Basis: Difference between revisions

Bases are mathematical structures that come from the field of [[Wikipedia:Linear algebra|linear algebra]], and are used in [[regular temperament theory]], where the most common example of a basis is a [[comma basis]]. The fact that a comma basis is a ''basis'' conveys how when a temperament [[tempers out]] the set of commas explicitly listed in a comma basis, then it also tempers out any interval that's equal to any combination of those commas. We could never possibly list the infinitude of commas tempered out, so instead we carefully choose a minimal set of commas that is capable of representing all of them.

For example, the comma basis {{bra|{{vector|4 -4 1}}}} only includes {{vector|4 -4 1}}, but it represents the subspace that also includes {{vector|8 -8 2}}, {{vector|12 -12 3}}, and all possible multiples of this vector, including negative ones like {{vector|-4 4 -1}}.  

The comma basis {{bra|{{vector|4 -4 1}} {{vector|7 0 -3}}}} only includes {{vector|4 -4 1}} and {{vector|7 0 -3}}, but it represents the subspace that also includes {{vector|4 -4 1}} + {{vector|7 0 -3}} = {{vector|11 -4 -2}}, and 2·{{vector|4 -4 1}} + -1·{{vector|7 0 -3}} = {{vector|1 -8 5}}, and many many more.

In mathematical language, a [[Wikipedia:Basis_(linear_algebra)|basis]] for a [[Wikipedia:Linear_subspace|subspace]] of a [[Wikipedia:Vector_space|vector space]] is a minimal set of [[Wikipedia:Vector_(mathematics_and_physics)|vectors]] that [[Wikipedia:Linear_span|span]] the subspace.  

Importantly, a set of vectors that spans a subspace but is not [[full-rank|full-grade]], that is, includes [[linear dependence|linearly dependent]] vectors, or in less technical terms "redundant" vectors, is not considered a basis; in that case, it is merely a spanning set.

Bases are a concept in vector spaces, the subject of linear algebra. The analogous concept for [[Wikipedia:Group_(mathematics)|groups]] (and [[Wikipedia:Module_(mathematics)|modules]]), which are more general structures within the broader field of [[Wikipedia:Abstract_algebra|abstract algebra]], is a [[Wikipedia:Generating_set_of_a_module|minimal generating set]].

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The sense of "subgroup" in this table is different than [[Just_intonation_subgroup|the specialized meaning it has taken on in RTT]]. Also, the sense of "generator" in this table is different than the one used for [[MOS scale]]s in the context of [[period]]s; for further disambiguating information, see [[generator]].

Subspaces and bases have a close relationship. A basis, even in its everyday dictionary definition, is an underlying support or foundation ''for something'', and in this mathematical case, that something is a subspace. Without bases, it would be much more challenging to communicate about subspaces; they're quite specific objects, but they happen to be infinitely large, and so bases were developed to be finite representations of them, for convenience.