Meet and join: Difference between revisions

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== Subgroup Temperament Meet and Join ==

If A and B are two ~~subgroup~~ on different ~~temperaments~~, then there are similarly two natural operations that we can use to combine them: informally, we can look for the ~~simplest~~ temperament ~~that supports~~ both, and ~~most complex~~ temperament ~~supported by~~ both.

If A and B are two temperaments on different subgroups, then there are similarly two natural operations that we can use to combine them: informally, we can look for the "largest" temperament supported by both, and the "smallest" temperament that supports both, in a sense to be made precise below.

The first is found by taking the intersection of the two subgroups and the intersection of the two temperament kernels, independently, producing another subgroup temperament. This is the '''join''' of the two subgroup temperaments, which reduces to the prior definition of the join if the two subgroups are equal. The join is the ~~simplest~~ temperament that ~~supports~~ both A and B, in the sense that any other temperament that ~~supports~~ both A and B also ~~supports~~ the join. Every comma tempered out by ''both'' A and B is also tempered out in the join, and vice versa.

The first is found by taking the intersection of the two subgroups and the intersection of the two temperament kernels, independently, producing another subgroup temperament. This is the '''join''' of the two subgroup temperaments, which reduces to the prior definition of the join if the two subgroups are equal. The join is the "largest" temperament that both A and B support, in the sense that any other temperament that both A and B support is also supported by the join. Every comma tempered out by ''both'' A and B is also tempered out in the join, and vice versa.

The second is found by extending the two subgroups to the simplest subgroup which includes both, and then repeating with the two kernels. This is the '''meet''' of the two subgroup temperaments. The meet is the ~~simplest~~ temperament that ~~is supported by~~ both A and B, in the sense that if any other temperament is also ~~supported by~~ both A and B, it ~~is supported by~~ the meet. Every comma tempered out by ''either'' A or B is also tempered out in the meet, and vice versa.

The second is found by extending the two subgroups to the simplest subgroup which includes both, and then repeating with the two kernels. This is the '''meet''' of the two subgroup temperaments. The meet is the "smallest" temperament that supports both A and B, in the sense that if any other temperament also supports both A and B, it supports the meet. Every comma tempered out by ''either'' A or B is also tempered out in the meet, and vice versa.

== Examples ==

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 == Subgroup Temperament Meet and Join ==
-If A and B are two subgroup on different temperaments, then there are similarly two natural operations that we can use to combine them: informally, we can look for the simplest temperament that supports both, and most complex temperament supported by both.
+If A and B are two temperaments on different subgroups, then there are similarly two natural operations that we can use to combine them: informally, we can look for the "largest" temperament supported by both, and the "smallest" temperament that supports both, in a sense to be made precise below.
-The first is found by taking the intersection of the two subgroups and the intersection of the two temperament kernels, independently, producing another subgroup temperament. This is the '''join''' of the two subgroup temperaments, which reduces to the prior definition of the join if the two subgroups are equal. The join is the simplest temperament that supports both A and B, in the sense that any other temperament that supports both A and B also supports the join. Every comma tempered out by ''both'' A and B is also tempered out in the join, and vice versa.
+The first is found by taking the intersection of the two subgroups and the intersection of the two temperament kernels, independently, producing another subgroup temperament. This is the '''join''' of the two subgroup temperaments, which reduces to the prior definition of the join if the two subgroups are equal. The join is the "largest" temperament that both A and B support, in the sense that any other temperament that both A and B support is also supported by the join. Every comma tempered out by ''both'' A and B is also tempered out in the join, and vice versa.
-The second is found by extending the two subgroups to the simplest subgroup which includes both, and then repeating with the two kernels. This is the '''meet''' of the two subgroup temperaments. The meet is the simplest temperament that is supported by both A and B, in the sense that if any other temperament is also supported by both A and B, it is supported by the meet. Every comma tempered out by ''either'' A or B is also tempered out in the meet, and vice versa.
+The second is found by extending the two subgroups to the simplest subgroup which includes both, and then repeating with the two kernels. This is the '''meet''' of the two subgroup temperaments. The meet is the "smallest" temperament that supports both A and B, in the sense that if any other temperament also supports both A and B, it supports the meet. Every comma tempered out by ''either'' A or B is also tempered out in the meet, and vice versa.
 == Examples ==