Support: Difference between revisions

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A [[regular temperament]] is '''supported''' by an [[equal temperament]] that [[tempers out]] all of its [[comma]]s. The ET thus '''supports''' this temperament.

A [[regular temperament]] is '''supported''' by an [[equal temperament]] that [[tempers out]] all of its [[comma]]s<ref>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."</ref>. The ET thus '''supports''' this temperament.

For example, [[22edo|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]].

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== 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 limit|prime-]] or [[odd-limit]], etc.

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 limit|prime-]] or [[odd-limit]] or [[interval basis|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'''. 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.

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-A [[regular temperament]] is '''supported''' by an [[equal temperament]] that [[tempers out]] all of its [[comma]]s. The ET thus '''supports''' this temperament.
+A [[regular temperament]] is '''supported''' by an [[equal temperament]] that [[tempers out]] all of its [[comma]]s<ref>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."</ref>. The ET thus '''supports''' this temperament.
 For example, [[22edo|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]].
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 == 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 limit|prime-]] or [[odd-limit]], etc.
+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 limit|prime-]] or [[odd-limit]] or [[interval basis|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'''. 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.