Re: [tuning-math] Digest Number 420

HI Gene,

> Presumably, 5-limit beats 3-limit, unless you mean 5 with no 3.

Well, kind of vague here. Two ideas in mind - the relative proportions
of each prime if you leave them free and find the best one for
a given value of product of denominator and denumerator, and
then the idea that one could maybe leave some of them out altogether
and see which such methods work well.

With that other idea you could e.g. find out if the golden ratio
is better approximated using say 2^a*3^b*7^c or using
2^a*3^b*5^c in the limit as the numbers tend to infinity
- from the data so far, one would rather suspect that the 7 will win
here. (while I suppose if you think in terms of density the
5 should win). Of course any finite sequence doesn't really give
much indication of what will happen in the infinite without some
proof backing it.

> Did this use integer relations algorithms, brute force, or what?

Pretty much just brute force search of the lattice. It's sparse
enough so that is quite effective. I wanted to do something
more sophisticated but couldn't find out how to do it at the
time, anyway easily goes up to 10^50 to 10^100 kind of a range
so not much incentive to look for a better method. I was
originally interested in using it to search
for xenharmonic bridges.

> This sounds more like a topic for the number theory list or sci.math, but
> in any event I'm skeptical.

Well, just an intriguing idea at present. :-).

Robert

----- Original Message -----
From: <tuning-math@yahoogroups.com>
To: <tuning-math@yahoogroups.com>
Sent: Friday, July 05, 2002 1:06 AM
Subject: [tuning-math] Digest Number 420

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> There is 1 message in this issue.
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> Topics in this digest:
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> 1. p-limit approximations
> From: Gene W Smith <genewardsmith@juno.com>
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>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 1
> Date: Thu, 4 Jul 2002 11:24:58 -0700
> From: Gene W Smith <genewardsmith@juno.com>
> Subject: p-limit approximations
>
> --- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
>
> > I may be taking it out of context, but the thought I had is
> > that you could characterise the golden ratio say as 3 or 5 limit
> > etc. depending on whether it is more rapidly approximated by a sequence
> of
> > 3 limit or 5 limit ratios.
>
> Presumably, 5-limit beats 3-limit, unless you mean 5 with no 3.
>
> > I wrote a program a while back to look for ratio approimations
> > to another ratio, and just updated it to accept arbitrary decimals.
> > so that it can look for approximations to golden ratio etc. too.
>
> Did this use integer relations algorithms, brute force, or what?
>
> > Obviously these huge numbers aren't of immediate musical relevance,
> > but kind of interesting. It rather looks as if there is enough
> > of a trend there so that with some work one could define
> > a mathematically precise notion of the relative proprotions
> > of the various priimes needed to approximate an irrational,
> > which mightn't necessarily converge, so next thing would
> > be to see if one could prove it did converge, and if
> > every irrational has a definite flavour in the n-limit
> > or if only some do and so forth.
>
> This sounds more like a topic for the number theory list or sci.math, but
> in any event I'm skeptical.
>
>
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