Re: large prime ratios

Hi Dave (Keenan)

I'm trained as a mathematician too.

I've seen so many surprising connections between things turn up in mathematical
investigations. I'm sure you must be able to think of a few too.

I don't know, maybe it is something to do with a kind of pure / applied ethos or
something.

I've absorbed the attitude that it is worth while to try applying anything to anything in
maths, even if it seems there is no connection at all.

Anyway, seems to me, why not apply prime numbers to scales, follow it up as far as one
can, and see what turns up.

Clearly one can do it as art, but one can do it as maths too! Explore the numbers, look
for patterns, - those are all things mathematicians do in the pre-theorem proving stage.
Then if one finds something that looks interesting, then one calls in any mathematicians
one knows, to see if they have come across anything like it before, or can explain it.

Also, I think one needn't make such a sharp distinction between maths and art.
Mathematicians play around with numbers, and look for patterns in them, and they aren't
proving theorems necessarily, can be just for fun. Are you going to call that art then?
Maybe it is in a way, but I don't think many theorems would have been proved if
mathematicians hadn't had that kind of fun attitude to what they are doing, certainly, not
the more "surprising" ones.

Perhaps you are right that there are no particularly deep mathematical results to be found
in this way, and that the only real relevance is for art. Could be so. But i think there
is only one way to find out, one can't say in advance.

One can also distinguish between the maths used to discover results, and the maths used to
prove them. For example, Newton discovering results using calculus, but then re-casting
them all in geometrical form in order to satisfy the notions of rigour of his time. Or
earlier Greek mathematicians doing the same with pre-calculus ideas.

One may well discover interesting scale patterns using large primes that "could" have been
discovered by other methods too - and in fact we seem generally agreed that one will find
the same things at least musically speaking, as very large primes can't be heard as such.

Perhaps another lead into this is to think of it as a kind of "recreational mathematics"
like the Martin Gardner columns when he was writing for NS and Sci. Am.

There, numbers and patterns were and are explored wherever they turn up, and there is very
much the ethos of applying anything to anything and seeing what comes up. Indeed, that's
probably where I've absorbed the attitude as I have a keen interest in recreational
mathematics.

Some of it later became mainstream mathematics, including the Penrose Tilings that I
study. If I remember right, I think they were first published in a Cambridge student maths
magazine, but the first widely read publication certainly was in Martin Gardner's column.
So they begun as an area of recreational mathematics (though also having roots in
mainstream mathematics and the early results proving existence of non periodic tilings)
and are now applied in materials science in study of new and unpredicted crystals with
perfect five-fold x-ray crystallographic patterns (something "impossible" in classical
x-ray crystallography theory).

anyway, that's another point of view on the matter, and it is the point of view of a
"mathematician" for what that's worth...

}:-)

Robert

--- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:
> Hi Dave (Keenan)
> I'm trained as a mathematician too.
> I've seen so many surprising connections between things turn up in
mathematical
> investigations. I'm sure you must be able to think of a few too.

Robert,

I applaud you for coming forth in a rare moment of community spirit
with this wonderful post here! Your vision energizes my musical
heart toward the goals at hand, and I feel a great sense of kindred
sprit with this; a manner which joyfully opens the doors to all
creative exploration.

> I've absorbed the attitude that it is worth while to try applying
anything to anything in
> maths, even if it seems there is no connection at all.
> Anyway, seems to me, why not apply prime numbers to scales, follow
it up as far as one
> can, and see what turns up.
> Clearly one can do it as art, but one can do it as maths too!
Explore the numbers, look
> for patterns, - those are all things mathematicians do in the pre-
theorem proving stage.
> Then if one finds something that looks interesting, then one calls
in any mathematicians
> one knows, to see if they have come across anything like it before,
or can explain it.

You have so precisely and eloquently described a part of my
explorative process, as if you had spoken my thoughts. My approach
was without preconceptions, and just done in a spirit of pure
exploration.

> Also, I think one needn't make such a sharp distinction between
maths and art.
> Mathematicians play around with numbers, and look for patterns in
them, and they aren't
> proving theorems necessarily, can be just for fun. Are you going to
call that art then?
> Maybe it is in a way, but I don't think many theorems would have
been proved if
> mathematicians hadn't had that kind of fun attitude to what they
are doing, certainly, not
> the more "surprising" ones.
> Perhaps you are right that there are no particularly deep
mathematical results to be found
> in this way, and that the only real relevance is for art. Could be
so. But i think there
> is only one way to find out, one can't say in advance.

Here I must insert my Music Mantra again (sorry). When mathematics
comes to study at the door of the ancient practice of music making,
it will find itself uncomfortably intertwined in "art", emotion,
culture and feeling and many other mathematically unqualifiable
things which are directly *experienceable* in human existence. This
is where it finds itself but an outside observer of the mysterious
cycle of creation and participation in the experience of musical
enjoyment; existing outside of cold mathematical analysis.

