Talk:POTE tuning

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"Find POTE = T/T[1]"

It doesn't seem to handle the case where the period is a fraction of an octave. For example, try this diaschismic mapping: [⟨2, 2, 7, 15], ⟨0, 1, -2, -8]]. We know the period is half an octave. Indeed,

T = [599.44659699 703.03185125]

Now POTE = T/T[1] gives

POTE = [1200. 1407.36176623]

Is this correct?

Besides, since I don't see a definition of TE generator, I don't know if it should be octave reduced, period reduced or left as is. Changing the basis definitely changes the result. Does TE generators specify a basis or whatever maps are accepted?

From my perspective, this seems more intuitive:

• Find the TE octave: (TV)₁, that is, the first entry of TV.
• POTE = T/(TV)₁

resulting in

POTE = [600. 703.68088312]

Any idea? FloraC (talk) 09:14, 6 June 2020 (UTC)

OK everything's solved. It should be made more clear that TE generators and TE map are different. While TE gens change on basis, never does TE map. FloraC (talk) 05:55, 25 June 2020 (UTC)

Video tutorial

Is someone able and willing to make a video? Or is there an article with illustrations that enables even non-mathematicians like me to follow the process? Thanks in advance for your help. --Xenwolf (talk) 13:49, 6 December 2020 (UTC)

Justification

This article fails to justify why POTE is a good idea. The Kees Height is a complexity measure on JI intervals, not an error metric on tempered intervals, so I can't see how the resulting tuning is optimal in this sense.

As I understand it, the POTE tuning simply finds the least-squares tuning under the TE norm, and then does a completely ad-hoc adjustment to get the octaves just. In constrast, the constrained TE tuning is actually optimal under the TE norm, in the subspace where the octave is just. It seems like a much better candidate for a 'standard' tuning.

-Sintel (talk) 20:45, 18 December 2021 (UTC)

Make sure to ask Paul Erlich. It's a shame he isn't here to help. I added some of his words but that was a tip of the iceberg. FloraC (talk) 01:39, 19 December 2021 (UTC)

OK, after reading up on exterior algebra, POTE just seems wrong. On the page Tenney-Euclidean tuning a justification is given (which should be moved here anyway):

"The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object."

This only works for rank-1 (eg edos) though. In general, for a rank-r temperament, you have some subspace/hyperplane of dimension r. By only looking at the line from 0 to T, you are missing all the other degrees of freedom that you can optimize. (Again, something CTE does correctly.)

- Sintel (talk) 16:11, 11 April 2022 (UTC)

Is it possible (for you) to repair this? --Xenwolf (talk) 06:22, 12 April 2022 (UTC)

No, not really. I've been rallying for CTE to be the default tuning given for temperaments for this reason. (And I've built a web tool that calculates it.)

- Sintel (talk) 12:59, 12 April 2022 (UTC)

How about Kees metric? It's still the optimal tuning by that metric isn't it, set aside why Kees metric is a thing. FloraC (talk) 13:08, 12 April 2022 (UTC)

Good point. We don't know if that's true though. It seems like a strong claim that it's optimal over all intervals. I find it strange that it's the default but nobody can show me why. - Sintel (talk) 17:36, 12 April 2022 (UTC)

What it says is correct. The tuning map can be thought of as a point in projective space or defining a line in tuning space. The TE and POTE tunings are both on the same line and so the same point in projective space. The TE tuning can be defined as an angle in tuning space so that it equally defines TE and POTE. --X31eq (talk) 20:07, 8 July 2025 (UTC)

Just because you can think of it that way doesn't mean you should. Switching to projective space has no benefits when talking about optimization. – Sintel🎏 (talk) 00:00, 9 July 2025 (UTC)

No, it doesn't, does it? That page does look overly mathematical, as I'd expect from its original author, sadly no longer here to defend it. A lot of this wiki is like that and fixing it will certainly be a big task. This argument does, however, make sense from the point of view of geometry and "TE" is a geometric term. It does help to explain (I don't like the term "justify") why "POTE" should follow from "TE". Projective space has good theoretical properties. I noticed a diagram on the front page of this wiki that looks a lot like a projective space plot. --X31eq (talk) 19:23, 10 July 2025 (UTC)

Are there any temperaments in particular where you prefer the POTE tuning to the CTE tuning? We can also compare with the tunings from Path-based goodness. --hkm (talk) 19:03, 9 July 2025 (UTC)

There's a thing. I haven't spent the past 25 years comparing different octave-equivalent tunings. I'll note that a lot of my reasons aren't to do with the sound of POTE in isolation. I think TE is the best tuning in the TE family and POTE works as a supplement to it. I happen to have a guitar fretted to 11-limit POTE magic, but the TE octave is only 0.14 cents sharp so I don't expect CTE would come out much different. I do find it useful to click the "POTE" link in my app so that I can see how the odd-prime errors compare with that small octave tempering removed. There are examples on the CTE page that are best discussed there. As you asked here, I'll say that with meantone it doesn't look right to tune the 7 sharp when it's better to balance the 7 and 5 (even disregarding 5:3) but I don't think it'll make much difference in practice. The Blackwood example also looks like it's giving too much error to the 5:3 but I haven't tuned it up to try. I also think it's bad that the 5 gets tuned just. I think one of the advantages of TE and other RMS optimizations over TOP is that none of the intervals, including octaves, are tuned pure and so have phase locking. It's best to spread the errors around. But in this case it turns out that the TE tuning does leave 5:1 just anyway. I'll also mention the classic example of 5-limit 19EDO where POTE doesn't apply, but it shows a weakness of metrics that only consider the absolute errors of the odd primes. I think (and this used to be a community consensus) that 19EDO is better than it looks in lists that don't consider the error in the minor third. --X31eq (talk) 19:57, 10 July 2025 (UTC)