Talk:Patent val: Difference between revisions

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::::::: 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)

:::::::: Ah! Sorry. I did not mean to insult your intelligence or otherwise offend. As soon as I saw your reply, I remembered that when I drafted my previous post, I had the flicker of a thought that I should say something like "or maybe you do totally understand the concept but I just need you to clarify what you mean by 'exact' and 'proper'". I regret failing to follow through on that now. And you've gone ahead and done it for me. Thank you for that. Your full explanation lets me see that we're actually just about on the same page, so that's great.

:::::::: To be clear, I am the one who made the original post here arguing against the terminology "generalized patent val". You don't have to convince me that it's bad; we agree on that. I do think FloraC here still supports it to some extent, so arguments presented against it are still useful. I myself have elsewhere made the point you make above about how putting the words "generalized" and "patent" together is already confusing; I see that I didn't make it myself above in my original post, so I'd like to explicitly say that I second that point.

:::::::: You are correct so far when you say that the "generalized" part of "generalized patent val" means "from integers to reals". (And I'll take this opportunity to reiterate my position that it's more logical to structure it the other way around: defining a basic concept for the reals, i.e. uniform maps, and then specify integers when appropriate, i.e. integer uniform maps, AKA simple maps / patent vals / nearest edomappings.)

:::::::: However, I think you have the wrong idea about which numbers exactly that "generalized patent vals" are supposed to be thought of as generalizing from integers from reals. So, to be clear, I am choosing to assist my opponents here by clarifying their reasoning, but of course I want us to have a fair consideration of each other's ideas at their best and clearest. And as far as I can tell, it seems to be the case that you think "generalized patent vals" generalize ''all of the entries in the map'' from integers to reals, but that is not correct. ''Only the EDO number'' that is used in the otherwise identical process for computing a patent val is generalized. Or as I prefer to think about it, what is generalized is the ''multiplier'' that the terms of the JIP (i.e. {{map|log₂2 log₂3 log₂5 ...}}) are uniformly scaled by (except, again, I prefer to say that the default case is that this uniform multiplier is real, and if you need it to be integer, you say so).

:::::::: You say that "nobody actually uses these real-number edomappings (directly)" and I agree with that. That's a big reason why structures like {{map|17 26.944 39.473 47.72}}, {{map|17.1 27.103 39.705 48.006}}, {{map|17.45 27.658 40.518 48.988}} are ''not'' what we've given the name "generalized patent val"/"uniform map" for. You say that those ''are'' the structures that the name "generalized patent vals" makes sense for, and while I agree that generalized patent val ''could'' be a reasonable name for them if anyone wanted to name them, I don't believe there's a wide desire or any good reason to name these such structures at all. They don't have enough value in and of themselves. They are merely intermediate states before being rounded to the structures of value that are worth being named, i.e. "generalized patent vals"/"uniform maps" (I had to check your numbers here to make sure I understood what points you were illustrating with them — and yes, these are all indeed valid uniform scalings of the JIP which upon rounding would therefore give you GPVs/uniform maps).

:::::::: Consider the case of a non-generalized patent val, or as I would say, an integer uniform map. What is the "integer" of the name here is the EDO or uniform scalar; note that other than the first term of the map pre-rounding, all of the other numbers in the map will always be reals, even for integer uniform maps / patent vals. Let's compare using the examples you gave yourself. {{map|17 26.944 39.473 47.72}} rounds to {{map|17 27 39 48}} and so that is the "patent val"/"(integer) uniform map" for 17, and {{map|17.1 27.103 39.705 48.006}} rounds to {{map|17 27 40 48}} and it is the "generalized patent val"/"uniform map" for 17.1. In both cases, all the entries in the final maps are integers. And in both cases, all of the entries in the not-yet-rounded maps besides the entry for prime 2 are reals. The only difference is whether the entry for prime 2 in the map before it is rounded is an integer or a real. Does it make sense now why "generalized patent val" is a reasonable term, and I why I'd disagree with you that there is "nothing generalized" about the final results? (Though again, I am defending our opponents' reasoning here; I think generalized patent val is ''reasonable'', sure, but that's hardly enough for it to qualify as ''good'' or ''great'' at its job... just look how much confusion it has caused here alraedy, and how much work we're having to do in order to get on the same page about it!)

