Talk:Direct approximation: Difference between revisions

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::::: Good stuff. Then the master tuning space would be, I guess, a countably infinite direct sum of JI's (or para-JI's or whatever). [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 00:10, 23 December 2021 (UTC)

== Redirect lemmas and relation to patent val ==

How wrong would you consider the term '''patent interval'''? I mean, there is an (at least to me) obvious relation to [[patent val]] that led me to the sloppy ''[[patent fifth]]'' of an EDO in the [[circle-of-fifths notation]] article. Shouldn't one or the other still be mentioned in this article? --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 22:47, 23 February 2022 (UTC)

: FYI: The ''patent fifth'' is [https://en.xen.wiki/index.php?title=Circle-of-fifths_notation&type=revision&diff=86834&oldid=72315 now replaced] by ''direct approximation of the fifth. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 22:58, 23 February 2022 (UTC)

:: From what I gather, the Direct Approximations of harmonics in a given EDO come together as generators which comprise that EDO's Simple Map or Patent Val. However, the approximations of other intervals derived indirectly by means of the patent val (which may be arguably termed the "Patent Approximations" or, more ambiguously, "patent intervals") may not be identical to said intervals' direct approximations. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:21, 24 February 2022 (UTC)

::: I confirm Aura's previous statement; the only intervals whose direct approximations are guaranteed to match their mapping by the simple map in RTT are the prime harmonics (or whichever intervals are the generators of the JI subgroup), because direct approximation of each of them is how the simple map is defined.

::: Although I'd prefer we keep "patent" out of it. Why not "direct intervals", such as a "direct fifth", if you're looking for something pithy like that? --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 02:53, 24 February 2022 (UTC)

:::: The reason I'm going the way I'm going with it is so that I can set up a contrast as needed between the two versions of the interval in question. Having a term for the approximations of other intervals derived indirectly by means of the patent val is important in helping to define things like [[telicity]]. While I agree with the idea of the term "direct fifth" as a replacement for "patent fifth" as Xenwolf initially meant it, being able to use the term "patent" for these other, non-direct approximations would at least be useful to me and other people who want to deal in telicity. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 03:23, 24 February 2022 (UTC)

::::: Oh! My bad. Yes, I see what you're saying now. Sorry I didn't read carefully enough the first time. Of course, you would need to contrast a direct fifth with a patent fifth. That makes sense and is totally fine with me. As you may know I prefer "simple map" to "patent val" and so would prefer "simple fifth" to "patent fifth", but I won't begrudge you for using "patent", as it's patently ;) well-established. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 03:51, 24 February 2022 (UTC)

== Rounding function symbol ==

I never heard of "⌈⌋". No occurrence in [[Wikipedia: Rounding]] either. Is it attested anywhere? I reckon round () isn't too verbose? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 09:56, 24 February 2022 (UTC)

: I've definitely seen this before. See: https://mathworld.wolfram.com/NearestIntegerFunction.html Hoewever I think <math>\text{round}(x)</math> is much clearer. -- [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 12:12, 24 February 2022 (UTC)

:: Feel free to change it if you think it's best. My reasoning for using was this: because "round" was spelled out in the preceding text, the context was sufficient to make <math>⌈·⌋</math> acceptable despite not being well-established. And I like <math>⌈·⌋</math> because it's concise, and also because it's logical w/r/t to the well-established ceil <math>⌈·⌉</math> and floor <math>⌊·⌋</math> notations. I knew it was used at least sometimes but couldn't remember specifically where I'd seen <math>⌈·⌋</math> before, but a web search confirmed that it's at least not entirely unheard of. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 21:22, 24 February 2022 (UTC)

::: Not gonna lie, I prefer spelling out all of them – floor, ceil, and round. Maybe it's cuz I do codes more than math. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 22:13, 24 February 2022 (UTC)

:::: FWIW I like \lfloor \rfloor and \lceil \rceil for floor and ceil in maths; my only slight issue with them is that they seem to imply rectangular brackets are "round()" which doesn't feel right because there is no obvious choice for which way to round half-integers. Therefore I often lean to \operatorname{floor}() and \operatorname{ceil}() for this reason. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 09:40, 27 May 2024 (UTC)

== Standardize "direct mapping" to "direct approximation" across wiki? ==

I noticed Flora correcting one copy of "direct mapping" to "direct approximation" today, and it piqued my curiosity, so I searched the wiki for occurrences of "direct mapping". There's a bunch! I suggest, per the reasons covered in discussion topics on this page which led to the reworking of this page from "direct mapping" to "direct approximation", that we should change these all to "direct appoximation". --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 16:51, 23 May 2024 (UTC)

: Good idea. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:56, 24 May 2024 (UTC)

:: I saw you did most of the work already. Thanks. I did what I think to be the remainder just now.

