Support

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A regular temperament is supported by an equal temperament that tempers out all of its commas[1]. 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[2][3]. 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 temperament may be said to support a tuning of another temperament, or scale may be said to support a chord or harmony within a certain prime- or odd-limit or interval subspace, 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. The technical RTT sense of "supports" defined above is strictly mathematical and makes no such stipulation of aesthetic goodness, however. According to it, the 2c map for 2-ET supports meantone despite tuning the fifth to 600c, well outside the diamond tuning ranges for meantone. Furthermore, since 0-ET tempers out every comma, it therefore supports every temperament, though clearly it does not do so in any musically useful sense. Due to this key difference between the technical RTT definition and the informal general definition, occasionally usages may conflict and surprise some readers.

According to the page EDO vs ET, the technical definition of "support" given on this page has met with some contention since at least 2011, though the present author is not aware of links to places where it has been debated. The original definition from Gene Ward Smith dates from 2005, and has been generally accepted and propagated throughout this wiki.

See also

Footnotes

  1. The original definition of support in this RTT sense was given by Gene Ward Smith here: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_13712#13967, "Suppose T is a wedgie, and v is an equal temperament val. Then v supports T if and only if T^v = 0."
  2. 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.
  3. 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.