:::::::: So the act of generalizing from integers to reals is not what "allows" things be exact, as you say. I can see now that you are using the word "exact maps" to name those not-yet-rounded maps such as {{map|17 26.944 39.473 47.72}}, {{map|17.1 27.103 39.705 48.006}}, {{map|17.45 27.658 40.518 48.988}} which I suggested above don't need to be named. So, no, this is not what I mean by "uniform". My term "uniform map" is completely synonymous with "generalized patent val", which is the name for maps like these ''after'' they have been rounded. As you put it, "the whole point of exact real-number edomappings is that you can derive ordinary inexact integral edomappings from them". Right. I agree. If I had to rewrite that statement myself I would write "the point of maps that are uniform scalings of the JIP is that you can derive uniform maps from them".

:::::::: You then say '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".' I agree with all of your points here, and I now see what you mean by proper. But I think my proposed "uniform" is an even better term than "proper". That's because "uniform" is more descriptive. "Proper" is vague. When you used the word "proper" I had no idea what you meant. "Propriety" always requires further explanation about what qualities a thing is proper with respect to, while "uniformity" already gives you a sense of what the thing looks/feels/sounds like; uniform gives you the image of a perfectly straight vertical line drawn across the charts that I shared in my previous post, i.e. perfectly straight and not curved or wiggly or anything, which would be "unnatural" mapping choices. This is the same type of reasoning, actually, why FloraC above convinced me that your term "nearest" was superior to my term "simple" as a replacement for "patent"!

:::::::: As you say, "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 ''my'' proposed terminology, then, 17edo has exactly 9 uniform 7-limit maps, one of which is the integer uniform map AKA the nearest map. I think my terminology is slightly more self-explanatory than yours.

:::::::: It looks like you're using the word "stretched" in the same sense as I'm using "scaled" in my explanations of "uniform", i.e. uniform scaling of the JIP before rounding. My word "scaled" supports both stretching and compressing already, so perhaps you would like to use it. But I don't think we should discuss "nearest stretched edomappings" until we've tied up some discussion points we've already got open. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 17:19, 29 September 2021 (UTC)