> One may well discover interesting scale patterns using large primes
that "could" have been
> discovered by other methods too - and in fact we seem generally
agreed that one will find
> the same things at least musically speaking, as very large primes
can't be heard as such.
> Perhaps another lead into this is to think of it as a kind
of "recreational mathematics"
> like the Martin Gardner columns when he was writing for NS and Sci.
Am.
> There, numbers and patterns were and are explored wherever they
turn up, and there is very
> much the ethos of applying anything to anything and seeing what
comes up. Indeed, that's
> probably where I've absorbed the attitude as I have a keen interest
in recreational
> mathematics.

It makes one wish to ask: "Must the answer to any investigation be
known in advance, or should one always have a clear goal in mind at
the onset?"

To proceed without preconception can sometimes open doors to things
which may be hidden by having a clear initial goal. One might tend to
not notice a particular useful pattern or the like.

> anyway, that's another point of view on the matter, and it is the
point of view of a
> "mathematician" for what that's worth...

Refreshed and exhilarated,

Jacky Ligon

Robert Walker wrote,

<<anyway, that's another point of view on the matter, and it is the
point of view of a "mathematician" for what that's worth...>>

I just wanted to let you know that I for one appreciate your creative
and open-minded participation here at the tuning list of late...

Because I must admit that I've long felt decidedly out of step with
the overriding mindset here as personified by the likes of Paul Erlich
and Dave Keenan -- even though I in fact have great respect for both
and wouldn't want either of them to do anything but exactly what it is
that they already do... But hopefully the occasional vocal appearance
of someone like yourself, a specialist of sorts, will help give the
tenor of the list something it has (to my mind anyway) sorely needed
for a long time -- a little bit more of a balanced persona in the fact
and figure department.

--Dan Stearns

--- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:
> Hi Dave (Keenan)
>
> I'm trained as a mathematician too.

Strictly speaking, my training is in Physics and (mostly) Computer
Science, but that's difficult to do without picking up some
mathematics.

> I've seen so many surprising connections between things turn up in
mathematical
> investigations. I'm sure you must be able to think of a few too.

Sure.

> I don't know, maybe it is something to do with a kind of pure /
applied ethos or
> something.
>
> I've absorbed the attitude that it is worth while to try applying
anything to anything in
> maths, even if it seems there is no connection at all.

I agree with your general point, but life is too short to try applying
anything to anything, one usually follows some kind of intuition about
where it might be best to look.

> Anyway, seems to me, why not apply prime numbers to scales, follow
it up as far as one
> can, and see what turns up.

Sure. But if someone asks me for an opinion, and my intuition tells me
they are wasting their time, then I'm obliged to say so. They are
welcome to ignore me. I guess I should make it very clear when this is
based on intuition, not proof.

> Clearly one can do it as art, but one can do it as maths too!
Explore the numbers, look
> for patterns, - those are all things mathematicians do in the
pre-theorem proving stage.
> Then if one finds something that looks interesting, then one calls
in any mathematicians
> one knows, to see if they have come across anything like it before,
or can explain it.

Indeed.

> Also, I think one needn't make such a sharp distinction between
maths and art.
> Mathematicians play around with numbers, and look for patterns in
them, and they aren't
> proving theorems necessarily, can be just for fun. Are you going to
call that art then?
> Maybe it is in a way, but I don't think many theorems would have
been proved if
> mathematicians hadn't had that kind of fun attitude to what they are
doing, certainly, not
> the more "surprising" ones.

Very good point. The boundary between mathematics and art is quite
fuzzy in places, and I'm grateful for that, because I feel a need in
my life to help further not only truth (maths science philosophy
spirituality) and goodness (morality ethics politics) but also beauty
(art).

> Perhaps you are right that there are no particularly deep
mathematical results to be found
> in this way, and that the only real relevance is for art. Could be
so. But i think there
> is only one way to find out, one can't say in advance.

No, one can't say in advance. But life is short and so we'd probably
like to avoid dead-ends if we can, that is unless we are enjoying the
path for its own sake.

> One can also distinguish between the maths used to discover results,
and the maths used to
> prove them. For example, Newton discovering results using calculus,
but then re-casting
> them all in geometrical form in order to satisfy the notions of
rigour of his time. Or
> earlier Greek mathematicians doing the same with pre-calculus ideas.

Sure.

> One may well discover interesting scale patterns using large primes
that "could" have been
> discovered by other methods too - and in fact we seem generally
agreed that one will find
> the same things at least musically speaking, as very large primes
can't be heard as such.

Yes.

> Perhaps another lead into this is to think of it as a kind of
"recreational mathematics"
> like the Martin Gardner columns when he was writing for NS and Sci.
Am.
>
> There, numbers and patterns were and are explored wherever they turn
up, and there is very
> much the ethos of applying anything to anything and seeing what
comes up. Indeed, that's
> probably where I've absorbed the attitude as I have a keen interest
in recreational
> mathematics.

Me too. I expect you have a zometool kit http://www.zometool.com. Got
the green struts yet?

> anyway, that's another point of view on the matter, and it is the
point of view of a
> "mathematician" for what that's worth...

It's worth a lot. Thankyou.

Regards,
-- Dave Keenan

Hi Dave Keenan

> Strictly speaking, my training is in Physics and (mostly) Computer
> Science, but that's difficult to do without picking up some
> mathematics.

