# Talk:Direct approximation

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## 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 are what establishes the patent val for an EDO. However, the best approximations of other intervals are not necessarily identical with those approximations 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)
Yes, that does make sense to me; it's precisely the reason why I objected to the term in the first place. At least, that's what I was trying to convey. I'll say it another way now, in case it helps further: I don't think this relates to the type of regular mapping that we do in RTT; the meaning of the word regular in its name is that this theory shows how to regularize these approximations so that those sorts of inconsistencies don't happen. So I'm not sure why you then use the word "nevertheless" when you say that you can see the value in making a change; it seems like you should instead use the word "therefore". But I am nevertheless (haha, see what I did there?) glad that you can see this value, one way or another.
If you re-read my earlier comment, you should see that I actually only suggested changing "direct mapping" to "direct approximation"; I didn't suggest that you change "patent interval" to "direct approximation", as you've said. However, now that you've said it, I would support changing "patent interval" to "direct approximation" even more than I would support changing "direct mapping" to "direct approximation". Here's why:
So I don't suppose there's anything inconsistent with your use of the word "patent" in the xenharmonic community, but I am disappointed to see that usage propagated any further, because I think it was a bad choice in the first place. The main reason is that it may mislead people into thinking that "patent vals" are the best or only maps for their EDO.
And as for "interval", it doesn't look like you're actually using "patent interval" to refer to the interval itself, but actually a measurement of it. That is, you wouldn't say "10 is the patent interval of 3/2 in 17edo", right? 10\17edo would be an interval, but 10 is just a number of steps. And a "number of steps" are the words you use to define this thing in the first sentence. So that's a reason to choose "approximation" over "interval", because 10 does make sense to call an EDO's approximation. --Cmloegcmluin (talk) 13:07, 22 December 2021 (UTC)
The thing is that when you temper out multiple commas, you also temper out not only multiples of each comma, but also the sums and differences of those commas. The sums and multiples of those tempered out commas can get pretty big, and thus, the inconsistences I'm talking about with respect to JI are guaranteed to happen at some point no matter what you do. To me, the term "regular" in "regular temperament" is more about regularizing the approximations of prime intervals and building a consistent mapping of various other intervals based on that regularization, however, I'm under no illusion that this process always results in the best approximation of all intervals within any given EDO. Therefore, there is a valid reason for me to have used the term "nevertheless" rather than the term "therefore" in my original reply.
At any rate, it seems that we are mostly on the same page regarding the need to change "patent interval" and "direct mapping" to "direct approximation". However, as for the term "patent", we need some way of designating the maps which utilize the best approximations of all prime intervals. --Aura (talk) 14:33, 22 December 2021 (UTC)
Re: "patent" for designating such maps, my preference is "simple": simple map. --Cmloegcmluin (talk) 14:52, 22 December 2021 (UTC)
Nice. Now that I see what's being talked about on that page, it makes sense. Regardless, I think we should wait for Sintel to weigh in, just to make sure I at least have all my facts straight. --Aura (talk) 14:55, 22 December 2021 (UTC)
Just to give a sort of overview here, there are two ways you could go about constructing a mapping for 12edo.
The first is the construct a regular temperament, with mapping M = [12 19 28]. To calculate the output you just take the prime factorization and multiply, as we're used to:
• M(5/4) = 4
• M(81/80) = 0
• M((81/80)^3) = 0
Now because M is linear, it satisfies the conditions:
• M(a) + M(b) = M(a * b) (for two JI intervals a, b)
• n*M(a) = M (a^n) (for some integer n)
So "linear map" really means: a map that conserves interval arithmetic. I think this is what the word "regular" refers to. ("linear temperament" already means something else sadly.)
(Factoring through the map, addition becomes multiplication because of the way interval arithmetic works on $\mathbb{Q}(\times)$)
The second way to define the 12edo temperament is to just round. The map is then:
F(x) = round(12 * log2(x))
• F(5/4) = 4
• F(81/80) = 0
• F((81/80)^3) = 1 (!)
Because F is not a linear function, it does not satisfy any of the conditions above. So even though rounding sometimes gives us better approximations, I think most people prefer conserving interval interval arithmetic, which is why we care about RTT in the first place.
Note that all of this is unrelated to the fact that we found the regular 12edo map by rounding. It's just a sort of shortcut to find linear maps that have reasonable errors. Note also that in the definition of the linear map above, you never have to take any logarithms, they're also just convenience.
"Direct approximation" is a good name imo. Regarding "simple map" and "integer uniform map", I have some more thoughts on that but I'll talk about those there.
- Sintel (talk) 18:26, 22 December 2021 (UTC)
I second that this page had better be moved to direct approximation. FloraC (talk) 16:39, 22 December 2021 (UTC)
Thanks for that fantastic explanation, Sintel. You said it better than I could have. I look forward to your thoughts on simple and integer uniform map. —Cmloegcmluin (talk) 18:44, 22 December 2021 (UTC)
Aura, are you still considering the change to direct approximation, per my, Flora, and Sintel's suggestion? --Cmloegcmluin (talk) 20:01, 23 February 2022 (UTC)
Yes. I still think this page should be moved to direct approximation. --Aura (talk) 21:02, 23 February 2022 (UTC)
The move is now done, I think the redirects should be kept, but there is a need for some rewording now ... --Xenwolf (talk) 22:05, 23 February 2022 (UTC)

