Talk:EDO vs ET: Difference between revisions

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:::::::: Perhaps, though, all you mean is, again, not that EDOs and ETs are both regular temperaments, but that they are both ''generated tunings'', as Fredg999 described, or in other words, that they can be visualized on a lattice. Even more specifically, they can both be visualized on 1D lattices. And one other thing they have in common is that both lattices' single dimension's generators represent an interval 21/n, whether that's for n-ET or n-EDO, and whether or not this interval is tuned exactly. That much I can certainly agree with.

:::::::: --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:55, 25 June 2023 (UTC)

::::::::: > the domain is our JI lattice

::::::::: I first said it was a non-JI. However, "completely non-JI" "temperament" may have been crazy idea. At that time, I was thinking about the validity of naming the pitches 2i/n in near-octave n-EDO.

::::::::: And thanks for the discussion. I feel like I'm starting to understand something, but let me confirm.

:::::::::* Is just intonation a RT?

:::::::::*# Yes. It maps ideal 2/1 to actual 2:1.

:::::::::*# No. It doesn't have any tempered out comma.

:::::::::*# No. It is not a temperament because it does not call itself a temperament.

:::::::::*# ...But yes, it is useful for introductory.

:::::::::* Is 12-EDO a RT?

:::::::::*# ...Unknown. How do you think of "Octave" as?

:::::::::*# No. It doesn't have any tempered out comma.

:::::::::*# No. It is not a temperament because it does not call itself a temperament.

:::::::::* Is 2-limit 12-ET (= contorted (order 12) 2-limit) a RT?

:::::::::*# Yes. It maps ideal 2/1 to actual 2:1.

:::::::::*# No. It doesn't have any tempered out comma.

:::::::::*# Yes. It is a temperament because it calls itself a temperament and well-defined.

::::::::: --[[User:Dummy index|Dummy index]] ([[User talk:Dummy index|talk]]) 14:27, 26 June 2023 (UTC)

:::::::::: Just intonation and 12-EDO are tuning systems (sets of concrete pitches or intervals), not regular temperaments. However, for both of these tuning systems, you could choose a domain and calculate the [[simple map]] associated with that domain, and now you would have a regular temperament. For instance, p-limit just intonation with the identity matrix as the temperament map is a regular temperament (usually considered trivial for obvious reasons). 12-ET is indeed a regular temperament. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 15:55, 26 June 2023 (UTC)

::::::::::: Thanks for that, Fredg999. Though I'd tweak that slightly. The simple map for an n-EDO is one simple way to get a regular temperament from an EDO, specifically an n-ET, but there are other possible maps you might want (optimal with respect to the tunings of a consonance set, for one of the very many definitions of optimal popularly used), and I think this is more straightforwardly understood as finding an n-ET from an n-EDO so much as it is finding an n-ET from n itself. Also, I don't think there's any sense in which one could calculate a simple map for JI, but maybe you didn't intend to suggest that. (Oh, except that the 1×1 identity matrix that interprets 2-limit JI as a trivial regular temperament would qualify as the simple map for an equal temperament, but that's decidedly an edge case!)

::::::::::: Yes, Dummy index, thank you for the discussion too. This is helping me refine my understanding of the similarities and differences between all of these structures, too. Fredg999 has already given direct answers to your questions, but I'll supplement what they wrote by answering using your own choices:

::::::::::: Is just intonation a RT? No. It is not a temperament because it does not call itself a temperament. (But I'll add: we can interpret JI as a trivial regular temperament, i.e. one that doesn't make any commas vanish.)

::::::::::: Is 12-EDO a RT? No. It is not a temperament because it does not call itself a temperament. (But I'll add: more importantly, it's not a regular temperament because it doesn't map a domain such as JI.)

::::::::::: Is 2-limit 12-ET (= contorted (order 12) 2-limit) a RT? Yes. It is a temperament because it calls itself a temperament and well-defined. (But I'll add: I've suggested we call this sort of thing a "temperoid"; I prefer to use the term "enfactored" rather than "contorted", and the point is that {{map|12}} doesn't bring anything more to the table ''as a regular temperament'' than {{map|1}} does, so if we wanted to make a list of all the unique regular temperaments in existence, we would include {{map|1}} but we might not want to include {{map|12}}.)

