Generator: Difference between revisions

In [[MOS]] and [[rank]]-2 temperament contexts, the '''generator''' of a MOS or a rank-2 temperament is an interval that you stack up and reduce by the [[period]] of the MOS to construct the MOS pattern within each period. Along with the [[period]], it is one of two defining intervals of the [[MOS]]. For example:  

* In diatonic (LLLsLLs), the perfect fifth is a generator: stacking 6 fifths up from the tonic and reducing by the octave produces the pattern LLLsLLs, the Lydian mode. Note that the perfect fourth and the perfect twelfth are also generators.

* One example for a MOS with multiple periods per octave: for [[pajara]][10] (ssLssssLss), the perfect fifth ([[~]][[3/2]]) is a generator and the half-octave is the period.

A '''generating set''' of a [[Wikipedia: Group (mathematics)|group]] (such as a [[JI subgroup]] or a [[regular temperament]] based on it) is a subset of the elements of the group which is not contained in any [[Wikipedia: Subgroup|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 [[Wikipedia: Abelian group|abelian group]], it is called a [[Wikipedia: Finitely generated abelian group|finitely generated abelian group]]. An element of a generating set is called a '''generator'''.

A '''minimal generating set''' is a generating set which has no "redundant" or "unnecessary" generators. For example, {2, 3, 5} and {2, 3, 5/3} are minimal generating sets for the JI subgroup 2.3.5. However, {2, 3, 5, 15} is not a minimal generating set: 15 = 3 · 5 so we can take out 15 from this generating set.

If the abelian group is written additively, then if <math>\lbrace g_1, g_2, \ldots g_k \rbrace</math> is the generating set, every element <math>g</math> of the group can be written