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 ::::::: 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)
+:::::::: Ah! Sorry. I did not mean to insult your intelligence or otherwise offend. As soon as I saw your reply, I remembered that when I drafted my previous post, I had the flicker of a thought that I should say something like "or maybe you do totally understand the concept but I just need you to clarify what you mean by 'exact' and 'proper'". I regret failing to follow through on that now. And you've gone ahead and done it for me. Thank you for that. Your full explanation lets me see that we're actually just about on the same page, so that's great.
+:::::::: To be clear, I am the one who made the original post here arguing against the terminology "generalized patent val". You don't have to convince me that it's bad; we agree on that. I do think FloraC here still supports it to some extent, so arguments presented against it are still useful. I myself have elsewhere made the point you make above about how putting the words "generalized" and "patent" together is already confusing; I see that I didn't make it myself above in my original post, so I'd like to explicitly say that I second that point.
+:::::::: You are correct so far when you say that the "generalized" part of "generalized patent val" means "from integers to reals". (And I'll take this opportunity to reiterate my position that it's more logical to structure it the other way around: defining a basic concept for the reals, i.e. uniform maps, and then specify integers when appropriate, i.e. integer uniform maps, AKA simple maps / patent vals / nearest edomappings.)
+:::::::: However, I think you have the wrong idea about which numbers exactly that "generalized patent vals" are supposed to be thought of as generalizing from integers from reals. So, to be clear, I am choosing to assist my opponents here by clarifying their reasoning, but of course I want us to have a fair consideration of each other's ideas at their best and clearest. And as far as I can tell, it seems to be the case that you think "generalized patent vals" generalize ''all of the entries in the map'' from integers to reals, but that is not correct. ''Only the EDO number'' that is used in the otherwise identical process for computing a patent val is generalized. Or as I prefer to think about it, what is generalized is the ''multiplier'' that the terms of the JIP (i.e. {{map|log₂2 log₂3 log₂5 ...}}) are uniformly scaled by (except, again, I prefer to say that the default case is that this uniform multiplier is real, and if you need it to be integer, you say so).
+:::::::: You say that "nobody actually uses these real-number edomappings (directly)" and I agree with that. That's a big reason why structures like {{map|17 26.944 39.473 47.72}}, {{map|17.1 27.103 39.705 48.006}}, {{map|17.45 27.658 40.518 48.988}} are ''not'' what we've given the name "generalized patent val"/"uniform map" for. You say that those ''are'' the structures that the name "generalized patent vals" makes sense for, and while I agree that generalized patent val ''could'' be a reasonable name for them if anyone wanted to name them, I don't believe there's a wide desire or any good reason to name these such structures at all. They don't have enough value in and of themselves. They are merely intermediate states before being rounded to the structures of value that are worth being named, i.e. "generalized patent vals"/"uniform maps" (I had to check your numbers here to make sure I understood what points you were illustrating with them — and yes, these are all indeed valid uniform scalings of the JIP which upon rounding would therefore give you GPVs/uniform maps).
+:::::::: Consider the case of a non-generalized patent val, or as I would say, an integer uniform map. What is the "integer" of the name here is the EDO or uniform scalar; note that other than the first term of the map pre-rounding, all of the other numbers in the map will always be reals, even for integer uniform maps / patent vals. Let's compare using the examples you gave yourself. {{map|17 26.944 39.473 47.72}} rounds to {{map|17 27 39 48}} and so that is the "patent val"/"(integer) uniform map" for 17, and {{map|17.1 27.103 39.705 48.006}} rounds to {{map|17 27 40 48}} and it is the "generalized patent val"/"uniform map" for 17.1. In both cases, all the entries in the final maps are integers. And in both cases, all of the entries in the not-yet-rounded maps besides the entry for prime 2 are reals. The only difference is whether the entry for prime 2 in the map before it is rounded is an integer or a real. Does it make sense now why "generalized patent val" is a reasonable term, and I why I'd disagree with you that there is "nothing generalized" about the final results? (Though again, I am defending our opponents' reasoning here; I think generalized patent val is ''reasonable'', sure, but that's hardly enough for it to qualify as ''good'' or ''great'' at its job... just look how much confusion it has caused here alraedy, and how much work we're having to do in order to get on the same page about it!)
+:::::::: So the act of generalizing from integers to reals is not what "allows" things be exact, as you say. I can see now that you are using the word "exact maps" to name those not-yet-rounded maps such as {{map|17 26.944 39.473 47.72}}, {{map|17.1 27.103 39.705 48.006}}, {{map|17.45 27.658 40.518 48.988}} which I suggested above don't need to be named. So, no, this is not what I mean by "uniform". My term "uniform map" is completely synonymous with "generalized patent val", which is the name for maps like these ''after'' they have been rounded. As you put it, "the whole point of exact real-number edomappings is that you can derive ordinary inexact integral edomappings from them". Right. I agree. If I had to rewrite that statement myself I would write "the point of maps that are uniform scalings of the JIP is that you can derive uniform maps from them".
+:::::::: You then say '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".' I agree with all of your points here, and I now see what you mean by proper. But I think my proposed "uniform" is an even better term than "proper". That's because "uniform" is more descriptive. "Proper" is vague. When you used the word "proper" I had no idea what you meant. "Propriety" always requires further explanation about what qualities a thing is proper with respect to, while "uniformity" already gives you a sense of what the thing looks/feels/sounds like; uniform gives you the image of a perfectly straight vertical line drawn across the charts that I shared in my previous post, i.e. perfectly straight and not curved or wiggly or anything, which would be "unnatural" mapping choices. This is the same type of reasoning, actually, why FloraC above convinced me that your term "nearest" was superior to my term "simple" as a replacement for "patent"!
+:::::::: As you say, "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 ''my'' proposed terminology, then, 17edo has exactly 9 uniform 7-limit maps, one of which is the integer uniform map AKA the nearest map. I think my terminology is slightly more self-explanatory than yours.
+:::::::: It looks like you're using the word "stretched" in the same sense as I'm using "scaled" in my explanations of "uniform", i.e. uniform scaling of the JIP before rounding. My word "scaled" supports both stretching and compressing already, so perhaps you would like to use it. But I don't think we should discuss "nearest stretched edomappings" until we've tied up some discussion points we've already got open. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 17:19, 29 September 2021 (UTC)