Talk:EDO vs ET: Difference between revisions

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# '''Re: finding the approximation of JI intervals: EDs round, and ETs map.''' As we just mentioned, EDs have nothing in particular to do with JI; that said, whenever we do want to know what an ED's closest approximation of a JI interval is, we simply ''round'' each interval's pitch (as measured e.g. in cents) to the nearest step count. With ETs, however, we ''map'' each interval's frequency ratio — using its prime composition — to some step count, which is not necessarily nearest by pitch rounding.

# '''For each integer <math>n</math>, there's only one <math>n</math>-ED2, but there are many <math>n</math>-ETs.''' An <math>n</math>-ED2 is already a fully-specified pitch set, but <math>n</math>-ET is not quite there yet. For starters, there are different <math>n</math>-ETs for each prime limit, while EDs have nothing in particular to do with primes or prime limits. And more interestingly, we have cases such as <math>n</math> = 17 at the 5-limit where we find more than one reasonable equal temperament: we have 17p-ET with map ⟨17 27 39], but we also have 17c-ET with map ⟨17 27 40].

== "Supports" ==

Toward the bottom of this page it brings up EDOs "supporting" temperaments. This could be a bit confusing because it's using "support" in the informal, generic sense (see: https://en.xen.wiki/w/Support#Other_informal_usage) while "support" also has a technical, specific sense for ETs (https://en.xen.wiki/w/Support).

In other words, answering the question, "Does 11-ED2 support hanson?" is indeed ambiguous. However, answering the question "Does 11-ET support hanson?" is not. This is just cold hard numbers. Hanson is a 5-limit temperament, and there are three different reasonable 5-limit 11-ETs (according to http://x31eq.com/cgi-bin/rt.cgi?ets=11&limit=5), so this is different than the case for 12 where there's only one reasonable 5-limit 12-ET. Right, so we have 11c with map {{map|11 17 25}}, 11b with map {{map|11 18 26}}, and 11p with map {{map|11 17 26}}. To know whether these support hanson, we just need to ask whether or not they make the hanson comma vanish. The hanson comma is also known as the kleisma, and is 15625/15552, which as a vector is {{vector|-6 -5 6}}. 11c maps this interval to -1 steps, 11b maps it to 0 steps, and 11p maps it to 5 steps. So 11b-ET supports hanson, but 11c and 11p do not. There are infinitely more 5-limit 11-ETs (which are increasingly unreasonable), and some of these support hanson, and some of them don't.

I suggest that some explanation along these lines should be added to this part of the page, in place of where it says "has been debated without a consensus having been reached", which is not substantiated with links to discussion, and I think the answer is clear, as I've just described.

--[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:54, 20 June 2023 (UTC)

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 # '''Re: finding the approximation of JI intervals: EDs round, and ETs map.''' As we just mentioned, EDs have nothing in particular to do with JI; that said, whenever we do want to know what an ED's closest approximation of a JI interval is, we simply ''round'' each interval's pitch (as measured e.g. in cents) to the nearest step count. With ETs, however, we ''map'' each interval's frequency ratio — using its prime composition — to some step count, which is not necessarily nearest by pitch rounding.
 # '''For each integer <math>n</math>, there's only one <math>n</math>-ED2, but there are many <math>n</math>-ETs.''' An <math>n</math>-ED2 is already a fully-specified pitch set, but <math>n</math>-ET is not quite there yet. For starters, there are different <math>n</math>-ETs for each prime limit, while EDs have nothing in particular to do with primes or prime limits. And more interestingly, we have cases such as <math>n</math> = 17 at the 5-limit where we find more than one reasonable equal temperament: we have 17p-ET with map ⟨17 27 39], but we also have 17c-ET with map ⟨17 27 40].
+== "Supports" ==
+Toward the bottom of this page it brings up EDOs "supporting" temperaments. This could be a bit confusing because it's using "support" in the informal, generic sense (see: https://en.xen.wiki/w/Support#Other_informal_usage) while "support" also has a technical, specific sense for ETs (https://en.xen.wiki/w/Support).
+In other words, answering the question, "Does 11-ED2 support hanson?" is indeed ambiguous. However, answering the question "Does 11-ET support hanson?" is not. This is just cold hard numbers. Hanson is a 5-limit temperament, and there are three different reasonable 5-limit 11-ETs (according to http://x31eq.com/cgi-bin/rt.cgi?ets=11&limit=5), so this is different than the case for 12 where there's only one reasonable 5-limit 12-ET. Right, so we have 11c with map {{map|11 17 25}}, 11b with map {{map|11 18 26}}, and 11p with map {{map|11 17 26}}. To know whether these support hanson, we just need to ask whether or not they make the hanson comma vanish. The hanson comma is also known as the kleisma, and is 15625/15552, which as a vector is {{vector|-6 -5 6}}. 11c maps this interval to -1 steps, 11b maps it to 0 steps, and 11p maps it to 5 steps. So 11b-ET supports hanson, but 11c and 11p do not. There are infinitely more 5-limit 11-ETs (which are increasingly unreasonable), and some of these support hanson, and some of them don't.
+I suggest that some explanation along these lines should be added to this part of the page, in place of where it says "has been debated without a consensus having been reached", which is not substantiated with links to discussion, and I think the answer is clear, as I've just described.
+--[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:54, 20 June 2023 (UTC)