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<ref>The original definition of support in this RTT sense was given by [[Gene Ward Smith]] in [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 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]].

A [[regular temperament]] is '''supported''' by an [[equal temperament]] that makes all of its [[comma]]s [[vanish]].<ref>The original definition of support in this RTT sense was given by [[Gene Ward Smith]] in [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 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, [[22edo|22-ET]] supports [[pajara]], because pajara makes [[225/224]] and [[64/63]] vanish, and so does 22-ET. The supporting temperament will make at least one additional comma vanish; in this example, 22-ET makes [[245/243]] vanish.

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== 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 also makes all the commas that the lower-nullity temperament vanishes vanish.<ref>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.</ref><ref>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]].</ref> 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 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 also makes all the commas that the lower-nullity temperament vanishes vanish.<ref>As an edge case, JI is conceptualized as a temperament where no intervals are made to vanish, so any temperament "supports" JI; in other words, any temperament is a temperament of JI.</ref><ref>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]].</ref> 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 22-ET and pajara example above, we can see that pajara has a mapping form {{rket|{{map|12 19 28 34}} {{map|22 35 51 62}}}}, which contains 22-ET {{map|22 35 51 62}} as its second row.

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-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]] in [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 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]].
+A [[regular temperament]] is '''supported''' by an [[equal temperament]] that makes all of its [[comma]]s [[vanish]].<ref>The original definition of support in this RTT sense was given by [[Gene Ward Smith]] in [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 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, [[22edo|22-ET]] supports [[pajara]], because pajara makes [[225/224]] and [[64/63]] vanish, and so does 22-ET. The supporting temperament will make at least one additional comma vanish; in this example, 22-ET makes [[245/243]] vanish.
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 == 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 also makes all the commas that the lower-nullity temperament vanishes vanish.<ref>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.</ref><ref>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]].</ref> 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 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 also makes all the commas that the lower-nullity temperament vanishes vanish.<ref>As an edge case, JI is conceptualized as a temperament where no intervals are made to vanish, so any temperament "supports" JI; in other words, any temperament is a temperament of JI.</ref><ref>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]].</ref> 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 22-ET and pajara example above, we can see that pajara has a mapping form {{rket|{{map|12 19 28 34}} {{map|22 35 51 62}}}}, which contains 22-ET {{map|22 35 51 62}} as its second row.