:: It was interesting to find a separate case on the wiki where "direct mapping" and "consistent mapping" were used as terminology, in the context of some theory about tunings of temperaments (where mapping ''does'' apply). I found it on two wiki pages. Here those two pages are, if you're curious:

::* [[The Riemann zeta function and tuning#Interpretation of results: "cosine relative error"]]

::* [[TOP tuning#TOP with "inconsistent" rational tuning extensions]]

:: I was also interested to find a couple places where the author was using the term "consistent mapping" along with "direct mapping". I changed "direct mapping" to "direct approximation", but I hesitated to change "consistent mapping" to "consistent approximation". Unlike "direct approximation", the name "consistent approximation" is not self-explanatory; in fact, I would say it's a bit confusing. From what I could understand, what was meant by "consistent mapping" (and what would be meant by "consistent approximation") was this: the sum of the direct approximations of an interval's constituent odd harmonics (such as would be used in a check for [[consistency]] of an EDO). If you like, you may feel free to push for this terminology, but if so, I think it should be bolded and defined on the consistency page. I'm not so sure about it myself; it seems a bit strained. For now, I simply unpacked the meaning in both places:

::* https://en.xen.wiki/index.php?title=4th-octave_temperaments&type=revision&diff=145002&oldid=138306

::* https://en.xen.wiki/index.php?title=1984edo&type=revision&diff=145003&oldid=144911

:: --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 01:52, 25 May 2024 (UTC)

::: I agree ''conistent mapping'' isn't quite self-explanatory. I know it means the same as ''mapping with a val'', ''val mapping'', ''RTT mapping'' or whatever that describes the opposite of direct approximation. Personally I find ''val mapping'' clear enough, but if you don't like ''val'', and replace it with ''map'', you get ''map mapping'' which is absurd. So idk. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 08:12, 26 May 2024 (UTC)

:::: FWIW I prefer "a/b is mapped to [x, y, ...]" where a/b is the interval and [x, y, ...] are the coordinates in the tempered space of the temperament. In other words, "mapped" already by default implicates a regularity to me if you don't specify that it's a "direct mapping". For example [[32/27]] is mapped to 285{{cent}} in [[80edo|80et]]. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 09:33, 27 May 2024 (UTC)

