I don’t know if I’m inventing the wheel, but I devised a quick way to order the “complexity” of rational numbers because it made sense in a specific context I was working on. It seems like it makes sense, but at the same time, maybe I’m doing something very wrong, or it’s a well-known procedure that I ignore. I’d be glad if anyone wants to take a look, comment, or even point me to another way to do this. I’m sure many did something similar before, but I just don’t know.
The code calculates a rational number’s “complexity” (for the lack of a better word) or “simplicity” to sort a list of rationals. It is calculated based on the number of terms (continued fractions), the average coefficient size, and whether the continued fraction representation of the number has a repeating pattern or not (two numbers and a boolean). I use those three measures to compare and generate an ordering type for rationals.
Reading the link, I think you mean strict total order, right? Yes, I thought that people in the JI community would have done something equivalent, but I didn’t know where to look. The practical application includes rhythmic stuff.
Your relation is missing the antisymmetric property (3/2 R 2/3, 3/2 R 2/3 are both true, but 2/3 != 3/2), so it doesn’t define a strict or non-strict total order on rationals, only on these int-rational-bool triples.
Yes, but isn’t it the nature of continued fraction representation, since it’s not always possible to exactly reconstruct the original number? I did some tests and it seems not possible. It seems it can be helpful in a loose way, since the quality itself it is measuring is difficult to define.
Maybe it can be improved if it considers another aspect of the rational number to compare.
I did not know about the Benedetti height, but it seems to be similar to a positive definite cosine similarity applied to natural numbers a,b which could be the numerator and denominator of a rational number:
k(a,b) := gcd(a,b)^2/(a*b)
Since it is a positive definite kernel, it means that you can do Cholesky decomposition on a Gram matrix if you wanted to measure the similarities between two numbers a,b : 0 < k(a,b) <= 1, where 1 means identical, and near 0 means very different.
I should mention that I have used this to define the simplicity of a ratio and so to measure consonance in two pitches in Western music:
So basically it is a mapping from pitches to a vector space, where we associate to each pitch in music a vector in such a way, that two pitches which sound consonant when played together are more similar ( have a larger inner product k(a,b) then two pitches with a lower inner product which should sound more dissonant.)
Here is a piece for piano which has been done with the method described above:
Thanks for your explanation. I noticed that it is the inverse of the Benedetti height in this case.
The function k(a,b) = gcd(a,b)^2/(ab) is positive definite over the natural numbers, which means, that if you consider the Gram matrix G_n = (k(i,j))_{1 <= i,j <= n} of pairwise k(i,j), it will be positive definite and this means one can do the Cholesky decomposition on this matrix to get the feature vector phi(i) for each natural number $i$: You might want to look here ( nt.number theory - A geometric proof that there are infinitely many primes? - MathOverflow ) for a direct definition of the feature vector phi, which does not use the Cholesky decomposition : < phi(a) , phi(b) > = k(a,b) = gcd(a,b)^2 / (ab).
You might be interested in Euler’s Gradus Suavitatus function, which attempts to score consonance of rational numbers. (See for example Gradus Suavitatis, Music and Mathematics)
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