Re: Irrational time signature and tuplets

[Hans Aberg]

On 12 Jun 2014, at 11:54, David Kastrup <address@hidden> wrote:

> Hans Aberg <address@hidden> writes:
> 
>> How might one display the time signature 8 + 2 sqrt(5) over 8, and the
>> tuplet 1:1 + sqrt(5), where “sqrt" is the square root sign? - It is
>> for experimenting with a popular meter in 12 [1-2].
>> 
>> 1. https://en.wikipedia.org/wiki/Leventikos
>> 2. https://www.youtube.com/watch?v=vYR_pvRWO_g
> 
> \documentclass{article}
> \usepackage[active,textmath,displaymath,tightpage]{preview}
> \begin{document}
> $8+2\sqrt{5}\over8$
> $1\over1+\sqrt{5}$
> \end{document}
> 
> run through LaTeX, run
> dvips -E -i -Pwww
> 
> on the resulting DVI file, work with the resulting EPS files.  It might
> be cute to integrate something like that into LilyPond like
> preview-latex <URL:http://www.gnu.org/software/auctex/preview-latex>
> integrates it into Emacs editing, but that would end up to be a bunch of
> work for somebody™.

One has to wrestle a bit, but that was a part of the reason I got curious about 
it, as you note, an interesting example.

The mathematical background is that I replace the proportion 3:2 of the 
original meter in 12 with the slightly larger golden ratio x = (1 + sqrt 5)/2, 
which has as continued fraction convergents the successive ratios of the 
Fibonacci sequence 1, 1, 2, 3, 5, 8, 13,... , that is, 1/1, 2/1, 3/2, 5/3, 8/5, 
13/8, ...; so here 3/2 occurs naturally.

Tying this back to LilyPond, the graphical representation will look the same, 
only that the meter and the tuplet numbers are changed. There is another 
problem in the MIDI file: how to make a better approximation than 3:2. The next 
is 5:3, but then one ends up with tuplets in the notation for the 2 over 8 
parts in the meter that must be suppressed. Doable, but a lot of typing.