Meet and join: Difference between revisions

Line 57:

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.

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 above is the larger picture of both, and the duality between the two, as that is really what is interesting. Different conventions may be used in different writings, where people can make clear 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.

@@ Line 57: / Line 57: @@
 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.
+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 above is the larger picture of both, and the duality between the two, as that is really what is interesting. Different conventions may be used in different writings, where people can make clear 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.