Support: Difference between revisions

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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 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 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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 == 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 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 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.