Meet and join: Difference between revisions

Line 54:

Similarly, if we had gone with the val-join instead of kernel-join, we would have gotten the same result, except the minimal and maximal temperaments would be flipped.

== A Note on ~~Historical~~ Terminology ==

== A Note on 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~~. ~~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~~.

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 terms "meet" and "join" to refer to what is above called the "val-meet" and "val-join," along with the caveat that it doesn't really matter which you call meet or join.

The simple reality 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, as the larger picture of both, and the duality between the two is really what is interesting. People can be clear in their own writings if they are joining vals or kernels.

When looking at subgroups on different temperaments, it is a little bit awkward to literally do things in terms of vals, as the svals you are "joining" or "meeting" don't even exist in the same vector space - rather they are *quotients* of the same vector space. It is still possible to do by looking at how you can "co-temper" vals to get svals, but the simpler way to proceed is simply to do everything in terms of subgroups and kernels, define the kernel-meet and kernel-join in the straightforward way as above, and then simply ''define'' the "val-meet" of two subgroup temperaments as equal to the kernel-join and vice versa. As long as the subgroup temperaments you are joining have no torsion or contorsion, and you remove anything like that from the result, all of these things are equivalent.

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.

@@ Line 54: / Line 54: @@
 Similarly, if we had gone with the val-join instead of kernel-join, we would have gotten the same result, except the minimal and maximal temperaments would be flipped.
-== A Note on Historical Terminology ==
+== A Note on 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. 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.
+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 terms "meet" and "join" to refer to what is above called the "val-meet" and "val-join," along with the caveat that it doesn't really matter which you call meet or join.
+The simple reality 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, as the larger picture of both, and the duality between the two is really what is interesting. People can be clear in their own writings if they are joining vals or kernels.
+When looking at subgroups on different temperaments, it is a little bit awkward to literally do things in terms of vals, as the svals you are "joining" or "meeting" don't even exist in the same vector space - rather they are *quotients* of the same vector space. It is still possible to do by looking at how you can "co-temper" vals to get svals, but the simpler way to proceed is simply to do everything in terms of subgroups and kernels, define the kernel-meet and kernel-join in the straightforward way as above, and then simply ''define'' the "val-meet" of two subgroup temperaments as equal to the kernel-join and vice versa. As long as the subgroup temperaments you are joining have no torsion or contorsion, and you remove anything like that from the result, all of these things are equivalent.
 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.