RE: [tuning] Digest Number 3522

Robert,

You wrote:
...
Yes of course I agree that its common
use is as an ordered scale, and that
it is necessary of course to accept that.
It is only once one thinks about
the interval structures of scales
that one may come to think that
an unordered set be a more
appropriate way to think of a scale.

[YA] You're right, of course: both descriptions
are just different ways of thinking about a scale,
and it makes sense to choose the way - and
description - that is most appropriate to your
particular purpose at any given time.

You continued:
BTW I was thinking, a really clear
case for a scale with no natural ordering on
it would be a 2D (or higher) doubly
infinite scale - well it does
have an infinite ordering of type
omega squared - i.e. so that each
pitch in a row is counted as occurring
before the the one to its right,
and each entire row is counted as
occurring before the next entire row.

It means that if you order the scale
that way you have to accept that there
are some pitches in the scale
that can never be reached in a finite
number of steps by just walking up through
its notes in increasing order.
Rather than think of a scale in
that way one might prefer to think
of it as not having a fixed
natural ordering.

[YA] Then again, if we're only talking about
two countable infinities - ones that correspond
to the natural (counting) numbers 1, 2, 3, ... -
then we can always apply the Cantor diagonal
ordering, can't we? This shows, incidentally,
that omega squared is just omega itself - that
is, that your doubly infinite scale is no "bigger"
than a singly infinite scale.

Could you please give an example of the sort of
doubly infinite scale you had in mind? Your
later comments on "density" suggest you may
have been thinking of a continuum, rather than
a countable infinity.

Regards,
Yahya

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