[Why you don't really want] irrational tuplets [nor CF approximations]

[Alexander Kobel]

On 2016-11-08 18:15, Hans Åberg wrote:
> I gave an example of a true irrational time signature [1]. The code
> is actually written in 12/8, with a MIDI approximation in 19/8. Such
> meter approximations can be obtained using continued fractions
> convergents [2]. For the rendering, it suffices with an approximation
> that separates and orders the notes that occur at different times.
> For the MIDI it suffices to be within its accuracy, or alternatively,
> what performers might do.

Seriously: I'm a half-mathematician by training (probably a lesser one 
than Hans, but still), and work all day in computer algebra; I share the 
admiration for the elegance of CF approximations. But for this purpose, 
I really think you should use dyadic fractions, say, 2^-20. Or at least 
fractions with a common denominator, say, 1,000,000, if you're anything 
like other humans and consider them easier to parse.

Let's do the math: For rendering purposes, with a resolution of 1200 
dpi, you have 12,000 dots on a 10-inch wide paper. You will not be able 
to distinguish any resolution higher than 1 ppm for sure.
For musical purposes (by a computer, not a human - that's out of 
question), assume 192 kHz frequency at max. Account for Shannon's 
theorem and aliasing issues, that makes it 384 kHz. With a reasonable 
base tempo of one quarter per second, that's appx. (1.6e6 s)^-1 max 
resolution. Granted: that's slightly more than what you get with a 
million as a denominator, but just.
(BTW: I didn't check the above calculations; probably the numbers are 
off, because the mathematician in me ignores constant factors, and the 
computer scientist in me doesn't do calculations in his head... ;-))

There are multiple benefits: No blow-up of the least common multiple of 
the denominators - it simply stays the same. No issues with accuracy - 
fits well to machine integers used for the rationals. In particular, no 
crashes because you suddenly exceed the implementation-based range. Even 
exactly convertible to hardware floats (double prec) if you use dyadic 
fractions. Finally: Significantly easier calculations for you to do to 
figure out the right durations to sync at barlines.

Of course, it's much more satisfying to do the "right thing" (tm). But 
that's using the exact values.
Exact computation is non-trivial even for arbitrarily large rationals. 
Exact calculations with algebraic numbers are possible, but not quite 
around the corner (that's good; otherwise, I'd need a new job). Exact 
representation and /some/ operations with an /a priori/ chosen subset of 
/some/ of the transcendentals is just possible, but a long way around 
several corners (that's even better, just in case I need a new job at 
some point). Exact computation with arbitrary real numbers is infeasible.
And that's not a "today's computers are not fast enough". It's simply 
fundamentally impossible.

And the most interesting numbers for a creative mind will be those that 
are the most complicated to capture and implement. Which reminds me of 
the phonograph in Hofstadter's wonderful "Gödel, Escher, Bach - an 
Eternal Golden Braid". If you didn't read it yet, here's a hint for a 
christmas gift...

On a side note, CF approximations are optimal in the sense that they 
give the best approximation with numerator and denominator that do not 
exceed any given bound. But they are /not/ better than dyadic 
approximations if your measure is the asymptotic growth of the 
worst-case bitlength of numerator and denominator that is required to 
achieve a certain approximation quality.


Cheers,
Alexander