As the others here have said, irrational time signatures are just about impossible to perform without some kind of audio cue (similar to click track).
But - some do have potential for interesting sounds I think. I've explored irrational time signatures in my Bounce Metronome program and that feature does interest composers though I don't yet know of anyone who has used it in a piece of music.
The golden ratio is especially interesting, because in a certain sense it is the "most out of sync" two rhythms can be. That's because the sequence of ratios that approximate the golden ratio approximate it more slowly than any other irrational number.
Here is a play list of some of the bounce animations for the golden ratio:
Notice how the two rhythms almost coincide at consecutive Fibonacci numbers such as 5:8, 8:13 and 13:21. That's because those are approximations to the goldenr ratio.
As you get further up the Fibonacci series, the approximations get better, and the two rhythms match more and more closely, but of course, never coincide exactly, and the closest points here approach each other more slowly than for any other tempo ratio (except some other numbers constructed from the golden ratio). For the mathematical result see continued fraction - a property of the golden ratio φ
For more about this and the separate videos, see also Golden Ratio Inharmonic Polyrhythms
I also used the golden ratio as a frequency ratio in these videos too. The golden ratio pitch interval is in a similar way "as far as possible out of tune as you can be" if you consider "in tune" as meaning in a pure frequency ratio such as 3/2, 5/4 etc. i.e. ratio between two frequencies in the harmonic series.
Now - though it is pretty impossible to play an exact golden ratio - another advantage of the golden ratio sequence is that you get lots of approximations. So you can start with 2:3, played as a polyrhythm, then 3:5, then 5:8, then 8:13, all well within a normal human player's capabilities.
It continues 21:13, 34:21, 55:34, 89:55 and 144:89 - not sure how far a human player could go up that series accurately without a click track. Surely a fair bit further than 8:13, a professional musician could probably go quite far up the series with practise,
You have all the previous Fibonacci numbers as landmarks, for instance when you play 144:89 then along the way you need to play an almost exact 89:55, and along the way to that you play an almost exact 55:34 and so on. You could get used to how far the beats need to miss each other for each number of beats in the Fibonacci series, and in which direction (the direction of the miss alternates incidentally), and indeed the amount and direction of the "miss" for the 144:89 as well, and could have other landmarks along the way too.
If anyone could get that far then they could play an almost perfect golden ratio time signature for 89, or 144 beats respectively.
See
More about this: Approaching the golden ratio with the Fibonacci sequence
You could also practise with click tracks that go silent like this one
More about it here: Golden Ratio Rhythm with each part silent for a few measures at a time independently
To go even further you probably need some kind of audio or visual cue to keep your two rhythms in the golden ratio with each other to that level of precision.
Only a computer could keep going with an irrational time signature indefinitely.
However, let's say, 55 beats, up to the 55:34 of the golden ratio rhythm or 55 seconds of playing at 60 bpm - it seems to me possible that a human performer might be able to achieve a golden ratio rhythm that long with practise, close enough to it to "sound like the same rhythm" - and possibly further. It would be interesting to hear if anyone has attempted performance of polyrhythms for the higher number ratios in the golden ratio fibonacci sequence or a near perfect golden ratio rhythm like this.
There's also the possibility of human players playing a piece of music along to a click track - or perhaps a visual metronome like Bounce (it can be easier to play in a freeer and musically expressive way along with a visual metronome), and so keep going indefinitely or as long as is required for the composition.
I've got another page here Other Inharmonic Polyrhythms though I think the golden ratio ones are in some ways the most interesting.
BTW there is also an interesting mathematical relationship between the beating partials in a perfect golden ratio dyad which might interest you:
A curious connection between the golden ratio rhythms and the golden ratio pitch interval
So - for all these reasons, I feel the golden ratio rhythm and pitch interval have some interest as a rhythm and pitch interval that could be of use in composition. I've had interested comments from composers who listen to these rhythms, but so far, I haven't yet come across any compositions that use them.
