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Time Signatures and polyrhythms

Polyrhythms (Cross rhythms)

Fractional beats per bar

Fractional beats per bar audio clips

Intro - Audio examples - More examples - What are fractional numbers of beats per bar? - Maths background - how to explore this further

Intro

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This page is for you if you are interested in an unusual type of rhythm with fractional numbers of beats per bar. For instance you may have PI beats to the bar or the golden ratio as the number of beats so that the beats drift in and out of phase with the bar line. If that's what you are looking for or it sounds intriguing, read on.

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Audio examples - a clip which is as polyrhythmic and as inharmonic as you can get

Here you can listen to the most inharmonic possible pitch ratio combined with the most polyrhythmic possible rhythm (I use g for the golden ratio phi).
g / 4.mid

Here it is again as an mp3: g / 4.mp3 (4.8 MB)

It's a steady beat and I've just added some volume variation to help bring it out a bit.

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More of these rhythms

To listen these rhythms at any tempo you can use the Fractional Harmonics metronome which is part of Bounce Metronome Pro. You get a free 30 day Test drive - with all the features completely unlocked. Get your free trial here.

If you liked that one, what about adding a couple more rhythms and pitches in the same way?

Here I added a couple more rhythms and pitches in the same way - as g^2 (another interval of phi and speed up of tempo by phi) and g^3 - the same again.

This time I've used varying tempo and a lilt for the beats:

g^3_/_4_with_g^2_/_4_with_g_/_4_with_lilt_varying_tempo.mid

or as an mp3: g^3_/_4_with_g^2_/_4_with_g_/_4_with_lilt_varying_tempo.mp3

Here it is again with all three rhythms, and with the lilt, but without the varying tempo
g^3_/_4_with_g^2_/_4_with_g_/_4.mid

or as an mp3: g^3_/_4_with_g^2_/_4_with_g_/_4.mp3 (4.8 MB)

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PI/4 example

And here is Pi/4 which has closer beats more often because of close rational
approximations like 22/7

PI / 4.mid

or as an mp3: PI / 4.mp3 (4.8 MB)

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What are fractional numbers of beats per bar?

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The idea there is that the number of beats to a bar isn't a whole number. So the beats drift in and out of phase with the bar.

If the number is irrational, i.e. a number like PI or the golden ratio which you can only show approximately using fractions, then the beats never quite hit the bar line exactly, though they may get very close to it after many bars.
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Why is the golden ratio rhythm the most polyrhythmic and inharmonic possible rhythm?

If you play phi/4 i.e. a golden ratio number of beats to the bar, then it is the most polyrhythmic rhythm possible in a certain sense - you have to wait most bars for the beats to get close to the bar beat.

That works because the golden ratio is as far as you can get from any simple fraction.

For a nice demo of this, see Nature golden ratio - a nice demo of how the golden ratio is as far as you can get from any simple fraction, and how this explains the way sunflowers form using the golden ratio and fibonacci number related spirals

As ratios it is approached closer and close by 3/2, 5/3, 8/5, 13/8, 21/13, ... After 5 bars (i.e. 5*phi) and 8 beats the difference is 0.09 of a beat. After 8 bars (8*phi) and 13 beats (i.e. on the 14th beat), it's 0.06 of a beat, then after 13 bars (13*phi) and 21 beats, the difference is 0.035 of a beat, and so on, so the numbers get closer together. When you get to 21 beats (i.e. on the 22nd beat) then you have to pay close attention to hear the drift at all - but the numbers come together more slowly than for any other fractional number of beats per bar.

For similar reasons two pitches at a frequency ratio of phi are as far as you can get from pure ratio type harmonies, so the musical interval is the most inharmonic possible (approached as pure ratio harmonies by 3/2, 5/3, 8/5, 13/8, 21/13, ... again). Yet it can sound very pleasant, and repeatable too, not even needing to be resolved into a more harmonious interval.

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How to explore this further

To play more rhythms like this or make your own audio clips get your free 30 day Test drive of the Fractional Harmonic metronome in Bounce Metronome Pro - with all the features completely unlocked.

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