[tuning-math] p-limit approximations
- From: Gene W Smith <genewardsmith@xxxxxxxx>
- To: tuning-math@xxxxxxxxxxxxxxx
- Date: Thu, 4 Jul 2002 11:24:58 -0700
--- In tuning@xxxx, "Robert Walker" <robertwalker@xxxx> 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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