Support

From Xenharmonic Wiki
Jump to navigation Jump to search

A regular temperament is supported by an equal temperament that tempers out all of its commas[1]. The equal temperament thus supports this temperament, or equivalently stated, the equal temperament is a temperament of this temperament, in the same sense as a temperament is a temperament of just intonation.

For example, 22et supports pajara, because pajara tempers out 225/224 and 64/63, and 22et tempers out both of those. The supporting temperament will temper out at least one additional comma; in this example, 22et tempers out 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 the most common use case as of 2022. However, in general, we can say that any lower-nullity (higher-rank) temperament is supported by a higher-nullity (lower-rank) temperament if the higher-nullity temperament tempers out all the commas the lower-nullity temperament does[2][3]. Technically speaking, we would 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 maps all intervals the same way as the higher-rank temperament does. 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 22et and pajara example above, we can see that pajara has a mapping form [12 19 28 34] 22 35 51 62], which contains 22et 22 35 51 62] as its second row.

Other informal usage

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 2et supports meantone despite tuning the fifth to 600¢, well outside the diamond tuning ranges for meantone. Furthermore, since 0et 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 in [1], "Suppose T is a wedgie, and v is an equal temperament val. Then v supports T if and only if T^v = 0."
  2. As an edge case, JI is conceptualized as a temperament where nothing is tempered out, so any temperament "supports" JI; in other words, any temperament is a temperament of 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.