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

[Hans Åberg]

> On 8 Nov 2016, at 21:00, Alexander Kobel <address@hidden> wrote:
> 
> 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. 
...
> 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.

The user does not see the rational approximation, so just use whatever does the 
job. (Guile has some functions for that, I think.)

And a reason of writing a complex time signature might be to make it impossible 
for the performer to follow it: In Balkan music, one plays by ear, and the 
variation is greater than the irrational time signature examples I gave. A 
Western musician when seeing 12/8, 12 = 3+2+2+3+2 with quadruplets on them, 
might try to play it as exactly as possible, but that is not how it should be 
performed. Check out the metric time bends indicated here:
  https://en.wikipedia.org/wiki/Leventikos

Otherwise, if one really wants it to be played exactly, a sequencer track would 
be necessary.