The Collatz conjecture is a well-known, unsolved problem. It poses a question about the Collatz map \(f\) from the set of positive integers to the same set \[f(x) = \begin{cases}3 x + 1 & \text{if } x \text{ is odd};\\\frac{x}{2} & \text{if } x \text{ is even}.\end{cases}\]A weaker subproblem, called weak Collatz conjecture, asks the (non-)existence of non-trivial Collatz cycles i.e. whether there exists a positive integer \(n\) such that \(f^m(n) = n\) for some positive integer \(m\), besides ones from the trivial cycle \(1-2-4\).
Here we point out that such a non-trivial cycle, if any, will yield a variant of the Pythagorean tuning. It is the following near miss that the twelve-tone Pythagorean temperament is based on: \[\frac{3^{12}}{2^{19}} \equiv 1.0136432647705078\]So the comma is sufficiently small.
If we have a non-trivial cycle \(C = \{c_1, c_2, ..., c_k\}\) of length \(k\), then any \(c_i\) must be bigger than \(2^{68}\) [1]. It means that, by dividing each \(c_i\) by 2 enough times, we find a \(k\)-tone temperament without any comma at the expense of negligible \(+1\) every time the generator is called.
Of course the tuning is never of practical use as \(k\) will be huge.
[1] Tao, T., 2022. Almost all orbits of the Collatz map attain almost bounded values. Forum of Mathematics, Pi 10, e12. doi:10.1017/fmp.2022.8