Given the rhythmic fragment
I model it as the following:
0 mod 2 = 0
1 mod 2 = 1
(0 + 2) mod 2 = 0
(1 + 2) mod 2 = 1
(3 + 4) mod 4 = 3
In 0 mod 2 why do I chose 2? It’s because given C = 4/4, I get 4 in the upper part and I factorize 4 into 2^p1*3^p2*5^p3... I get p1 > 0. That’s enough to select 2 as the only prime in this case. I could begin trying to fit the numbers into n mod 1 but that’s not interesting. It would only capture the first notes of the bars. The main reason is that the half-note is the maximum subdivision of the example. So I begin modeling it as n mod 1 * 2 (remember that p1 > 0). Then I try n mod 2 * 2 and I stop there because the quarter note is the finest quantization I can get for the segment. And there you go! I just modeled the rhythmic fragment in a discrete way.
Now we can group the notes according to their congruence. Now things get interesting! Mr Thomas Noll, Andreatta Moreno and Mazzola, why didn’t you model it this way? Why come up with functions for meter that are too procedural in nature? Peraí um pouquinho, tchê! It’s much better to model rhythm using algebra! And you guys are the math gods. I remember reading one paper by Noll and another by Elaine Chew that just create a procedural function to model meter. This is much closer to how a musician sees rhythm in a purer mathematical form.
If we group the notes above according to sharing modules we get
( 0 mod 2, 1 mod 2, (0 + 2), (1 + 2) ) mod 2 which are the 4 notes, fitting into the half-note quantization.
Then we get (3 + 4) mod 4 for the last note.
In the next post, congruence will help us correlate these notes.
