Talk:Constrained tuning: Difference between revisions

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::::: 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. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 09:14, 19 March 2024 (UTC)

:::::: 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. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 09:50, 19 March 2024 (UTC)

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 ::::: 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. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 09:14, 19 March 2024 (UTC)
+:::::: 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. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 09:50, 19 March 2024 (UTC)