Talk:Direct approximation

Applicability

I wished to express that the concept of "patent interval" is not only useful for just intervals which can be represented by rational numbers, but any given real value treated as an interval an be approximated this way. The overall intention of the article was to link "patent fifth" to something more practical than the more abstract patent val. But maybe this wasn't such a great idea... --Xenwolf (talk) 09:11, 18 January 2021 (UTC)

I would say the page is still useful.

(And you can in principle make a patent val for any set of intervals, say {3/2, 236 cents, φ}, not necessarily just intervals like {2/1, 3/1, 5/1}. Just remember what interval corresponds to which entry. The notation used on the wiki is p-limit-centric though; we could talk about extending the notation for JI subgroups.) Inthar (talk) 10:00, 18 January 2021 (UTC)

Yes, that's basically what I meant. Thanks for adjusting the wording in the article. --Xenwolf (talk) 11:20, 18 January 2021 (UTC)

Did I mention that this concept appears to be one of the concepts used for mapping in the Hunt System IQGPA calculator? --Aura (talk) 15:15, 18 January 2021 (UTC)

I should also mention that this is how I make retunings of other EDOs like in "Space Tour". Most of what I had to do for this is map the intervals comprising the steps of the smaller EDOs to the nearest step in 159edo, through there were times where I had to choose which mapping to use. --Aura (talk) 15:39, 18 January 2021 (UTC)

I was trying to get at this before, but I didn't know about direct mappings before, so I didn't know how to communicate it properly. Anyhow, I notice that according to Wolfram Alpha, 49/32 effectively has two separate mappings in 159edo. The first, given by round(log2(49/32)*159), is 98 steps, while the second, given by {159, round(log2(3)*159), round(log2(5)*159), round(log2(7)*159)}.{-5, 0, 0, 2}, is 97 steps. The first one is the "direct mapping", but what is the proper term for the second, more traditional mapping? I think this article could be expanded by describing the relationship between these two different types of mappings. --Aura (talk) 23:32, 18 January 2021 (UTC)

Plea for direct mapping

The patent val notation (like val notation in general) contains approximations of prime intervals. That these can be combined additively will be understood by readers who are familiar with primes and fractions. But the implicit consistency of this method can lead to confusion (and also the impression of inconsistency), see for example the divergence between direct and consistent mappings of 7/5 in 23edo:

7/5 == 11,16…\23 ~ 11\23

vs.

(7/4)/(5/4) ~ (19\23)-(7\23) == 12\23

Personally, I'm not really convinced by the concept of consistency which builds on the concept of odd limit. Since consistent mapping is so present in the wiki, would it be too confusing if the term "patent interval" implied direct mapping? --Xenwolf (talk) 11:57, 18 January 2021 (UTC)

I certainly don't mind the idea of patent interval implying "direct mapping". I mean my "complete consistency" concept from earlier- which I have since renamed "telicity"- hinges on "direct" and "consistent" mappings of intervals in a given prime chain being identical up until the prime chain itself connects with an interval of a lower p-limit prime chain. For this to work, the prime chain must not exceed 50% relative error from the starting point up until its connection with the lower prime chain. Thus, you can be sure that when I finalize this concept of mine, I'll be utilizing this very notion of the patent interval implying direct mapping. --Aura (talk) 14:41, 18 January 2021 (UTC)

Connection between direct mapping and patent interval

I don't know about you, but it seems to me that "direct mapping" is the mapping procedure that generates "patent intervals". Would you mind me trying to fix this to make the relationship more clear? --Aura (talk) 23:54, 18 January 2021 (UTC)

Never mind. I realized as I was trying to fix some of it that "direct mapping" makes sense as a synonym for "patent interval". Sorry about the confusion. Still, I did manage to reword that opening sentence a bit. --Aura (talk) 00:08, 19 January 2021 (UTC)

Sorry, only now I realized that you already addressed this. But maybe the new topic helps. --Xenwolf (talk) 08:24, 19 January 2021 (UTC)

Distinguish between patent interval and direct mapping

To reduce confusion: the patent interval for an ideal interval is obtained by direct mapping. So these things are not synonymous. You use the word patent interval to distinguish one interval from alternative renditions of the same ideal interval. Maybe the direct mapping concept is much more productive and the term patent interval may be dispensable but the latter is more obvious (to musicians) than the first one. --Xenwolf (talk) 08:16, 19 January 2021 (UTC)

Thanks. I really was wondering about that, but if that's the case, then "patent interval" really is not a dispensable term because it allows you to distinguish the tempered version of interval like 49/32 obtained by direct mapping from the tempered version obtained from stacking two tempered 7/4 intervals and octave-reducing. The question is how to distinguish these two tempered versions. --Aura (talk) 08:31, 19 January 2021 (UTC)

Not about mapping?

I'm unclear on what the concept of "best approximation" has to do with prime mappings or vals at all, patent, best, or otherwise. It's inconsistent with any val if it is the case that best_approximation(5/4, 17edo) + best_approximation(6/5, 17edo) ≠ best_approximation(3/2, 17edo). Each individual result is the best approximation of a given interval in a given EDO, but it is not the same thing as a val-mapped interval in a given EDO, such as a patent-val-mapped interval or a best-val-mapped interval.

The sentence: 'Just as the patent val itself can be referred to as the "nearest edomapping", so a patent interval can be referred to as a "direct mapping"' seems to be making a category mistake based on the "ing"-ending words being usable as both a name for a process and the end-result of that process. To make it clear, when one is referring the end result, one can instead use the ending "-ed interval". e.g. "patent-val-mapped interval", as I've done above.

I believe what the author may be trying to do in this sentence is to contrast the patent-val-mapped interval with the "directly-mapped interval". But this might suggest the existence of a "direct val". So I suggest referring to it as the "direct approximation", since this implies that it does not go via the intermediary of any prime mapping or val. --Cmloegcmluin (talk) 22:35, 28 June 2021 (UTC)

The best approximations of prime intervals specifically establishes the patent val for an EDO. However, the best approximation of other intervals is not necessarily identical with the mapping established by the patent val. Does that make sense? It may not be obvious that this is the case, but you can begin to see what I mean when you compare the best approximation of 49/32 in 159edo with a stack of two instances of the best approximation of 7/4 in that same EDO. Nevertheless, I can see the value in using the term "direct approximation" instead of "patent interval". --Aura (talk) 03:08, 22 December 2021 (UTC)