Re: [tuning-math] Digest Number 1294

Hi Gene,

> If R+ is your topological space, then 0 isn't in it and isn't an
> accumulation point; if R is your space, then it might be; in fact it
> probably is. One way or another I think it needs to be made clear what
> is going on and what is allowed in a scale.

It's a while since I did much topology but isn't a discrete
subset one that doesn't have any accumulation points
that are members of the subset? Surely accumulation points
that aren't members of the subset are irrelevant
to the definition in this particular case.

So since no scale includes 0 it doesn't matter if you
think of it as a subset of R or R+ or if you use
logarithmic pitches (such as cents) or just
the ordinary relative magnitude to the 1/1
(i.e. based on hertz measurements).

Example. I would have thought that
1/1 1/2 1/3 1/4 1/5 ... 1/n ...
counts as a discrete subset of R.
But add in 0 as a member, and it is no longer
discrete.

I can understand the use of logarithms as
a way to avoid dense scales - if you want to
say that your scale isn't dense then it makes
sense to say that you mean that the logarithms
of the pitches don't form a dense set because if you don't take logarithms then
the 0 will be an accumulation point and though
not included in the set, it will still
make the set dense there.

I suppose the point is that a set can be dense
and still discrete. In fact, I think it would
be possible to have a set that is dense at
countably many accumulation points and
yet discrete and finitely bounded.
Just take the union of sets with accumulation
points at say 2^1/2, 2^1/3, 2^1/4 etc, then
no rational multiple of any of those
is in any others, use 2^1/n + 1/1, 2^1/n + 1/2, ... 2^1/n + 1/m...
as the nth set - and then the union of them
all is a discrete bounded set with countably many accumulation points.
Well I'd have thought so anyway...

> > It would be legitimate to restrict oneself to finite scales too,
> > since in practice every scale is finite too
> > because eventually they either go above
> > human hearing or below, or else the pitches
> > get too fine for anyone to ever discriminate
> > them.

> That simply makes life difficult to no purpose.

Could you not say the same about the restriction
to scales that aren't dense?
At least I can't see the purpose of that
restriction particularly. Except that
it delineates an interesting subset
of all possible scales of course.
While another interesting subset
is one with a single accumulation point
- or one with two accumulation points
etc.

If you don't allow dense scales, it seems
you may as well stop at the minimum
pitch distinction that a human with
best pitch sensitivity can reliably
perceive. So that was the sort
of reasoning there - and there would
be a purpose perhaps, if it was to do with
understanding human perception.

Robert

--- In tuning-math@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:
> Hi Gene,
>
> > If R+ is your topological space, then 0 isn't in it and isn't an
> > accumulation point; if R is your space, then it might be; in fact it
> > probably is. One way or another I think it needs to be made clear what
> > is going on and what is allowed in a scale.
>
> It's a while since I did much topology but isn't a discrete
> subset one that doesn't have any accumulation points
> that are members of the subset? Surely accumulation points
> that aren't members of the subset are irrelevant
> to the definition in this particular case.

If you take R+ with the relative topology, it is homeomorphic to R
under the log map, and a sequence tending to zero has no limit. If you
take it to be a set of points in R, then 0 is the limit of such a
sequence. I'm simply trying to say that isn't something the definition
is trying to rule out. Anyway it would be OK to say the notes must be
discrete, which just means there is a neighborhood containing only one
note of the scale.