Talk:POTE tuning: Difference between revisions

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:: 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 [https://sintel.pythonanywhere.com/ web tool] that calculates it.)

:: - [[User:Sintel|Sintel]] ([[User talk: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. [[User:FloraC|FloraC]] ([[User talk: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. - [[User:Sintel|Sintel]] ([[User talk: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. --[[User:X31eq|X31eq]] ([[User talk: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. – [[User:Sintel|Sintel🎏]] ([[User_talk: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. --[[User:X31eq|X31eq]] ([[User talk: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]]. --[[User:Hkm|hkm]] ([[User talk: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 [[Constrained_tuning|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. --[[User:X31eq|X31eq]] ([[User talk:X31eq|talk]]) 19:57, 10 July 2025 (UTC)

@@ Line 53: / Line 53: @@
 :: 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 [https://sintel.pythonanywhere.com/ web tool] that calculates it.)
 :: - [[User:Sintel|Sintel]] ([[User talk: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. [[User:FloraC|FloraC]] ([[User talk: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. - [[User:Sintel|Sintel]] ([[User talk: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.  --[[User:X31eq|X31eq]] ([[User talk: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. – [[User:Sintel|Sintel🎏]] ([[User_talk: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.  --[[User:X31eq|X31eq]] ([[User talk: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]]. --[[User:Hkm|hkm]] ([[User talk: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 [[Constrained_tuning|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.  --[[User:X31eq|X31eq]] ([[User talk:X31eq|talk]]) 19:57, 10 July 2025 (UTC)