Scales again

That is a striking definition of `seyir` of a `maqam`, which would otherwise
be synonymous with `melodic pattern` here.

I would rather consider any unordered sequence of notes to fall into the
above-mentioned category, and spare `scale` for a perfectly ordered
ascending-descending sequence.

Cordially,
Ozan

----- Original Message -----
From: "Robert Walker" <robertwalker@ntlworld.com>
To: <tuning@yahoogroups.com>
Sent: 21 May�s 2005 Cumartesi 18:56
Subject: [tuning] Re: question about just intonation scales

> Hi Gene,
>
> > I tried to fix that once by defining "scale" so that the notes were
> logarithmically discrete, but people were not happy. I suspect most
> people would prefer that it not be given a strict mathematical
> definition, which of course would also require stating whether it was
> a set, and if so, a set of what.
>
> Well I'm game to suggest a definition of scale, as a point
> for discussion, mainly to show up the difficulties.
>
> How about - a scale is an infinite countable unordered
> subset of the positive reals which must have 1 as a
> member.
>
> Then if someone says that scales are usually finite
> - well that is because you usually give them an
> interval of repetition. So what you have there
> isn't a scale as such, but a scale generator.
>
> A bit like saying
> Here are the natural numbers:
>
> 1, 2, 3, 4, ...
>
> But you haven't written them all down of course.
>
> Here is a scale
>
> 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 ...
>
> It's just that normally we leave out the
> ... as "understood". There the method we
> are using to generate the scale uses
> a particular ordering, and of course
> has to because that's how it works.
>
> However, the scale we
> are using it to generate doesn't have
> to be thought of as ordered in that
> way - the same scale can be
> ordered in whatever way we please.
>
> So
>
> 1/1 4/3 3/2 9/8 5/4 5/3 15/8 2/1 ...
> would also generate the same scale
> and if anyone wants to use that ordering
> or whatever ordering they please, then fine.
>
> That then avoids the problem of getting into such
> things as intervals of equivalence and the question
> of how to treat non octave scales, or all
> the different types of scales there could be
> - any kind of scale is allowed by this definition
> and you don't need to enumerate them in advance.
> It means someone can come up with some completely
> new kind of scale construction idea and so long as
> it generates countably many positive numbers
> then it is included in our mathematical definition
> so we won't need to keep updating our definition
> to take account of new scale construction ideas.
>
> It does have a bit of a drawback though because
> you might want to allow a range of values
> for an element in a scale.
>
> More generally, maybe a scale should be
> regarded as a set of line segments rather
> than a set of points. Normally each line
> segment is of zero length so you lose nothing
> by treating them as points. But they can
> be of non zero length corresponding to a
> range of values. Someone might want to go
> further and say that each line segment
> should also have a weighting attached to
> it so that e.g. the line segment around
> 3/2 might have 3/2 itself as the most
> 3/2 like interval, and then tail off
> quite rapidly to either side so that
> say 50 cents flat or sharp is very low
> probability or zero probability 3/2
> (in some particular context).
>
> But the one around 5/4 might have
> a more slowly tapered weighting so that
> even an 11/9 or 14/11 is within the
> range of possible 5/4s.
>
> What is your idea about defining a scale
> so that its notes are logarithmically
> discrete?
>
> Robert
>
>
>
>