One way to think of this approach
> is to think of JI as some sort of "perfectly intoned category system"
> which has exactly one category per ratio, and then to think of tvals
> as different category systems which have more than one ratio per
> category. I think it's an exciting prospect to start looking how
> different tvals relate to one another and how tvals in differed
> tempered spaces relate to one another. Perhaps we can reason through
> some things that a 12-EDO listener ought to expect when moving to
> 19-EDO, or what a meantone listener should expect when moving to
> porcupine.
>
> -Mike
>

It's interesting that ultimately one always ends up having to deal with category theory if you push the math far enough. I just ran across this the other day, with pushouts and pullbacks in conjunction with something else I was working on. I always tried to avoid it, kind of back off and deal with something else, but it was there like a monster under the bed and I think a person just has to deal with category theory in the long run. Are you familiar with the term "Abstract Nonsense" in conjunction with Category Theory? Or do you mean "category" in some other sense?

PGH

On Fri, Jul 6, 2012 at 12:10 AM, Paul <phjelmstad@msn.com> wrote:
>
> It's interesting that ultimately one always ends up having to deal with
> category theory if you push the math far enough. I just ran across this the
> other day, with pushouts and pullbacks in conjunction with something else I
> was working on. I always tried to avoid it, kind of back off and deal with
> something else, but it was there like a monster under the bed and I think a
> person just has to deal with category theory in the long run. Are you
> familiar with the term "Abstract Nonsense" in conjunction with Category
> Theory? Or do you mean "category" in some other sense?

I meant category in the sense of learned perceptual interval
categories, such as "minor third" and so on.

-Mike