Scale definition

(1) For some index set I of integers, there is a scale map
s:I --> F, where F is the set of compactly supported positive fuzzy
numbers. That is, an element of F is a map f:R+ --> [0, 1], such that
the set of values for which f(x)>0 is compact.

(2) If i and j are not equal and are elements of I, then if we take
the interval from the glb to the lup of the support of s(i), and do
the same for s(j), the intervals do not intersect.

(3) The set of the logarithms of the glb and sup of the supports of
s(i), for i in I, is a discrete subset of R.

Now what use is such a definition?

Hi Gene,

I'm cross posting to the main tuning list as well since
I think this is something that doesn't _need_ so much
maths as this to be discussed, and should be quite
generally accessible, though of course it helps
the discussion to express it like this as well.

> (1) For some index set I of integers, there is a scale map
> s:I --> F, where F is the set of compactly supported positive fuzzy
> numbers. That is, an element of F is a map f:R+ --> [0, 1], such that
> the set of values for which f(x)>0 is compact.

(1) says that they are fuzzy reals i.e. with weightings on finite intervals for each pitch - and are countable
so that's just the same as I suggested - as you say
in your tuning post.

> (2) If i and j are not equal and are elements of I, then if we take
> the interval from the glb to the lup of the support of s(i), and do
> the same for s(j), the intervals do not intersect.

I'd want to leave out (2). I suppose the motivation for it is to
remove ambiguity, that if you are given an interval then
you should be able to figure out which scale degree
it was - but I'd want to say that's unnecessary. You could
have an instrument with fuzzy pitches and say that
the scale is the scale of that instrument and if
the instrument comes out with a squeaky 11/9
the player intended some flavour of a 4/3 and the context implies a 4/3 then it is a strange pitch
bended 4/3. I'd want to say, if the player intended a 5/4 then that's what it is
- with fuzzy pitches that is.

So the supports for the reals are taken as
a weighting factor that helps you to
interpret the pitches **within the current
harmonic and melodic context**

> (3) The set of the logarithms of the glb and sup of the supports of
> s(i), for i in I, is a discrete subset of R.

I don't understand what the logarithms are doing here.
With the normal one dimensional type scales, they
will be discrete anyway unless perhaps you can consider
0 as a point in the scale somehow.
Apart from 0, I can't see how taking
logarithms affects discreteness or otherwise
of the subset. I must be missing something
- are you perhaps using a subtly different
notion of discrete? I'm assuming that discrete means no isolated
points. As in this definition here:

http://planetmath.org/encyclopedia/Isolated.html

So the rationals are not discrete because
a single point is a closed set, not open,
and any interval around a single point
will contain other points, so
it isn't isolated.

Anyway I can't go along with 3) myself
since a scale which is everywhere dense will have
no isolated points at all, never mind
ensuring that all the points are isolated.

On 1 again, I think perhaps I'd want to say additionally that there can (optionally) be a point assigned
to each fuzzy real which is the intended ideal
pitch for that real. But this can also be left
out if there is no intended ideal pitch - in which
case the fuzzy weighting might well have a plateau
rather than a single peak - or may have multiple
peaks.

As to the use of this, I don't know if it is
actually practically useful, though who knows,
it is hard to anticipate what will be of
practical interest in maths. Even if not though,
it is maybe of philosophical interest or something that
is a useful basis for discussion of what
a scale is, not expecting to get a final
answer to that at all.

Robert

--- In tuning-math@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:

> > (3) The set of the logarithms of the glb and sup of the supports of
> > s(i), for i in I, is a discrete subset of R.
>
> I don't understand what the logarithms are doing here.
> With the normal one dimensional type scales, they
> will be discrete anyway unless perhaps you can consider
> 0 as a point in the scale somehow.

The point is simply to keep 0 from being an accumulation point.

> Anyway I can't go along with 3) myself
> since a scale which is everywhere dense will have
> no isolated points at all, never mind
> ensuring that all the points are isolated.

I don't think a scale should be anywhere dense, much less everywhere.

Hi Gene,

> The point is simply to keep 0 from being an accumulation point.

Well I still don't understand. Because (I'd have thought) 0 isn't going to be
allowed as a pitch, and a subset of the reals is defined to
be discrete so long as every point in it is isolated. Since 0 isn't
in the set, it doesn't matter if it is an accumulation
point or not - because every element in the set is
isolated so the set is discrete - so the same subsets will
count as discrete whether you use logarithms or not
- or am I missing something?

> I don't think a scale should be anywhere dense, much less everywhere.

A scale dense at a single point is interesting too.
Like your square steps scale which could be thought
of as a single infinite scale that you can stop anywhere
and which has an accumulation point, though the accumulation
point is not in the scale, but if you add 2/1 to it
then you get an infinite scale with an accumulation point
at 2/1.

