Generators

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A set of generators for a group is a subset of the elements of the group which is not contained in any proper subgroup, which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an abelian group, it is called a finitely generated abelian group.

If the abelian group is written additively, then if {g1, g2, ... gk} is the generating set, every element g of the group can be written

g = n1g1 + n2g2 + ... + nkgk

where the ni are integers. If the group operation is multiplicative,

g = g1^n1 g2^n2 ... gk^nk

An important example is provided by regular temperaments, where if a particular tuning for the temperament is written additively, the generators can be taken as intervals expressed in terms of cents, and if multiplicatively, as intervals given as frequency ratios. Another example is provided by just intonation subgroups, where the generators are a finite set of positive rational numbers. These two example converge when we seek generators for the abstract temperament rather than any particular tuning of it.