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 "direct mapping", but what is the proper term for the second one? Is it "consistent mapping"? I think this article could be expanded by describing the relationship between these two different types of mappings. --Aura (talk) 22:24, 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)