Re: [tuning-math] Digest Number 1295

Hi Gene,

> > So, why not just require that it has
> > no accumulation points if you want to rule out
> > such scales as this one.

> I thought that is what I did. I tried to clarify matters by saying I'm
> not counting 0 as an accumulation point, but instead that seems to
> have confused things.

No I don't think you did, not in the definition. You just
said the scale had to be discrete, not that it should have
no accumulation points. Of course it can be both
discrete and have no accumulation points. The discreteness
condition though is redundant really since
no accumulation points would force discreteness (unless of course you use the discrete topology
in which case everything is discrete including
the entire real line).

> > Yes you can rule out 0 that way by using logarithms.
> > Just that I found it a bit puzzling put that way
> > but it is a matter of taste. One could as well
> > just say that it has no non zero accumulation
> > points, and the two definitions are mathematically
> > equivalent, and I'd have thought it was mathematically simpler
> > to just say that it has no non zero accumulation
> > points.

> Well, perhaps it would be but this is music, and people all the time
> are thinking of notes as on a logarithmic scale.

Yes that's true.

Well one could make the whole thing a definition applied
to logarithmic scales and then use a map from R for the
fuzzy reals rather than R+.

So then it is still set theoretically simple and elegant
becausae the entire set theoretic definition is about
logarithmic pitches, and doesn't even need to mention
logarithms except in a preamble to explain the
intended interpretation of R as the logarithmic
pitch with 0 as the 1/1.

Robert