::::::::::: Let me know if you have further questions, and I'll do my best to answer. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 20:49, 26 June 2023 (UTC)

:::::::::::: So I tried to get a one-to-one correspondence with the output by "enfactoring" the input by 12, and the result was the 21/12 domain basis. But I have just come to the conclusion that this has no practical significance. Mapping an exponential function to the equal-step tuning is no different from the generator chain, and there is nothing to do in 1D domain. Barely the 2-limit case is only useful for explaining as a edge case. --[[User:Dummy index|Dummy index]] ([[User talk:Dummy index|talk]]) 15:00, 28 June 2023 (UTC)

== "Supports" ==

@@ Line 65: / Line 65: @@
 :::::::: Perhaps, though, all you mean is, again, not that EDOs and ETs are both regular temperaments, but that they are both ''generated tunings'', as Fredg999 described, or in other words, that they can be visualized on a lattice. Even more specifically, they can both be visualized on 1D lattices. And one other thing they have in common is that both lattices' single dimension's generators represent an interval 2<sup>1/n</sup>, whether that's for n-ET or n-EDO, and whether or not this interval is tuned exactly. That much I can certainly agree with.
 :::::::: --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:55, 25 June 2023 (UTC)
+::::::::: &gt; the domain is our JI lattice
+::::::::: I first said it was a non-JI. However, "completely non-JI" "temperament" may have been crazy idea. At that time, I was thinking about the validity of naming the pitches 2<sup>i/n</sup> in near-octave n-EDO.
+::::::::: And thanks for the discussion. I feel like I'm starting to understand something, but let me confirm.
+:::::::::* Is just intonation a RT?
+:::::::::*# Yes. It maps ideal 2/1 to actual 2:1.
+:::::::::*# No. It doesn't have any tempered out comma.
+:::::::::*# No. It is not a temperament because it does not call itself a temperament.
+:::::::::*# ...But yes, it is useful for introductory.
+:::::::::* Is 12-EDO a RT?
+:::::::::*# ...Unknown. How do you think of "Octave" as?
+:::::::::*# No. It doesn't have any tempered out comma.
+:::::::::*# No. It is not a temperament because it does not call itself a temperament.
+:::::::::* Is 2-limit 12-ET (= contorted (order 12) 2-limit) a RT?
+:::::::::*# Yes. It maps ideal 2/1 to actual 2:1.
+:::::::::*# No. It doesn't have any tempered out comma.
+:::::::::*# Yes. It is a temperament because it calls itself a temperament and well-defined.
+::::::::: --[[User:Dummy index|Dummy index]] ([[User talk:Dummy index|talk]]) 14:27, 26 June 2023 (UTC)
+:::::::::: Just intonation and 12-EDO are tuning systems (sets of concrete pitches or intervals), not regular temperaments. However, for both of these tuning systems, you could choose a domain and calculate the [[simple map]] associated with that domain, and now you would have a regular temperament. For instance, p-limit just intonation with the identity matrix as the temperament map is a regular temperament (usually considered trivial for obvious reasons). 12-ET is indeed a regular temperament. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 15:55, 26 June 2023 (UTC)
+::::::::::: Thanks for that, Fredg999. Though I'd tweak that slightly. The simple map for an n-EDO is one simple way to get a regular temperament from an EDO, specifically an n-ET, but there are other possible maps you might want (optimal with respect to the tunings of a consonance set, for one of the very many definitions of optimal popularly used), and I think this is more straightforwardly understood as finding an n-ET from an n-EDO so much as it is finding an n-ET from n itself. Also, I don't think there's any sense in which one could calculate a simple map for JI, but maybe you didn't intend to suggest that. (Oh, except that the 1×1 identity matrix that interprets 2-limit JI as a trivial regular temperament would qualify as the simple map for an equal temperament, but that's decidedly an edge case!)
+::::::::::: Yes, Dummy index, thank you for the discussion too. This is helping me refine my understanding of the similarities and differences between all of these structures, too. Fredg999 has already given direct answers to your questions, but I'll supplement what they wrote by answering using your own choices:
+::::::::::: Is just intonation a RT? No. It is not a temperament because it does not call itself a temperament. (But I'll add: we can interpret JI as a trivial regular temperament, i.e. one that doesn't make any commas vanish.)
+::::::::::: Is 12-EDO a RT? No. It is not a temperament because it does not call itself a temperament. (But I'll add: more importantly, it's not a regular temperament because it doesn't map a domain such as JI.)
+::::::::::: Is 2-limit 12-ET (= contorted (order 12) 2-limit) a RT? Yes. It is a temperament because it calls itself a temperament and well-defined. (But I'll add: I've suggested we call this sort of thing a "temperoid"; I prefer to use the term "enfactored" rather than "contorted", and the point is that {{map|12}} doesn't bring anything more to the table ''as a regular temperament'' than {{map|1}} does, so if we wanted to make a list of all the unique regular temperaments in existence, we would include {{map|1}} but we might not want to include {{map|12}}.)
+::::::::::: Let me know if you have further questions, and I'll do my best to answer. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 20:49, 26 June 2023 (UTC)
+:::::::::::: So I tried to get a one-to-one correspondence with the output by "enfactoring" the input by 12, and the result was the 2<sup>1/12</sup> domain basis. But I have just come to the conclusion that this has no practical significance. Mapping an exponential function to the equal-step tuning is no different from the generator chain, and there is nothing to do in 1D domain. Barely the 2-limit case is only useful for explaining as a edge case. --[[User:Dummy index|Dummy index]] ([[User talk:Dummy index|talk]]) 15:00, 28 June 2023 (UTC)
 == "Supports" ==