How to calculate the length or ratio of a polyrhythmic cycle
Here we explain what the resulting fraction of the voice is, and how to perform two calculations that are very useful for composing or analyzing cyclic polyrhythms.
THE FRACTION RESULTING FROM THE VOICE
By combining the two elements that define each rhythmic voice (the fraction and the fixed iT) we obtain a resultant fraction (frR). This is the fraction that we will use to calculate both the length of the polyrhythmic cycle and the ratio between voices.
We obtain the resulting fraction by multiplying the numerator and the fixed iT of the same voice (i.e. the two values grouping pulsations), and keeping the denominator as it is. It is important to reduce the resulting fraction to its minimum values whenever possible. Let’s look at an example:
The numerator of the resulting fraction is obtained by multiplying 2 (numerator of the fraction of voice 1) by 3 (the fixed iT), resulting in 6. Since the denominator remains the same, the resulting fraction is 6/5. In this case, the fraction cannot be further reduced.
We can also perform the inverse process: Given a resulting fraction, how to express it with different fractions and fixed iT? To calculate the different possibilities, we only have to decompose the numerator into its different factors, and combine them between the numerator and the fixed iT (the denominator will always be the same). Whenever possible, the final fraction should be reduced. In the reference section of this node you can use the calculator to decompose any number into factors.
Let’s look at an example:
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In the two-voice polyrhythmsthe length of the polyrhythmic cycle is the quantity of pulsations between the beginning of the cycle where the voices are together, until the end of the cycle where the voices coincide again, and is equivalent to the numerator of the fraction resulting from voice 1.
Let’s look at an example:
In this case the length of the polyrhythmic cycle is 6 beats since the resulting fraction is 6/5.
In the three-voice polyrhythms, the length of the polyrhythmic cycle is the number of pulsations between the beginning of the cycle where all the voices are together, until the end of the cycle where all the voices coincide again, and it is equivalent to the lowest common multiple (LCM) of the numerators of the resulting fractions of the voices.
In the reference section of this node you can use the calculator to obtain the lowest common multiple.
Let’s look at an example:
In this case the length of the polyrhythmic cycle is 12 pulsations (equivalent to the m.c.m. of numerators 4 and 6).