But - some do have potential for interesting sounds I think. I've explored irrational time signatures in my Bounce Metronome program and that feature does interest composers though I don't yet know of anyone who has used it in a piece of music.
The golden ratio is especially interesting, because in a certain sense it is the "most out of sync" two rhythms can be. That's because the sequence of ratios that approximate the golden ratio approximate it more slowly than any other irrational number.
Here is a play list of some of the bounce animations for the golden ratio:
Notice how the two rhythms almost coincide at consecutive Fibonacci numbers such as 5:8, 8:13 and 13:21. That's because those are approximations to the goldenr ratio.
As you get further up the Fibonacci series, the approximations get better, and the two rhythms match more and more closely, but of course, never coincide exactly, and the closest points here approach each other more slowly than for any other tempo ratio (except some other numbers constructed from the golden ratio). For the mathematical result see continued fraction - a property of the golden ratio φ
For more about this and the separate videos, see also Golden Ratio Inharmonic Polyrhythms
I also used the golden ratio as a frequency ratio in these videos too. The golden ratio pitch interval is in a similar way "as far as possible out of tune as you can be" if you consider "in tune" as meaning in a pure frequency ratio such as 3/2, 5/4 etc. i.e. ratio between two frequencies in the harmonic series.
Now - though it is pretty impossible to play an exact golden ratio - another advantage of the golden ratio sequence is that you get lots of approximations. So you can start with 2:3, played as a polyrhythm, then 3:5, then 5:8, then 8:13, all well within a normal human player's capabilities.
It continues 21:13, 34:21, 55:34, 89:55 and 144:89 - not sure how far a human player could go up that series accurately without a click track. Surely a fair bit further than 8:13, a professional musician could probably go quite far up the series with practise,
You have all the previous Fibonacci numbers as landmarks, for instance when you play 144:89 then along the way you need to play an almost exact 89:55, and along the way to that you play an almost exact 55:34 and so on. You could get used to how far the beats need to miss each other for each number of beats in the Fibonacci series, and in which direction (the direction of the miss alternates incidentally), and indeed the amount and direction of the "miss" for the 144:89 as well, and could have other landmarks along the way too.
If anyone could get that far then they could play an almost perfect golden ratio time signature for 89, or 144 beats respectively.
See
More about this: Approaching the golden ratio with the Fibonacci sequence
You could also practise with click tracks that go silent like this one
More about it here: Golden Ratio Rhythm with each part silent for a few measures at a time independently
To go even further you probably need some kind of audio or visual cue to keep your two rhythms in the golden ratio with each other to that level of precision.
Only a computer could keep going with an irrational time signature indefinitely.
However, let's say, 55 beats, up to the 55:34 of the golden ratio rhythm or 55 seconds of playing at 60 bpm - it seems to me possible that a human performer might be able to achieve a golden ratio rhythm that long with practise, close enough to it to "sound like the same rhythm" - and possibly further. It would be interesting to hear if anyone has attempted performance of polyrhythms for the higher number ratios in the golden ratio fibonacci sequence or a near perfect golden ratio rhythm like this.
There's also the possibility of human players playing a piece of music along to a click track - or perhaps a visual metronome like Bounce (it can be easier to play in a freeer and musically expressive way along with a visual metronome), and so keep going indefinitely or as long as is required for the composition.
I've got another page here Other Inharmonic Polyrhythms though I think the golden ratio ones are in some ways the most interesting.
BTW there is also an interesting mathematical relationship between the beating partials in a perfect golden ratio dyad which might interest you:
A curious connection between the golden ratio rhythms and the golden ratio pitch interval
So - for all these reasons, I feel the golden ratio rhythm and pitch interval have some interest as a rhythm and pitch interval that could be of use in composition. I've had interested comments from composers who listen to these rhythms, but so far, I haven't yet come across any compositions that use them.
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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