Talk:Direct approximation: Difference between revisions

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 :: Thanks for that fantastic explanation, Sintel. You said it better than I could have. I look forward to your thoughts on simple and integer uniform map. —[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:44, 22 December 2021 (UTC)
+: I will simply add that the regular temperament formalism can easily be extended to include some of this, which we've talked about on the Facebook xen math group in the past.
+: For instance, 16-EDO has a perfectly good direct mapping for 9, which happens to be different than it's representation of 3*3. You can always adjoin an extra 9 to the subgroup as though it were a prime, which I have notated in the past as 9' (read 9-prime). So if we were in the 2.3.5 subgroup before, we extend it with the extra 9' to 2.3.5.9'.
+: This is kind of weird as a JI subgroup as you have a bunch of pseudo-intervals like 9'/9, which simply exist as monzos in this larger subgroup, even if their JI tuning would be 0 cents. But the usefulness happens when looking at vals on this group, so that the 16-edo patent val would be ⟨16 25 37 51|, and here we note that the mapping of 51 steps for 9' is not the same as the mapping of 50 steps for 9. This is still a linear map; you can certainly play a chord like 4:5:7:9, for which the outer dyad is 19 steps, then keep the top note the same and build an 8:10:12:15:18 chord going down from it; if you do so then the lowest note of the second chord will be one step higher than the lowest of the first chord, and since you moved up by 9' once and down be 3 twice, this shows you that 9'/9 is mapped to 1 step of 16-EDO, which is now a musically meaningful statement.
+: This is the way I have always found most useful to look at inconsistent mappings - I don't think most people care about the direct mapping for every ratio all the time, like (81/80)^100 or whatever, but for any equal temperament there are usually a few of these direct mappings here and there which really are useful, which can then be added explicitly, and for which you can still modulate around "regular"ly as long as you keep track of which version of these rationals you are using.
+: If you like, you can view the almighty master space for this idea to be the free abelian group where every single rational (possibly greater than 1/1) is treated as having its own basis vector. This group extends JI so that not only do we have the primes as basis vectors, but also have added an extra "primified" version of every rational number in this way. 2.3.5.9' is a subgroup of this, as well as anything else you could dream up, and you can always do temperament searches on the subgroups of this "beyond-JI" group. This is a group I have bumped into again and again; in addition to the above, the set of generalized patent vals on this is closely related to the zeta function, and it also may be useful with some of the subgroup temperament stuff I have been doing. I am not sure what to call this although "paraconsistent JI," "paraconsistent subgroups," etc were suggested in the past. Maybe para-JI, para-subgroup, para-interval, para-mapping, or something. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 19:51, 22 December 2021 (UTC)