Subgroup temperament families, relationships, and genes: Difference between revisions
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= Preliminaries = | = Preliminaries = | ||
The various "subgroups" we talk about are all subgroups of the group of all possible JI intervals, which can be identified with the set of positive rational numbers, which forms an infinite-rank free abelian group. However, mathematically, it is often much easier for us to choose some arbitrarily large but still finite-rank subgroup to formalize everything in, typically the p-limit for some "large enough" choice of p. To that extent we will choose some sufficiently large JI group, which we will call the ''universe group'' or simply the ''universe'', which all the subgroups are subgroups of. | The various "subgroups" we talk about are all subgroups of the group of all possible JI intervals, which can be identified with the set of positive rational numbers, which forms an infinite-rank [[free abelian group]]. However, mathematically, it is often much easier for us to choose some arbitrarily large but still finite-rank subgroup to formalize everything in, typically the p-limit for some "large enough" choice of p. To that extent we will choose some sufficiently large JI group, which we will call the ''universe group'' or simply the ''universe'', which all the subgroups are subgroups of. | ||
Sometimes it can be useful to choose a deliberately small universe just to see how the general system is structured. Most of the regular temperaments on the wiki fit into a 13-limit universe, with a few instances of 17 and 19 here and there. However, in theory, you can simply go as high as you want, with any sufficiently large prime-limit as the universe. | Sometimes it can be useful to choose a deliberately small universe just to see how the general system is structured. Most of the regular temperaments on the wiki fit into a 13-limit universe, with a few instances of 17 and 19 here and there. However, in theory, you can simply go as high as you want, with any sufficiently large prime-limit as the universe. | ||