Of course a scale dense everywhere and ordered in
increasing order isn't particularly interesting.
What makes the Lambdoma a useful scale is the way it is
presented, as a two dimensional array.

If you don't allow it as a scale, then it would
seem to suggest that scales can only be used
to study fragments of the complete Lambdoma
and not the entire array as a whole. Which
is fair enough.

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.

I prefer to take the widest possible
definition of a scale. But it's not a big
issue one way or another, just a matter
of mathematical taste.

Robert

--- In tuning-math@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:
> Hi Gene,
>
> > The point is simply to keep 0 from being an accumulation point.
>
> Well I still don't understand. Because (I'd have thought) 0 isn't
going to
> be
> allowed as a pitch, and a subset of the reals is defined to
> be discrete so long as every point in it is isolated. Since 0 isn't
> in the set, it doesn't matter if it is an accumulation
> point or not - because every element in the set is
> isolated so the set is discrete - so the same subsets will
> count as discrete whether you use logarithms or not
> - or am I missing something?

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 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.

Gene,

A fine start.

Of what use? I would hope that music software writers
(like Robert :-) )would be able to implement the most
general possible definition of a scale, allowing users to
come up with their own, but ensuring that anything which
_won't_ work is NOT definable as a scale. For example,
your use of logs to, as you say, prevent 0 being an
accumulation point.

For myself, I don't need "fuzzy", tho others may. It
seems to mean unnecessary complexity.

Regards,
Yahya

-----Original Message-----
(1) For some index set I of integers, there is a scale map
s:I --> F, where F is the set of compactly supported positive fuzzy
numbers. That is, an element of F is a map f:R+ --> [0, 1], such that
the set of values for which f(x)>0 is compact.

(2) If i and j are not equal and are elements of I, then if we take
the interval from the glb to the lup of the support of s(i), and do
the same for s(j), the intervals do not intersect.

(3) The set of the logarithms of the glb and sup of the supports of
s(i), for i in I, is a discrete subset of R.

Now what use is such a definition?
________________________________________________________________________

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Hi Gene,

> 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.

Ah but that doesn't rule out the idea of a scale with accumulation
points - only ones with attained accumulation points.

Example:

... 1/1 9/8 6/5 5/4 9/7 21/16 ... 2/1 ...

where the general term is

3*n / 2*(n+1)

then that's a discrete scale with an accumulation point
at 3/2. As long as we don't include 3/2 itself as
an element of the scale, then it remains
discrete - every element of the scale has a neighbourhood
with only that one element in it, such as
the interval between the midpoints of the previous and
next scale steps. You can also have a discrete scale with countably many
unattained accumulation points between 1/1 and 2/1, which is routine to construct.

Of course I want to include such scales and also
everywhere dense ones too. So, why not just require that it has
no accumulation points if you want to rule out
such scales as this one. I see it as useful
too as a way of marking out a particular
set of scales.

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. But then as a logician and set theorist, I think of
a logarithm as a pretty complex thing while an
accumulation point is much simpler conceptually, so would
tend to use more basic concepts such as sets and
elements when it is at all possible. That's probably
partly why I wondered what the place was for a logarithm
in the definition. So partly a matter of taste and training probably.

I can imagine that just possibly the idea of fuzzy
scale pitches could be useful for a microtonal software developer at some point, since fuzzy logic is quite widely used in computing. I think it might be
useful to add to it the idea of an "intended
pitch" for each fuzzy real - perhaps as an
extra function v which is the intended pitch
if non zero and can be set to 0 if there
is no intended pitch. Maybe then the
f and v can both be optional so you include v only if you don't need fuzzy
reals and f only if you don't need the
dea of an intended pitch. Thanks,

Robert

--- In tuning-math@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:

> > 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.
>
> Ah but that doesn't rule out the idea of a scale with accumulation
> points - only ones with attained accumulation points.

It's still OK to say it.

> 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.

> 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.

Robert and Gene,

You've been discussing dense and infinite sets of notes as
possible scales. Robert has shown us the lambdoma, and
considered its possible use as a resource from which one
could draw finite scales, which makes sense to me.

I'm still pondering the possible practical _musical_ uses of an
infinite scale. I guess one could take the infinite scale
1/1, 1/2, 1/3, ... 1/n, ... and play (electronically, at least) those
notes in that order, for matching durations in terms of say,
beats at 60 bpm; so the first note plays for 1 second, the second
for 1/2 second and so on. Long before the capacity of the
electronics to further divide the time and the frequency, one
would surely pass the limits of human perception; for example,
starting at pitch A 1760 for 1 sec, we play
A 880 for 1/2 s
A 440 for 1/4 s
...
A220 for 1/8 s
...
A 110 for 1/16 s
...
A 55 for 1/32 s
...
A 27.5 for 1/64 s
...
A 13.75 for 1/128 s
- at which point, most people can't hear the note at all.

