# Talk:Tenney-Euclidean tuning

## Crazy math theory's dominating the article

Anybody can read this article in its current shape and learn how to derive the TE tuning, TE generators, etc.? I can't. How I learned it was by coming up with the idea of RMS-error tuning, posting it on reddit and get told that was actually called TE tuning.

That said, TE tuning is an easy problem if you break it down this way.

What's the problem?

It's a least squares problem of the following linear equations:

$(AW)^\mathsf{T} \vec{g} = W\vec{p}$

where A is the known mapping of the temperament, g the column vector of each generators in cents, p the column vector of targeted intervals in cents, usually prime harmonics, and W the weighting matrix.

This is an overdetermined system saying that the sum of (AW)Tij steps of generator gj for all j equals the corresponding interval (Wp)i.

How to solve it?

The pseudoinverse is a common means to solve least square problems.

We don't need to document what a pseudoinverse is, at least not in so much amount of detail, cuz it's not a concept specific in tuning, and it's well documented on wikipedia. Nor do we need to document why pseudoinverses solve least square problems. Again, that's not a question specific in tuning.

The only thing that matters is to identify the problem as a least square problem. The rest is nothing but manual labor.

I'm gonna try improving the readability of this article by adding my thoughts and probably clear it up. FloraC (talk) 18:52, 24 June 2020 (UTC)

Update: the page is clear enough now.
The standard way to write the equation is:
$G(AW) = J_0 W$
The targeted interval list is known as JIP and is denoted J0 here. The main difference from my previous comment is that the generator list and the JIP are presented as row vectors. It can be further simplified to
$GV = J$
which is pretty clearly displayed in the article. FloraC (talk) 17:39, 16 December 2021 (UTC)

## Damage, not error?

The article says, "Just as TOP tuning minimizes the maximum Tenney-weighted (L1) error of any interval, TE tuning minimizes the maximum TE-weighted (L2) error of any interval." But shouldn't it be "damage", not "error"? As far as I understand it, there would be no way to minimize the maximum error of any interval under a tuning, because you could always find a more complex interval with more error; minimaxing only makes sense for damage, which scales proportionally with the complexity of the interval. Or am I misunderstanding these concepts? --Cmloegcmluin (talk) 16:50, 28 July 2021 (UTC)

Ah, I think I see. "Damage" may be a bit of an outdated term. It's what Paul Erlich uses in his Middle Path paper. But it means error weighted (divided) by the Tenney height, which is equivalent to the L1 norm, and so "Tenney-weighted (L1) error" is the same thing as damage. And "TE-weighted (L2) error" means error weighted by the TE height, which is equivalent to the L2 norm, so it's similar to damage. --Cmloegcmluin (talk) 19:04, 28 July 2021 (UTC)

## "Frobenius" tuning

Frobenius tuning has nothing to do with the Frobenius norm. It's simply the unweighted Euclidean norm. I propose renaming it to simply that: "unweighted Euclidean tuning".

The article also says:

This leads to a different tuning, the Frobenius tuning, which is perfectly functional but has less theoretical justification than TE tuning.

What theoretical justifications? This is ironic since the next paragraph proceeds to list several theoretical advantages of this tuning.

Not weighting the primes leads to -on average- errors that are the same across primes. It is the Tenney-Euclidean tuning that is biased towards lower primes and not the opposite. This is not a problem at all but the article is in no way clear on this. (In fact, even unweighted norms usually result in temperaments with a slight bias towards low primes, simply because the way temperaments are usually constructed (e.g. stacking edo maps) already has this bias (and especially wrt octaves))

-Sintel (talk) 19:37, 18 December 2021 (UTC)

I'm not sure if the name Frobenius tuning is derived from Frobenius norm.
The next part may be explained more clearly, but I'd like to remind you "the same error across primes" itself is a bias towards higher primes. Notice if 2 and 7 are equally weighted, 8 would get about thrice the error of 7's (see Graham's primerr.pdf). And no, I haven't observed the bias towards lower primes due to the way temperaments are constructed. FloraC (talk) 01:06, 19 December 2021 (UTC)
I've read Breed's paper and I think its very good work. Let me be clear: I think giving more weight to lower primes is a very good idea. But it seems obvious that this is explicitly introducing a certain bias to get better results in practice.
8 is not a prime, so when talking about average errors for the primes it is kind of irrelevant. In the case where you work in some subgroup like 2.5.9, I don't see why you would tolerate twice the error in 9 as you do for 3 in 2.3.5, as we are treating 9 here as a 'formal prime' and not even considering 3. Reading Breed's arguments further he does actually imply that his weighting is biased towards lower primes, but that this is a good thing.
I realize that this is just arguing semantics so I will not lose any sleep over it.
- Sintel (talk) 02:23, 19 December 2021 (UTC)