Support
A regular temperament is supported by an equal temperament that tempers out all of its commas. The ET thus supports this temperament.
For example, 22-ET supports pajara, because pajara tempers out 225/224 and 64/63, and 22-ET tempers out both of those. The supporting temperament will temper out at least one additional comma; in this example, 22-ET tempers out a single additional comma, 245/243.
Generalization
An equal temperament is the same thing as a rank-1 temperament, and the initial definition given here where the supporting temperament is rank-1 is by far the most common use case. However, in general, we can say that any lower-nullity (higher-rank) temperament is supported by a higher-nullity (lower-rank) temperament if the lower-nullity temperament's comma basis is a subset of the higher-nullity temperament's[1][2]. Technically speaking, we'd say that the lower-nullity temperament's comma space is a subspace of the higher-nullity temperament's comma space.
An equivalent generalized definition of "support" would be to say that the lower-rank temperament's mapping is a subset of the higher-rank temperament's mapping. In this case, the technical definition would be that the lower-rank temperament's mapping-row space is a subspace of the higher-rank temperament's mapping-row space. Another way to say this is that one can find forms of the mappings for these two temperaments where the higher-rank mapping is identical to the lower-rank mapping but with additional mapping rows. To use the 22-ET and pajara example above, we can see that pajara has a mapping form [⟨12 19 28 34] ⟨22 35 51 62]⟩, which contains 22-ET ⟨22 35 51 62] as its second row.
Other informal definition
The word "supports" is also used in a more informal and generic sense, both in and outside of regular temperament theory: a pitch structure A supports another pitch structure B when A can be used for B. For example, a temperament or tuning may be said to support a scale or a chord, or a scale may be said to support a chord or harmony within a certain prime- or odd-limit, etc.
Most typically, this sense of "support" is reserved for such cases where not only is B merely possible or valid with A, but A is actually good for B. Due to the fact that the technical RTT sense of "supports" defined above is strictly mathematical and makes no such stipulation of aesthetic goodness, it may occasionally conflict with this informal definition and surprise some readers.
Footnotes
- ↑ This is an edge case, but a temperament should have at least one comma to satisfy this definition; JI may be conceptualized as a temperament where nothing is tempered out, but clearly it would be silly to say that any temperament "supports" JI.
- ↑ Example uses of this sense can be found on the following pages: Subgroup Temperament Families, Relationships, and Genes#Support, Meet and join#Intra-Subgroup Temperament Meet and Join, and Interior product#Applications.