It’s amazing how the research community in Computer Science misunderstands music and music modeling. They come up with non-discrete models! Holy cow! Even Elaine Chew’s spiral is a bad model that fails to capture the discrete nature of music theory. And all these other guys looking at statistics and Markov chains. Holy cow! They really don’t grasp music theory. These guys should take a close look at abstract algebra and topology abandoning everything that is continuous. Mainly the Computer Science research community. It should really abandon all this continuous thinking and get serious with discrete Math. Given that computers count more 'naturally' it would be natural for folks in Computer Science to make better use of Discrete Math. Maybe it's the engineers that just contaminate Computer Science with their continuous Euclidean view of the world.
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.