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Tuplets can transform a rhythm into an irrational (or irregular) rhythm, but how do we precisely define a rational rhythm in the first place?

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First off time signatures can be irrational or rational, but rhythms however cannot. Rhythms can be irregular, but that's not the same thing as a meter being irrational and is better described as syncopated .

Whether a time signature is rational or irrational depends on the denominator of the time signature. Any time signature where the denominator is a power 2, for example 2, 4, 8, and 16, is rational.

What this means is the underlying beat is a typical note like the half, quarter, eighth, and sixteenth .

An irrational meter has a denominator that is not a power of two. This is rare, but can happen if you use a lot of tuples like you said. For example if you let the quater note triples get the beat in a time signature you'll end up with a 6 as your denominator and however many quater note triples you have per measuer is the numerator. For more explication on other examples of this, see this answer.

  • That's how it looked to me. Seems that the wikipedia article for tuplet will need some editing to adhere to this definition. – nightcod3r Mar 26 '16 at 18:19
  • Is there any agreement on using 'rational' also for note values? (meaning, I assume, those that result from adding consecutive powers of two). 3/8 (dotted-quarter) is rational, 1/9 is not. – nightcod3r Mar 28 '16 at 22:39
  • @nightcod3r I've never head of it, but I wouldn't be surprised. Just stick with calling them tuples for simplicity. – Dom Mar 28 '16 at 22:50
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    This is the standard musical definition of "irrational time signatures," but it is mathematically illiterate, since all time signatures that can be written as a fraction are mathematically rational. There is no logical reason why music shouldn't be written with a unequal "beats" in a mathematically irrational proportion like the square root of 2 (= approx. 1.414...) to 1, but nobody has invented a musical notation for it. (Actually, music has been performed in such rhythms for centuries, using inexact descriptions like "notes inegales" or "swing". – user19146 Mar 29 '16 at 7:42
  • @alephzero I was just going to point out that very aspect. "Rational" here is not a proper term (if taken from mathematics). In the case of note values, I wonder what would be a term for notes that depart from sums of consecutive powers of two. Tuplets cover a part of it, but what about Pi? – nightcod3r Mar 29 '16 at 8:34

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