Basis
A basis (plural bases, pronounced BAY-sees, or /ˈbeɪ siz/) is a list of vectors that represents the infinite set of vectors that are combinations of them. The corresponding infinite set is called its subspace.
Bases are mathematical structures that come from the field of 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 makes the set of commas explicitly listed in a comma basis vanish, then it also makes any interval that's equal to any combination of those commas vanish. We could never possibly list the infinitude of commas made to vanish in this way, so instead we carefully choose a minimal set of commas that is capable of representing all of them.
Examples
For example, the comma basis [[4 -4 1⟩] only includes [4 -4 1⟩, but it represents the subspace that also includes [8 -8 2⟩, [12 -12 3⟩, and all possible multiples of this vector, including negative ones like [-4 4 -1⟩.
The comma basis [[4 -4 1⟩ [7 0 -3⟩] only includes [4 -4 1⟩ and [7 0 -3⟩, but it represents the subspace that also includes [4 -4 1⟩ + [7 0 -3⟩ = [11 -4 -2⟩, and 2·[4 -4 1⟩ + -1·[7 0 -3⟩ = [1 -8 5⟩, and many many more.
Mathematical details
In mathematical language, a basis for a subspace of a vector space is a minimal set of vectors that span the subspace.
For example, a mathematical word for the set of all commas made to vanish by a temperament is a "nullspace", and specifically this is the nullspace of its mapping matrix; "nullspace" uses the word "space" in this same sense of a "subspace".
The vectors that appear explicitly in a basis are called the basis vectors. More generally we can refer to these as basis elements; for example, if [-1 1⟩ was a basis vector, we could say [math]\frac32[/math] was a basis element.
The verb used for the process by which linear combinations of the basis vectors reach all of the subspace vectors is "spanning"; we say that the basis vectors span the subspace.
Importantly, a set of vectors that spans a subspace but is not full-rank, that is, includes linearly dependent vectors, or in less technical terms "redundant" vectors, is not considered a basis; in that case, it is merely a spanning set.
Relationship to groups
The basis is a concept used across vector spaces and free abelian groups. The analogous concept for groups and modules in general, which are structures within the broader field of abstract algebra, is known as the minimal generating set.
| "Within a {}, …" | "…a {}…" | "…consists of {}…" | "…which {}…" | "…a {}." |
|---|---|---|---|---|
| vector space | basis | basis vectors | span | subspace |
| group | minimal generating set | generators | generate | subgroup |
The sense of "subgroup" in this table is different than 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 scales in the context of periods; for further disambiguating information, see generator.
Basis vs subspace
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.
And so it is not disingenuous to call something like 2.3.7 or [[4 -4 1⟩ [7 0 -3⟩] a "subspace" — if we are indeed referring to the infinitely large thing spanned by the this basis, and not the basis itself — because the entire point of bases are to enable representation of these such subspaces.
And when we are referring to the basis itself, it's perfectly fine to refer to a "subspace basis" as a "basis" for short, as we have been doing throughout this article, because there's no other type of basis in this context; something being a "basis" here implies that it is a "subspace basis".
We do have to be careful, though, to remember that a subspace has infinitely many possible basis representations. This is why canonical forms are typically developed for them, as they have been for mappings and comma bases, so that each subspace does have a uniquely identifying basis.