# Talk:Constrained tuning

## KE is not CTWE

According to Mike (here: https://www.facebook.com/groups/xenharmonicmath/posts/2549402798533260/?comment_id=2561594020647471&reply_comment_id=2566545800152293), Kees tuning was defined as destretched Tenney-Weil (destretched-octave minimax-lils-S), so I have to assume that Kees-Euclidean would be destretched Tenney-Weil-Euclidean (destretched-octave minimax-E-lils-S), not constrained Tenney-Weil-Euclidean (held-octave minimax-E-lils-S).

I also have evidence that this is so, because after followed his advice, I was finally able to get my code to agree with historically posted Kees and Kees-Euclidean tuning results (here: https://www.facebook.com/groups/xenharmonicmath/posts/2363908480416027/?comment_id=2363994823740726 and here: https://www.facebook.com/groups/xenharmonicmath/posts/2086012064872338)

I think we (addressing Flora here) briefly covered this on Discord somewhat recently, but maybe you haven't had time to update the article since.

Yes, I certainly agree that it's unfortunate and confusing that Kees tuning does not correspond to minimizing the tuning with damage weight based on Kees semiheight (and KE on KE seminorm), but that's just apparently how it is.

This might require some updates here as well: Weil_norm,_Tenney-Weil_norm,_and_TWp_interval_and_tuning_space#Kees-Euclidean_seminorm

If you wanted to lobby to revise the definition, I wouldn't be opposed. But since I prefer Dave and I's systematic descriptive naming for tuning schemes, I don't really have a horse in the race.

--Cmloegcmluin (talk) 19:17, 6 February 2023 (UTC)

- I do prefer
*CTWE*to*KE*as its name, so I'll simply remove*KE*from this article. But I don't see why Mike has kept that part in*Weil norm, Tenney-Weil norm, and TWp interval and tuning space #Kees-Euclidean seminorm*, if the same term is historically associated with something else. FloraC (talk) 11:44, 7 February 2023 (UTC)

- Oh, my. Yes, I can see that he himself added that information: https://en.xen.wiki/index.php?title=Weil_norm%2C_Tenney-Weil_norm%2C_and_TWp_interval_and_tuning_space&type=revision&diff=91295&oldid=91294 Well, that's even
*more*confusing, then! Maybe he wasn't thinking about the inconsistency in naming between tuning and norm when he wrote that, and assumed that logically they would match. The tests in my code confirm that the historical KE results which he posted equal the destretched-octave version of this tuning, not the held-octave.

- Oh, my. Yes, I can see that he himself added that information: https://en.xen.wiki/index.php?title=Weil_norm%2C_Tenney-Weil_norm%2C_and_TWp_interval_and_tuning_space&type=revision&diff=91295&oldid=91294 Well, that's even

- However, I see that he used the pseudoinverse to find his results, which I'd suppose would correspond with the held-octave (constrained) version, not the destretched. So... perhaps in the special case of Euclideanized tunings, the destretched and held versions work out the same?

- Unfortunately I don't have time to look into this further. If you find anything out, please let me know, as I may need to amend D&D's guide accordingly. Thanks for letting me know about this complication. --Cmloegcmluin (talk) 16:23, 7 February 2023 (UTC)

OK, I see there's some confusion about what the term KE actually means: if it's constrained-octave or destretched-octave.

I'm not quite sure myself, actually. There seems to be a long history of these terms being used in conflicting ways, and going back through some of my notes I see that I've used both at one point (probably because someone else I was talking to at the time insisted it be one way or the other).

I do agree KE would make the most sense as constrained octave. In the recent discussions we've been having, I've been referring to the constrained octave version. I'd like to see if both/either of these results are similar to POTE. Mike Battaglia (talk) 02:39, 19 March 2024 (UTC)

- I've gone back through my notes and I am pretty sure the most common usage of "Kees-Euclidean tuning" is the constrained-optimization version. Actually, it looks like Graham had independently invented it a year or two before I did, which I hadn't realized, and he also called it constrained-optimization. I don't remember why I was calling the destretched WE tuning the KE tuning in that Facebook thread. But anyway, yes, KE = CWE. Mike Battaglia (talk) 05:10, 19 March 2024 (UTC)

## CTWE, CWE, KE, etc

The way the term "TWE" is being used on this page is not correct. I'm not sure where the confusion is from, but the Tenney-Weil norm is a general norm with two free parameters p and k that interpolate between Wp and Tp. For p=2, then there's one free parameter k, and if k=0 it's TE and if k=1 it's WE. The thing called KE would be the "CWE" tuning, not in general the "CTWE" tuning. It'd be a special case of CTWE with k=1, but of course you don't want to say CTWE = CWE by default. Mike Battaglia (talk)

