In conventional music, the octave is divided into the twelve frequency Chromatic Scale, but then only subsets of those twelve frequencies form the key used at any given time. I understand why the twelve frequencies are a good choice, as their overtones line up nicely, but suppose I want to choose my own set of frequencies to serve as notes for some music. Perhaps lining up the overtones cleanly is less important or even undesirable in this context.
Is there any reason I should want to build a larger set of notes, analogous to the Chromatic Scale, and then derive keys from it as subsets, rather than simply building a key directly from frequencies every time I need one? Is it valuable to have a few (such as twelve) repeated frequencies utilized from song to song and key change to key change, or is the only reason for such a restriction the overtone math and physical instrument logistics?