Meet and join: Difference between revisions

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== A Note on Historical Terminology ==

Originally, the "meet" and "join" of two temperaments was proposed [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19399.html on the tuning-math list] by Keenan Pepper, who used the opposite convention from the above: the "meet" was the meet of vals (and hence the join of kernels) and vice versa. If the two temperaments are on the same subgroup, one can think about "joining" or "meeting" either their kernels, or the subgroups of supporting vals (the "join" in one convention is the "meet" in the other and so forth), so it makes little difference which convention is chosen (as noted in the tuning-math post above). However, when looking at temperaments on different subgroups, the other convention makes things much more natural, since then the "join" of two temperaments is simply the "join" of kernels and subgroups independently. ~~We have chosen~~ that ~~convention here, although~~ it ~~should be noted that in older emails~~ and ~~posts,~~ the ~~"join" and "meet" were often chosen using~~ the ~~other convention~~.

Originally, the "meet" and "join" of two temperaments was proposed [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19399.html on the tuning-math list] by Keenan Pepper, who used the opposite convention from the above: the "meet" was the meet of vals (and hence the join of kernels) and vice versa. If the two temperaments are on the same subgroup, one can think about "joining" or "meeting" either their kernels, or the subgroups of supporting vals (the "join" in one convention is the "meet" in the other and so forth), so it makes little difference which convention is chosen (as noted in the tuning-math post above). However, when looking at temperaments on different subgroups, the other convention makes things much more natural, since then the "join" of two temperaments is simply the "join" of kernels and subgroups independently. The big picture is that it is possible to join either vals or kernels - and the relationship between both is significant in its own right - so we've simply added both to the page.

In mathematical order theory, meet and join are denoted by ∨ and ∧. We avoid doing that for two reasons; the first is to avoid confusion with the interior and wedge products of multivals. The second is that meet and join are operations on abstract temperaments; ordering by increasing size of the group of commas and decreasing size of the group of vals is regarded and notated as the same.

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 == A Note on Historical Terminology ==
-Originally, the "meet" and "join" of two temperaments was proposed [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19399.html on the tuning-math list] by Keenan Pepper, who used the opposite convention from the above: the "meet" was the meet of vals (and hence the join of kernels) and vice versa. If the two temperaments are on the same subgroup, one can think about "joining" or "meeting" either their kernels, or the subgroups of supporting vals (the "join" in one convention is the "meet" in the other and so forth), so it makes little difference which convention is chosen (as noted in the tuning-math post above). However, when looking at temperaments on different subgroups, the other convention makes things much more natural, since then the "join" of two temperaments is simply the "join" of kernels and subgroups independently. We have chosen that convention here, although it should be noted that in older emails and posts, the "join" and "meet" were often chosen using the other convention.
+Originally, the "meet" and "join" of two temperaments was proposed [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19399.html on the tuning-math list] by Keenan Pepper, who used the opposite convention from the above: the "meet" was the meet of vals (and hence the join of kernels) and vice versa. If the two temperaments are on the same subgroup, one can think about "joining" or "meeting" either their kernels, or the subgroups of supporting vals (the "join" in one convention is the "meet" in the other and so forth), so it makes little difference which convention is chosen (as noted in the tuning-math post above). However, when looking at temperaments on different subgroups, the other convention makes things much more natural, since then the "join" of two temperaments is simply the "join" of kernels and subgroups independently. The big picture is that  it is possible to join either vals or kernels - and the relationship between both is significant in its own right - so we've simply added both to the page.
 In mathematical order theory, meet and join are denoted by ∨ and ∧. We avoid doing that for two reasons; the first is to avoid confusion with the interior and wedge products of multivals. The second is that meet and join are operations on abstract temperaments; ordering by increasing size of the group of commas and decreasing size of the group of vals is regarded and notated as the same.