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As I understand it, an irrational meter is when the bottom number on the time signature is not a power of 2, i.e. 1,2,4,8,16, 32, etc. An example irrational meter that I've seen is 5/24. According to Wikpedia:

Irrational time signatures (rarely, "non-dyadic time signatures") are used for so-called irrational bar lengths, that have a denominator that is not a power of two (1, 2, 4, 8, 16, 32, etc.). These are based on beats expressed in terms of fractions of full beats in the prevailing tempo—for example 3/10 or 5/24

Can any positive integer be used for irrational meters or are there certain restrictions?

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    This is an interesting question, can you cite a reliable reference for this? I ask because "irrational" in math means cannot be written as n/m, for integer n, m. I have also seen question on this site about 2/3 or N/3 time signatures.
    – user50691
    Jan 4, 2021 at 16:24
  • @ggcg So far haven't found an "official" definition, but the usage does seem to be generally accepted. Aside from Wikipedia and internet discussions, there is, for example, this paper on the subject.
    – Aaron
    Jan 4, 2021 at 16:57
  • @ggcg I agree that the term is unfortunate, especially since time signatures began their lives as ratios, but I can't think of a better one. Can you? "Rational-with-a-prime-factor-other-than-two-in-the-divisor time signature" is too cumbersome.
    – phoog
    Jan 4, 2021 at 17:04

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There is no rule in this regard; however the Wikipedia article includes this important point:

These signatures are of utility only when juxtaposed with other signatures with varying denominators; a piece written entirely in 4/3, say, could be more legibly written out in 4/4.

This sort of irrational time signature is just a notational convenience to express speed relationships between measures.

For mathematically irrational "time signatures" (actually, "metric ratios" would be the better term), take a look at Conlon Nancarrow. He used truly irrational numbers to describe metric relationships in his cannons. This is discussed here on SE MP&T.

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    "any positive real number": the numerator and denominator of a time signature must themselves be rational. You could argue for 2/2⅔ instead of 6/8, but how would you write an infinite nonrepeating decimal in a time signature? If you use a symbol representing an irrational number, what note values would you use (for example, for music written in a 3/π time signature)? Also, cannons or canons?
    – phoog
    Jan 4, 2021 at 17:10
  • There are some obvious practical limits, but see the Nancarrow link.
    – Aaron
    Jan 4, 2021 at 17:12
  • From that answer: "This does not refer to a time signature, a regular grouping of beat stressing, but to the ratio between the two voices in the canon."
    – phoog
    Jan 4, 2021 at 17:14
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    I see no reason why you couldn't use polyrhythms to fill out non-algebraic or irrational lengths of time, thus allowing time signatures to use both kinds of numbers. But this has no practical use, since no one could tell if you really played a 3:pi rhythm or just a 3.14:3 rhythm.
    – Edward
    Jan 5, 2021 at 0:45
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    @Edward - yep. It should perhaps be pointed out (if anyone here doesn't know it already) that Conlon Nancarrow composed for the player piano, not for humans. I also don't see how irrational meters could be of any possible musical use for human-performed music. So trying to figure out how to notate them is not really a pressing issue.
    – Scott Wallace
    Jan 5, 2021 at 11:29

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