I suppose I've tended to pick up some physics too, and of course,
computing techniques. In the year before going to univ. worked for a year
at Culham laboratories in the computational physics dept, which was
my first programming, in fortran (using punch cards!).

At university, my favourite maths subjects were things like number
theory (prime numbers and partitions especially), algebraic topology,
group theory, and set theory.

My research was originally in maths foundations / logic,
then later, non periodic tilings, and areas of recreational maths.

Guess I might have been an academic by now. However, my
examiners just _wouldn't_ be satisfied with my doctoral thesis,
even after I rewrote the whole thing following their recommendations
as best I could (that happens very rarely, but does, especially in
Oxford somehow).

My supervisor recommended that I try publishing
a few papers instead and do a doctorate "by publication"
called a DSc.

So, though trained as a mathematician, I'm not actually employed
as an academic, and instead, am trying to make a way as a freelance
inventor.

For last couple of years, mainly been writing software for my inventions.
and mainly FTS for the last year or so, and kind of hoping eventually to
make a living from freeware / shareware programs like FTS,
though I know that is kind of hard to do.

The other main Win 95/98 program I've done by the way is VRML trees.
When FTS 1.09 is finished, plan to try to release it before starting on
FTS 1.10, as it has been final beta within a fortnight or so of finishing
for about a year or so.

http://www.rcwalker.freeserve.co.uk/vrml_tree_exs/index.htm

It will be pretty good for music too eventually I hope - will prob. add midi
clip sound nodes to it, and make the clips in FTS, and use it to make
some of the trees and lattices one sees so many posts about on
the TL. Maybe develop it somewhat as a musical lattice construction
kit (and hopefully with links to Joe Monzo's Just Music program too).

Also been wondering if there is any way to make fractal tunes that
generate themselves by interactions between nodes of a VRML tree.
Have to give that a try some day and see if it works.

I.e., you click on a note, it plays, then sets one of its neighbours
going, which then plays, and so on, according to some kind of
connection established between the nodes. Very vague idea at
present. Like, the leaves of the tree play notes.

> Very good point. The boundary between mathematics and art is quite
> fuzzy in places, and I'm grateful for that, because I feel a need in
> my life to help further not only truth (maths science philosophy
> spirituality) and goodness (morality ethics politics) but also beauty
> (art).

Thanks, Yes, glad you like the fuzziness of the boundaries too!

> No, one can't say in advance. But life is short and so we'd probably
> like to avoid dead-ends if we can, that is unless we are enjoying the
> path for its own sake.

I suppose, the main point is, there can be a lot of value in enjoying the path for
its own sake.

If ones aiming to prove mathematical results as ones main aim, yes, one needs
some way to cut down on the number of things to investigate, and will prob.
choose whatever seems most likely to produce results, and produce them
easily and quickly.

But if that is a secondary aim, and one is basically just having fun with
maths, one can enjoy recreational maths diversions just for the pleasure
of doing them, and it is surprising how often things that come out of such "fun doodling"
can be the ones that actually turn out to be the most interesting.

> Me too. I expect you have a zometool kit http://www.zometool.com. Got
> the green struts yet?

I know about it and had it recommended to me, but haven't actually got it.
Would be nice to get it some day. I imagine it is a pretty useful thing to have.

Actually, I did most of my models for regions of the 3D Penrose tiling in paper
and card - not so much for presentation, as to help with seeing what is going
on for the research. The 3D tiles are joined to each other to make regions of
the tiling using paper clips.

I've also done a few VRML models of them.

http://www.rcwalker.freeserve.co.uk/interactive_models_with_titles/gr_triacontrahedron.wrl
(+ to see the golden icosahedron and dodecahedron: previous button)

Ray traced:
http://www.rcwalker.freeserve.co.uk/cubeetc/rhombic.htm

Anaglyphs:
http://www.rcwalker.freeserve.co.uk/anaglyphs/index.htm
(see near bottom of page)

My paper models look pretty much the same as the VRML ones. I proved non periodicity
for a set of four tiles - two colourings of the oblate and two of the prolate
golden rhombohedron (this result is not published, and seemed to be unproved
when I started on it, but I imagine someone else will surely have proved it by now
- the set of four tiles was well known at the time I was doing the main research).

For the 2D work, I started by cutting out tiles + drawing coloured lines
on them, and eventually wrote a program to generate the tilings.

Perhaps my best (unpublished) result is the proof of non periodicity of
a set of tiles that makes tilings with seven fold symmetry, including this tiling:
http://www.rcwalker.freeserve.co.uk/robert/7g2o7detail.htm

- they were also discovered previously by Soccolar, but he didn't
prove non periodicity - as a crystallogropher, he would be more interested in proving
crystallographic results. I don't know whether or not anyone else has proved
non periodicity for them yet.

Some day, would be nice to set some time aside and complete a few
of those papers. But I seem to be getting more and more interested
and involved in musical activities nowadays!

Maybe one day will go back to it again, who knows,...

However there is so much of mathematical interest in scales and tunings too.
So not really in any hurry to do so (unless perhaps someone else is _very_
keen to see the results).

Thanks!

Robert