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The secondary cycle

What is the secondary cycle?

In 3-voice polyrhythms, a secondary cycle may appear only between voices 1 and 2, which is contained within the polyrhythmic cycle that encompasses all voices, including voice 0. This secondary cycle is generated when the fractions resulting from voice 1 and 2 have a common factor in both denominators, other than 1. The secondary cycle does not coincide with voice 0 (the pulsations), and its length is a decimal number.

 

In this case the common factor in both denominators, other than 1, is 3.

 

To calculate the common factor of two numbers, you can use the calculator in the reference section. The same is done in the two steps shown in the following example with the numbers 12 and 30:

  • First we decompose both numbers into prime factors:

The 12 = 2 x 2 x 3 =22 x 3, and the 30 = 2 x 3 x 5

  • Then we observe the prime factors that coincide in both numbers, if they have different exponents we choose the ones with the lowest exponent, and multiply them together:

In this case the coincident prime factors are 2 (2 and22 have different exponents, so we choose the smaller one, which is 2) and 3. And we multiply them together: 2 x 3 = 6. Thus we obtain that 6 is the common factor of 12 and 30.

 

How to calculate the length of the secondary cycle?

From the frRs of voices 1 and 2, we must calculate the least common multiple of the numerators, and the common factor of the denominators. The length of the secondary cycle is the result of dividing both results:

 

 

In the example of the previous graph the resulting fractions of voice 1 and 2 are 2/3 and 5/3. The least common multiple of the numerators is 10, and the common factor of the denominators is 3. If we apply this simple formula, we obtain that the length of the secondary cycle is the result of dividing 10 by 3, which is equal to 3.33 beats.

 

Let’s look at a new example:

 

 

The resulting fractions of voice 1 and voice 2 are 5/6 and 3/4. The least common multiple of the numerators is 15, and the common factor of the denominators is 2. Applying the formula we obtain that the length of the secondary cycle is the result of dividing 15 by 2, which is equal to 7.5 beats.