So I've constructed an example, as an attempt to use
all the notes of an infinite scale explicitly. Obviously,
we could build other examples, without this pitch
limitation, but the time constraint remains : I can't see
how one could fit the expression of an infinite number
of pitches into finite time. Can we actually hear sounds
shorter than about 5 ms anyway?

Using an infinite set makes some kind of sense to me,
but only as a super-gamut from which one can draw an
actual gamut for realisation in an actual piece of music.
I guess I'm still not convinced we need infinite scales,
even conceptually.

Regards,
Yahya

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--- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> Using an infinite set makes some kind of sense to me,
> but only as a super-gamut from which one can draw an
> actual gamut for realisation in an actual piece of music.
> I guess I'm still not convinced we need infinite scales,
> even conceptually.

If we have a finite set of notes in an octave, the conceptually simple
way to make it a scale is to take everything of the form 2^n s, for s
a note in the octave [1, 2). This strikes me as much cleaner than
saying you have to set some bound, so you get different scales
depending on what bound you set.

Gene,

If we have a finite set of notes in an octave, AND we
regard octaves as equivalent, we can indeed regard the
gamut of notes in a scale as all those with frequencies
of the form 2^t s, for s a note in the octave [1, 2) and
octave t in N = {1, 2, ...}.

Using any [other] interval of equivalence n/d, the gamut
could be generated as all notes of form (n/d)^t s, for s
a note in the octave [1, n/d) and "equitave" t in N = {1, 2, ...}.

Regards,
Yahya

-----Original Message-----
________________________________________________________________________

--- ... <yahya@m...> wrote:

> Using an infinite set makes some kind of sense to me,
> but only as a super-gamut from which one can draw an
> actual gamut for realisation in an actual piece of music.
> I guess I'm still not convinced we need infinite scales,
> even conceptually.

If we have a finite set of notes in an octave, the conceptually simple
way to make it a scale is to take everything of the form 2^n s, for s
a note in the octave [1, 2). This strikes me as much cleaner than
saying you have to set some bound, so you get different scales
depending on what bound you set.
________________________________________________________________________

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On 5/24/05, Yahya Abdal-Aziz <yahya@melbpc.org.au> wrote:

> Using an infinite set makes some kind of sense to me,
> but only as a super-gamut from which one can draw an
> actual gamut for realisation in an actual piece of music.
> I guess I'm still not convinced we need infinite scales,
> even conceptually.

I don't see any need to explicitly exclude infinite sets from being
scales. A theorem for equal tempered scales, for example, may as well
be extrapolated to infinity.

Another example is the Kornerup sequence of meantones 12, 19, 31, 50,
... If you use the phi-tuning, you can construct an MOS of any number
of notes in the sequence. Given two notes defined on the spiral of
fifths, you can always say what order they are in. So this is still
true of the infinityth MOS in the sequence, and you may as well still
call it a scale.

It's impossible to ever use such a scale, because you don't know what
number to assign to any given note. So they don't do any harm.

Graham

--- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> Gene,
>
> If we have a finite set of notes in an octave, AND we
> regard octaves as equivalent, we can indeed regard the
> gamut of notes in a scale as all those with frequencies
> of the form 2^t s, for s a note in the octave [1, 2) and
> octave t in N = {1, 2, ...}.

Why N? I'd say t is in Z.

Graham,

You wrote:
> I don't see any need to explicitly exclude infinite sets from being
> scales. A theorem for equal tempered scales, for example, may as well
> be extrapolated to infinity.

[YA] OK, fine as far as proving theorems go. I guess that qualifies as
a "conceptual" use.

> Another example is the Kornerup sequence of meantones 12, 19, 31, 50,
> ... If you use the phi-tuning, you can construct an MOS of any number
> of notes in the sequence. Given two notes defined on the spiral of
> fifths, you can always say what order they are in. So this is still
> true of the infinityth MOS in the sequence, and you may as well still
> call it a scale.

[YA] "you can always say what order they are in" - means you can
assign each one an ordinal number, yes? Or perhaps only that you
can say which note precedes the other? What's the relevant notion
of "order" here?

> It's impossible to ever use such a scale, because you don't know what
> number to assign to any given note. So they don't do any harm.

[YA] "what number" = "what _ordinal_ number"?
If you have an operational procedure by which you can select a
member from the scale, then you could use it - perhaps only in an
aleatoric way, � la John Cage.

Regards,
Yahya
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