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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?

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You wrote your question from the perspective of modern Western music based on 12-tone equal temperament (12TET). The main advantage of this choice is that all intervals in all keys have the same size. This allows to use complex harmonies and modulations. However, the "lining up of the overtones" in 12TET is only approximate, and some people dislike its sound for this reason.

You are asking about creativity, and since you want to depart from conventions, it's totally up to you what you can do. What I however can recommend is to study music that departs from 12TET. Learning about what has been explored may help you to make your choices. Example directions to explore could be baroque music, contemporary classical music, or traditional/classical music of cultures other than Western.

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