Talk:Patent val: Difference between revisions

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:::::: Please let me know if this explanation helps, or if you have any other questions. FloraC and I have also provided alternative explanations of the concept in our discussion above that aren't present in the article itself (yet... maybe we'll rework it soon), so if you are unfamiliar with those explanations, they may prove helpful as well. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 16:48, 28 September 2021 (UTC)

::::::: I understand the math perfectly well. I understand the diagram too. What I don't understand is the terminology. Generalized in this context means generalized from integers to real numbers, right? Which allows the numbers to be exact, e.g. 3/1 is exactly 26.944/17 the size of 2/1, hence ⟨17 26.944] is an exact edomapping. And exact is what you mean by uniform, right? But nobody actually uses these real-number edomappings. Instead, the real numbers are rounded to integers. Which makes them not generalized, and not exact. So I don't understand, or more bluntly I strongly object to, calling the rounded edomappings generalized.

::::::: In other words, here's three of what I would call generalized PVs: ⟨17 26.944 39.473 47.725] and ⟨17.1 27.103 39.705 48.006] and ⟨17.45 27.658 40.518 48.988]. And here's what you get when you round them: ⟨17 27 39 48] and ⟨17 27 40 48] and ⟨17 28 41 49]. But there's nothing generalized about those last three. They are ordinary edomappings. But the whole point of exact real-number edomappings is that you can derive ordinary inexact integral edomappings from them. And because these three have been derived this way, they are more "reasonable" or "natural" than others which can't be. And "proper" is a much better term for that than "generalized". So IMO any edomapping through which you can draw a vertical line on the diagram and hit every component rectangle should be called a proper edomapping, and any one that you can't do this with should be called an improper edomapping.

::::::: Using my proposed terminology, 17edo has exactly 9 proper 7-limit edomappings, one of which is the nearest edomapping. As you would expect, the nearest edomapping of any edo is always proper. Using the current terminology, 17edo has 9 GPVs. One of them is the patent val, plus there's 8 non-patent vals. But each of those non-patent val is also a generalized patent val. So even though they're non-patent, they still have "patent" in the name. Which is confusing. This confusion goes away when you use the term proper.

::::::: Interesting side point: 17edo's smallest (i.e. contains the smallest numbers) proper edomapping is ⟨17 26 38 46]. And 16edo's largest is ⟨16 26 38 46]. The last 3 numbers are the same because 16.5 can be rounded up to 17 or down to 16. In other words, 16.5-edo is both stretched 17edo and compressed 16edo. In fact the whole section on GPVs might be improved by explicitly discussing stretched edos. We could even call GPVs stretched edomappings, if we use the word stretched loosely to mean both stretched and compressed and also neither one. Then what I'm calling proper edomappings would be called nearest stretched edomappings. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 08:40, 29 September 2021 (UTC)

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 :::::: Please let me know if this explanation helps, or if you have any other questions. FloraC and I have also provided alternative explanations of the concept in our discussion above that aren't present in the article itself (yet... maybe we'll rework it soon), so if you are unfamiliar with those explanations, they may prove helpful as well. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 16:48, 28 September 2021 (UTC)
+::::::: I understand the math perfectly well. I understand the diagram too. What I don't understand is the terminology. Generalized in this context means generalized from integers to real numbers, right? Which allows the numbers to be exact, e.g. 3/1 is exactly 26.944/17 the size of 2/1, hence ⟨17 26.944] is an exact edomapping. And exact is what you mean by uniform, right? But nobody actually uses these real-number edomappings. Instead, the real numbers are rounded to integers. Which makes them not generalized, and not exact. So I don't understand, or more bluntly I strongly object to, calling the rounded edomappings generalized.
+::::::: In other words, here's three of what I would call generalized PVs: ⟨17 26.944 39.473 47.725] and ⟨17.1 27.103 39.705 48.006] and ⟨17.45 27.658 40.518 48.988]. And here's what you get when you round them: ⟨17 27 39 48] and ⟨17 27 40 48] and ⟨17 28 41 49]. But there's nothing generalized about those last three. They are ordinary edomappings. But the whole point of exact real-number edomappings is that you can derive ordinary inexact integral edomappings from them. And because these three have been derived this way, they are more "reasonable" or "natural" than others which can't be. And "proper" is a much better term for that than "generalized". So IMO any edomapping through which you can draw a vertical line on the diagram and hit every component rectangle should be called a proper edomapping, and any one that you can't do this with should be called an improper edomapping.
+::::::: Using my proposed terminology, 17edo has exactly 9 proper 7-limit edomappings, one of which is the nearest edomapping. As you would expect, the nearest edomapping of any edo is always proper. Using the current terminology, 17edo has 9 GPVs. One of them is the patent val, plus there's 8 non-patent vals. But each of those non-patent val is also a generalized patent val. So even though they're non-patent, they still have "patent" in the name. Which is confusing. This confusion goes away when you use the term proper.
+::::::: Interesting side point: 17edo's smallest (i.e. contains the smallest numbers) proper edomapping is ⟨17 26 38 46]. And 16edo's largest is ⟨16 26 38 46]. The last 3 numbers are the same because 16.5 can be rounded up to 17 or down to 16. In other words, 16.5-edo is both stretched 17edo and compressed 16edo. In fact the whole section on GPVs might be improved by explicitly discussing stretched edos. We could even call GPVs stretched edomappings, if we use the word stretched loosely to mean both stretched and compressed and also neither one. Then what I'm calling proper edomappings would be called nearest stretched edomappings. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 08:40, 29 September 2021 (UTC)