Yahoo Tuning Groups Ultimate Backup tuning re: Scale definition

Hi Gene,

Actually, - well everyone please correct me if I'm wrong -
it doesn't seem that this is a particularly technical
mathematical topic, at least it is non
techy to the extent that one could discuss it in parallel here.

So maybe I'll post to both lists, - the
more philosophical or practice orientated
thoughts here and mathematical ways of treating them on tuning-math.

Your condition 1 is just the same as my
definition - a countable set of
either discrete or fuzzy reals.

I'd want to say that it is okay to think
of them as discrete and one doesn't
have to think of them as fuzzy
even though in practice of course
they will be, because if the player
is playing the pitches of the scale
to the best of their ability, even if
very imperfectly, one can think
of the practical case as an attempt
at playing the desired ideal perfect
ratios. So the scale they are playing
is the ideal scale consisting of
mathematical points in the continuum.

Similarly vibrato would be
thought of as vibrato around the intended
ideal pitch.

However, if the player is deliberately
bending the notes around, maybe with
pitch glissandi too, and also intends them
to be understood as still the "same scale degree" then they are playing
fuzzy pitches centred on whatever
they are taking as the central pitches
for each note (if any).

Of course one could use a scale with
fuzzy pitches as a way to analyse
the playing after the event too,
which is another thing.

Anyway either way of understanding it,
your conditino 2) says that the fuzzy
reals can't overlap. But I'd want
to say that they can. I can understand
the reason - because you want to
get back from a point in the real line
to the scale degree it belongs to.
But you can do that even with
overlaps if it is possible to use
the harmonic and melodic context
to disambiguate the ambiguous
pitches.

As for 3), if by discrete you mean
that all the points in the scale
should be isolated points
(the way it is usually understood)
- so that there are gaps of
finite size about each point with
no neighbours in them - then
that would rule out everywhere
dense scales such as the Lambdoma.

I'd want to say that 3) picks out a particular subset of all possible scales.
Though I can understand why som ewould
be particularly interested in those
subsets. Similarly with the Lambdoma
one can choose to focus on a particular
finite subset of the infinite lambdoma.
In that case the infinite scale is a gamut from which one would normally
choose a finite (so discrete) subset to play
in. But one could write a piece which traversed
a dense scale endlessly in some way, and even
though it would in practice never
cover more than a small finite region of the
scale, it would be.

On the distinction between tuning and
tuning-math - I think it is a matter
of the presentation rather than the
topic. Suggestion for guidelines about where
a post belongs:

If something presented mainly in ordinary
language with as few as possible
mathematcial words and any unfamiliar
words explained, then if no-one objects
to it then I think it is reasaonable
to post it here. If it involves symbols
and mathematical terms that would typically
be encountered first at first year
undergraduate level or later, and with no
explanation or attempt to express
them in ordinary language, it belongs
on tuning-math.

But if anyone objects to it as too mathematical despite maybe several attempts to
make it accessible, then one again it belongs on
tuning-math because obviously it then
isn't as accessible as one thought
it was. Or if they just say - please
go away basically, then rather than
attempt to explain again, it is polite
to just go over to tuning-math even if one doesn't quite
understand why they are saying that
about the particular topic
and presentation.

Thanks,

Robert

Hi Gene,

Getting confused by using a kind of everyday language
notion of discrete here.

> I'd want to say that it is okay to think
of them as discrete and one doesn't
have to think of them as fuzzy

Perhaps points for "discrete" there:

I'd want to say that it is okay to think
of them as points, and one doesn't
have to think of them as fuzzy

because in some infinite scales they
won't actually be discrete, in the sense
of all points being isolated from neighbours if
you use a large enough scale of magnification
to see the gaps - but still as ideal scale
pitches they can be treated as single
points all of them.

Robert

re: Scale definition

🔗Robert Walker <robertwalker@ntlworld.com>

🔗Robert Walker <robertwalker@ntlworld.com>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>