@@ Line 128: / Line 128: @@
 ::::: Good stuff. Then the master tuning space would be, I guess, a countably infinite direct sum of JI's (or para-JI's or whatever). [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 00:10, 23 December 2021 (UTC)
+== Redirect lemmas and relation to patent val ==
+How wrong  would you consider the term '''patent interval'''? I mean, there is an (at least to me) obvious relation to [[patent val]] that led me to the sloppy ''[[patent fifth]]'' of an EDO in the [[circle-of-fifths notation]] article. Shouldn't one or the other still be mentioned in this article? --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 22:47, 23 February 2022 (UTC)
+: FYI: The ''patent fifth'' is [https://en.xen.wiki/index.php?title=Circle-of-fifths_notation&type=revision&diff=86834&oldid=72315 now replaced] by ''direct approximation of the fifth. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 22:58, 23 February 2022 (UTC)
+:: From what I gather, the Direct Approximations of harmonics in a given EDO come together as generators which comprise that EDO's Simple Map or Patent Val.  However, the approximations of other intervals derived indirectly by means of the patent val (which may be arguably termed the "Patent Approximations" or, more ambiguously, "patent intervals") may not be identical to said intervals' direct approximations. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:21, 24 February 2022 (UTC)
+::: I confirm Aura's previous statement; the only intervals whose direct approximations are guaranteed to match their mapping by the simple map in RTT are the prime harmonics (or whichever intervals are the generators of the JI subgroup), because direct approximation of each of them is how the simple map is defined.
+::: Although I'd prefer we keep "patent" out of it. Why not "direct intervals", such as a "direct fifth", if you're looking for something pithy like that? --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 02:53, 24 February 2022 (UTC)
+:::: The reason I'm going the way I'm going with it is so that I can set up a contrast as needed between the two versions of the interval in question.  Having a term for the approximations of other intervals derived indirectly by means of the patent val is important in helping to define things like [[telicity]].  While I agree with the idea of the term "direct fifth" as a replacement for "patent fifth" as Xenwolf initially meant it, being able to use the term "patent" for these other, non-direct approximations would at least be useful to me and other people who want to deal in telicity. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 03:23, 24 February 2022 (UTC)
+::::: Oh! My bad. Yes, I see what you're saying now. Sorry I didn't read carefully enough the first time. Of course, you would need to contrast a direct fifth with a patent fifth. That makes sense and is totally fine with me. As you may know I prefer "simple map" to "patent val" and so would prefer "simple fifth" to "patent fifth", but I won't begrudge you for using "patent", as it's patently ;) well-established. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 03:51, 24 February 2022 (UTC)
+== Rounding function symbol ==
+I never heard of "⌈⌋". No occurrence in [[Wikipedia: Rounding]] either. Is it attested anywhere? I reckon round () isn't too verbose? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 09:56, 24 February 2022 (UTC)
+: I've definitely seen this before. See: https://mathworld.wolfram.com/NearestIntegerFunction.html Hoewever I think <math>\text{round}(x)</math> is much clearer. -- [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 12:12, 24 February 2022 (UTC)
+:: Feel free to change it if you think it's best. My reasoning for using was this: because "round" was spelled out in the preceding text, the context was sufficient to make <math>⌈·⌋</math> acceptable despite not being well-established. And I like <math>⌈·⌋</math> because it's concise, and also because it's logical w/r/t to the well-established ceil <math>⌈·⌉</math> and floor <math>⌊·⌋</math> notations. I knew it was used at least sometimes but couldn't remember specifically where I'd seen <math>⌈·⌋</math> before, but a web search confirmed that it's at least not entirely unheard of. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 21:22, 24 February 2022 (UTC)
+::: Not gonna lie, I prefer spelling out all of them – floor, ceil, and round. Maybe it's cuz I do codes more than math. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 22:13, 24 February 2022 (UTC)
+:::: FWIW I like \lfloor \rfloor and \lceil \rceil for floor and ceil in maths; my only slight issue with them is that they seem to imply rectangular brackets are "round()" which doesn't feel right because there is no obvious choice for which way to round half-integers. Therefore I often lean to \operatorname{floor}() and \operatorname{ceil}() for this reason. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 09:40, 27 May 2024 (UTC)
+== Standardize "direct mapping" to "direct approximation" across wiki? ==
+I noticed Flora correcting one copy of "direct mapping" to "direct approximation" today, and it piqued my curiosity, so I searched the wiki for occurrences of "direct mapping". There's a bunch! I suggest, per the reasons covered in discussion topics on this page which led to the reworking of this page from "direct mapping" to "direct approximation", that we should change these all to "direct appoximation". --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 16:51, 23 May 2024 (UTC)
+: Good idea. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:56, 24 May 2024 (UTC)
+:: I saw you did most of the work already. Thanks. I did what I think to be the remainder just now.
+:: It was interesting to find a separate case on the wiki where "direct mapping" and "consistent mapping" were used as terminology, in the context of some theory about tunings of temperaments (where mapping ''does'' apply). I found it on two wiki pages. Here those two pages are, if you're curious:
+::* [[The Riemann zeta function and tuning#Interpretation of results: "cosine relative error"]]
+::* [[TOP tuning#TOP with "inconsistent" rational tuning extensions]]
+:: I was also interested to find a couple places where the author was using the term "consistent mapping" along with "direct mapping". I changed "direct mapping" to "direct approximation", but I hesitated to change "consistent mapping" to "consistent approximation". Unlike "direct approximation", the name "consistent approximation" is not self-explanatory; in fact, I would say it's a bit confusing. From what I could understand, what was meant by "consistent mapping" (and what would be meant by "consistent approximation") was this: the sum of the direct approximations of an interval's constituent odd harmonics (such as would be used in a check for [[consistency]] of an EDO). If you like, you may feel free to push for this terminology, but if so, I think it should be bolded and defined on the consistency page. I'm not so sure about it myself; it seems a bit strained. For now, I simply unpacked the meaning in both places:
+::* https://en.xen.wiki/index.php?title=4th-octave_temperaments&type=revision&diff=145002&oldid=138306
+::* https://en.xen.wiki/index.php?title=1984edo&type=revision&diff=145003&oldid=144911
+:: --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 01:52, 25 May 2024 (UTC)
+::: I agree ''conistent mapping'' isn't quite self-explanatory. I know it means the same as ''mapping with a val'', ''val mapping'', ''RTT mapping'' or whatever that describes the opposite of direct approximation. Personally I find ''val mapping'' clear enough, but if you don't like ''val'', and replace it with ''map'', you get ''map mapping'' which is absurd. So idk. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 08:12, 26 May 2024 (UTC)
+:::: FWIW I prefer "a/b is mapped to [x, y, ...]" where a/b is the interval and [x, y, ...] are the coordinates in the tempered space of the temperament. In other words, "mapped" already by default implicates a regularity to me if you don't specify that it's a "direct mapping". For example [[32/27]] is mapped to 285{{cent}} in [[80edo|80et]]. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 09:33, 27 May 2024 (UTC)