- I did set the default CTWE to
*k*= 1 in my code cuz in the case of other norms, like the equilateral one where each prime is weighted the same, it's useful to introduce a 30-degree skew like Weil and that needs a term. Calling it "equilateral-Weil-Euclidean" is a possibility and if so, the Tenney-weighted counterpart is automatically "Tenney-Weil". Otherwise I'd always have to say "equilateral-Weil[1]-Euclidean" with the number in the brackets specifying that*k*= 1. So what do you think? FloraC (talk) 03:27, 19 March 2024 (UTC)

- If I understand your question correctly, the name for the Tenney-Weil norm with k=1 is just the Weil norm. Regarding the unweighted version, I haven't looked at this in a long time, but isn't the Hahn norm the unweighted Weil norm, if that's what you mean? So I'd just call it Hahn-Euclidean. Mike Battaglia (talk) 05:06, 19 March 2024 (UTC)

- Isn't the "Hahn norm" a modification on Hahn distance and is dependent on a chosen maximum odd limit? For example in the 7-odd-limit 2, 3, 5, and 7 are weighted equal whereas in the 9-odd-limit 2, 9, 5, and 7 are weighted equal. At least that's what you documented in the
*Hahn distance*article. But there's a norm where all primes are weighted equal regardless of the odd limit and is fit for those who require higher accuracy in higher primes. - And what I noted isn't limited to one type of weight. There's also the skewed variant of Wilson-Euclidean. FloraC (talk) 06:26, 19 March 2024 (UTC)

- Isn't the "Hahn norm" a modification on Hahn distance and is dependent on a chosen maximum odd limit? For example in the 7-odd-limit 2, 3, 5, and 7 are weighted equal whereas in the 9-odd-limit 2, 9, 5, and 7 are weighted equal. At least that's what you documented in the

- I don't remember anymore; I thought Gene told me at one point on IRC that the unweighted Weil norm was the Hahn norm. Looking at it now, it seems the Hahn distance is only a seminorm and it equals the unweighted version of Kees expressibility up to the 7-limit and then something else thereafter. So I'm not sure. But Tenney-Weil already means something, and Weil is "Tenney-weighted by default." So I would call your norm something else. Paul Erlich may have some ideas about if anyone's used it before. If nobody's claimed it then I say go with the Canou norm. Mike Battaglia (talk) 08:44, 19 March 2024 (UTC)

- Then I'm simply gonna call the skew by 30 degrees (
*k*= 1) "skewed" in the next iteration, okay? So there's skewed-equilateral-Euclidean (SEE), skewed-Wilson/Benedetti-Euclidean (SBE). Weil-Euclidean is skewed-Tenney-Euclidean which probably isn't needed as an alias. Hahn is also skewed already so the unskewed variant I laid out in my essay needs a distinct name. I might happily claim that one instead. FloraC (talk) 09:14, 19 March 2024 (UTC)

- Then I'm simply gonna call the skew by 30 degrees (

- OK, so you want some kind of general name for adding the extra coordinate in, times some scaling factor, and then taking the norm? Similarly to going from Tenney to Weil. Is that the idea? I guess you could call that "skewed." I don't want to go change all references from Weil to skewed-Tenney or whatever throughout the Wiki, but I agree it would be good to have a systematic name.

- I'm not sure what unskewed Hahn would mean - wouldn't the unskewed version just be the regular L1 norm? This is sometimes called the "nopf" or "number of prime factors." Gene called the L2 version of this the Frobenius tuning, which I really think is a silly name.

- There is an important difference between the Wilson and Benedetti tunings - they happen to be the same for full-limits, but I don't think they will in general for arbitrary subgroups. The Wilson tuning basically weights intervals using the sum-of-prime-factors, whereas the BE tuning uses the exponential of the Tenney norm. They happen to be the same on prime-limits because the convex hull of all intervals, divided by the exponential of their norm, is also the Wilson unit sphere, so both optimizations give the same thing.

- The "exp-Weil" tuning, or whatever you want to call it, which weights intervals by max(n,d) instead of log(max(n,d)), is probably really important. I haven't thought through what the convex hull is yet. I'm not really sure if it's just the skewed Wilson norm. This would be good to figure out. Mike Battaglia (talk) 09:50, 19 March 2024 (UTC)

- Well it's not as systematic as in D&D's guide, but I'm not super fond of their terminology since it something completely new. I'm just trying to add the missing pieces with minimal changes to existing materials.

- Since Hahn is a norm with a parameter for the maximum relevant odd limit, it's not the same as the equilateral/nonweighted norm whenever the odd limit > 7, and so is the unskewed variant. I'm adding support for them in my code.