## Incorporating into RTT

I will simply add that the regular temperament formalism can easily be extended to include some of this, which we've talked about on the Facebook xen math group in the past.
For instance, 16-EDO has a perfectly good direct mapping for 9, which happens to be different than it's representation of 3*3. You can always adjoin an extra 9 to the subgroup as though it were a prime, which I have notated in the past as 9' (read 9-prime). So if we were in the 2.3.5 subgroup before, we extend it with the extra 9' to 2.3.5.9'.
This is kind of weird as a JI subgroup as you have a bunch of pseudo-intervals like 9'/9, which simply exist as monzos in this larger subgroup, even if their JI tuning would be 0 cents. But the usefulness happens when looking at vals on this group, so that the 16-edo patent val would be ⟨16 25 37 51|, and here we note that the mapping of 51 steps for 9' is not the same as the mapping of 50 steps for 9. This is still a linear map; you can certainly play a chord like 4:5:7:9, for which the outer dyad is 19 steps, then keep the top note the same and build an 8:10:12:15:18 chord going down from it; if you do so then the lowest note of the second chord will be one step higher than the lowest of the first chord, and since you moved up by 9' once and down be 3 twice, this shows you that 9'/9 is mapped to 1 step of 16-EDO, which is now a musically meaningful statement.
This is the way I have always found most useful to look at inconsistent mappings - I don't think most people care about the direct mapping for every ratio all the time, like (81/80)^100 or whatever, but for any equal temperament there are usually a few of these direct mappings here and there which really are useful, which can then be added explicitly, and for which you can still modulate around "regular"ly as long as you keep track of which version of these rationals you are using.
If you like, you can view the almighty master space for this idea to be the free abelian group where every single rational (possibly greater than 1/1) is treated as having its own basis vector. This group extends JI so that not only do we have the primes as basis vectors, but also have added an extra "primified" version of every rational number in this way. 2.3.5.9' is a subgroup of this, as well as anything else you could dream up, and you can always do temperament searches on the subgroups of this "beyond-JI" group. This is a group I have bumped into again and again; in addition to the above, the set of generalized patent vals on this is closely related to the zeta function, and it also may be useful with some of the subgroup temperament stuff I have been doing. I am not sure what to call this although "paraconsistent JI," "paraconsistent subgroups," etc were suggested in the past. Maybe para-JI, para-subgroup, para-interval, para-mapping, or something. Mike Battaglia (talk) 19:51, 22 December 2021 (UTC)
I hadn't heard of this 9' (9-prime) thing before. But it's really cool! And I have to say that using the ' symbol to literally prime-ify a number is a stroke of expressive genius. If there's any way you can could up a link to the old Facebook discussion that'd be fantastic. If not, no big deal, (Facebook isn't really good for that, is it), but I would just like to read more about it if I could. --Cmloegcmluin (talk) 20:33, 22 December 2021 (UTC)
I'm not sure how to search Facebook for something like "9'", lol, but it goes all the way back to 2011 in tuning-math #18580. The basic idea hasn't changed very much since then, except that people didn't like the term "inconsistent JI," "inconsistent mapping," etc, so I suggested "paraconsistent JI" at a later date sometime on Facebook. I am glad you like the prime notation although I have never been sure how to do things if you are adding multiple of these paraprimes, let's say - if you are in 2.3.5.9'.15', what is 135', 9'*15 or 9*15'? I guess you could just write it explicitly as 9'*15 or 9*15' if it's ambiguous. Mike Battaglia (talk) 20:44, 22 December 2021 (UTC)
Some of the Discord people are fond of "dual fifth" systems. They've made pages here: Dual-fifth tuning and Dual-fifth temperaments. I haven't got time to review all the data they entered tho. FloraC (talk) 21:02, 22 December 2021 (UTC)
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). 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? --Xenwolf (talk) 22:47, 23 February 2022 (UTC)

FYI: The patent fifth is now replaced by direct approximation of the fifth. --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. --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? --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. --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. --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? FloraC (talk) 09:56, 24 February 2022 (UTC)

I've definitely seen this before. See: https://mathworld.wolfram.com/NearestIntegerFunction.html Hoewever I think $\text{round}(x)$ is much clearer. -- 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 $⌈·⌋$ acceptable despite not being well-established. And I like $⌈·⌋$ because it's concise, and also because it's logical w/r/t to the well-established ceil $⌈·⌉$ and floor $⌊·⌋$ notations. I knew it was used at least sometimes but couldn't remember specifically where I'd seen $⌈·⌋$ before, but a web search confirmed that it's at least not entirely unheard of. --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. FloraC (talk) 22:13, 24 February 2022 (UTC)