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Re: [Why you don't really want] irrational tuplets [nor CF approximation
From: |
Hans Åberg |
Subject: |
Re: [Why you don't really want] irrational tuplets [nor CF approximations] |
Date: |
Tue, 8 Nov 2016 22:20:50 +0100 |
> 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.
Re: How to get irrational tuplets inside a regular meter like 9/8, Hans Åberg, 2016/11/08
- [Why you don't really want] irrational tuplets [nor CF approximations], Alexander Kobel, 2016/11/08
- Re: [Why you don't really want] irrational tuplets [nor CF approximations],
Hans Åberg <=
- Re: [Why you don't really want] irrational tuplets [nor CF approximations], Thomas Morley, 2016/11/08
- Notational conventions, Hans Åberg, 2016/11/08
- Re: Notational conventions, Thomas Morley, 2016/11/08
- Re: Notational conventions, Hans Åberg, 2016/11/09
- Re: Notational conventions, David Kastrup, 2016/11/09
- Re: Notational conventions, Hans Åberg, 2016/11/09
- Re: Notational conventions, David Kastrup, 2016/11/09
- Re: Notational conventions, Hans Åberg, 2016/11/09
- Re: Notational conventions, David Kastrup, 2016/11/09
- Re: Notational conventions, Hans Åberg, 2016/11/09