What I am trying to achieve: Since I have exhausted everything I wanted to know, with respect to Enumerative Combinatorics on Necklaces, my goal now is to...
Hi Paul, I regret to say that i haven't followed any of your Necklace posts (for lack of time to get really involved), and am especially sorry about this since...
... Hi Monz, Thanks. I will look at this. I don't know why I am so obssessed with applying musical set theory to lattices, but then again, why not. So far...
... Polya theory can be applied to everything that is finite. So at least the original theory won't work for infinite lattices - maybe there is some...
I've been working on a new PDF: http://x31eq.com/composite.pdf It's a complement to the Prime Errors and Complexities paper I spent a long time working on. It...
I've been looking at LLL reduction again. I tried to use LLL reduction with parametric badness before and it didn't work. I've now found the bug in my code...
... least ... is ... finite ... Yes, that is the sort of thing I am looking for. My book still hasn't arrived, so I cannot look any of this up yet. Of course,...
Now in a "preliminary finished" state. http://x31eq.com/composite.pdf Also a single column format which may be better for reading electronically but also...
Theorem 3 looks closest to what I'm after. I don't follow the matrices. Can you confirm that it states that the RMS error over all unique intervals in prime...
... No. It's the same as *a* prime weighted error. There are n free parameters for n primes (one redundant). ... Why are you using RMS error for taxicab...
... Huh? ... We both take the unweighted RMS of intervals in a prime limit unioned with something else. You use max Tenney height. I use max taxicab distance....
... You can choose how you weight each prime interval. Tenney weighting is one way of doing it. An unweighted Tenney limit is always the same as a weighted...
Graham wrote... ... Do you prove that? I'd love to understand how you normalize. ... For taxicab distance zero you have only 1/1 and the error is zero....
... No. If I proved it then it'd be a theorem, not a conjecture. The normalization here is to divide the matrix by the number in the top left-hand corner....
Graham wrote... ... Yes. ... No no no. The above formula is the MS *error* over all unique intervals within a certain taxicab distance. y=2 gives two...
... No it doesn't. You missed 1. And if it's 1, 3 and 9 you don't have 3/2 or 9/8. ... Yes, that's how you can ignore them. ... How come? What exactly is...
... The error is over the intervals. There's no error on 1. ... I put those labels there to make it easier to follow. There are no 2s in this system. ... The...
... The average is over the intervals and 1:1 is an interval. ... Okay. <snip> ... So you have a simplified system where the RMS error of a temperament is the...
Graham wrote... ... It seems completely academic whether it's included or not. ... I don't think it's significant for most applications. Gene seemed to agree. ...
... For what applications? What did he agree about and why? Theorem 3 does hold for a Euclidean cutoff. All you need is that the "circles" are symmetrical...
... You can compare the errors if you want. For Tenney limits the result is that it doesn't matter much which limit you choose. For Farey limits I don't know...
... Since the RMSE of all intervals isn't bounded and the RMSE of primes erases sign information, I don't see why I should believe such a comparison would be...
... It's true that I only care about all the consonant chords I could ever form. But then I'd have to say what those are. ... No matter what I'm interested...
... Then why did you ask about comparing the errors of different sets of intervals? There's no way to avoid saying what intervals you want with an RMS. An...
Just a tidbit on the All-Interval scale in 22-tET. It is a 10-set, namely (0,1,2,4,5,7,11,12,15,18) with Interval Vector <5,5,5,5,5,5,5,5,5,5,5> It's the only...
... <5,5,5,5,5,5,5,5,5,5,5> Sorry it is really: (0,1,2,4,7,11,12,15,18) with Interval Vector <4,4,4,4,4,4,4,4,4,4,5> So not Perfect. If you add the 5 you get...
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