BOP tuning: Difference between revisions

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where the supremum is taken over all monzos <math>M</math> in the ''inverse-weighted'' coordinate system on the dual space. That is, if <math>m</math> were a monzo in unweighted coordinates, its representation in this space would be <math>M=W^{-1}\cdot m</math>, where we are left-multiplying by the inverse of the weighting matrix from before.

This tells us that the max weighted error on the primes has the property of also being the max weighted error on ''all'' monzos in the prime-limit, ~~and~~ that ~~this~~ weighted error will always be obtained at a prime~~, where the weighting is given by the dual L1 norm~~.

This tells us that the max weighted error on the primes has the property of also being the max weighted error on ''all'' monzos in the prime-limit, where this weighting is given by the dual L1 norm. Furthermore, this shows that the weighted error will always be obtained at a prime.

Now, if our weighting matrix is the usual <math>1/log(p)</math> Tenney-weighting matrix, then the above is equivalent to [[Paul Erlich]]'s theorem that minimizing the max Tenney-weighted error on the primes minimizes the max Tenney-weighted error on all intervals. However, if we instead change the weighting matrix to <math>1/p^s</math> instead, then our Linf norm will be dual to a different, somewhat unusual weighted L1 norm on monzos: the one where the weighting on the primes is given by <math>p^s</math>, and the weighting for an arbitrary monzo <math>m = |a\, b\, c\, ...\rangle</math> is given by

Now, if our weighting matrix is the usual <math>1/log(p)</math> Tenney-weighting matrix, then the above is equivalent to [[Paul Erlich]]'s theorem that the tuning that minimizing the max Tenney-weighted error on the primes also minimizes the max Tenney-weighted error on all intervals. This is called the [[TOP tuning]]. However, if we instead change the weighting matrix to <math>1/p^s</math> instead, then our Linf norm will be dual to a different, somewhat unusual weighted L1 norm on monzos: the one where the weighting on the primes is given by <math>p^s</math>, and the weighting for an arbitrary monzo <math>m = |a\, b\, c\, ...\rangle</math> is given by

<math>\text{sopfr}^s(m) = 2^s|a| + 3^s|b| + 5^s|c| + ...</math>

@@ Line 24: / Line 24: @@
 where the supremum is taken over all monzos <math>M</math> in the ''inverse-weighted'' coordinate system on the dual space. That is, if <math>m</math> were a monzo in unweighted coordinates, its representation in this space would be <math>M=W^{-1}\cdot m</math>, where we are left-multiplying by the inverse of the weighting matrix from before.
-This tells us that the max weighted error on the primes has the property of also being the max weighted error on ''all'' monzos in the prime-limit, and that this weighted error will always be obtained at a prime, where the weighting is given by the dual L1 norm.
+This tells us that the max weighted error on the primes has the property of also being the max weighted error on ''all'' monzos in the prime-limit, where this weighting is given by the dual L1 norm. Furthermore, this shows that the weighted error will always be obtained at a prime.
-Now, if our weighting matrix is the usual <math>1/log(p)</math> Tenney-weighting matrix, then the above is equivalent to [[Paul Erlich]]'s theorem that minimizing the max Tenney-weighted error on the primes minimizes the max Tenney-weighted error on all intervals. However, if we instead change the weighting matrix to <math>1/p^s</math> instead, then our Linf norm will be dual to a different, somewhat unusual weighted L1 norm on monzos: the one where the weighting on the primes is given by <math>p^s</math>, and the weighting for an arbitrary monzo <math>m = |a\, b\, c\, ...\rangle</math> is given by
+Now, if our weighting matrix is the usual <math>1/log(p)</math> Tenney-weighting matrix, then the above is equivalent to [[Paul Erlich]]'s theorem that the tuning that minimizing the max Tenney-weighted error on the primes also minimizes the max Tenney-weighted error on all intervals. This is called the [[TOP tuning]]. However, if we instead change the weighting matrix to <math>1/p^s</math> instead, then our Linf norm will be dual to a different, somewhat unusual weighted L1 norm on monzos: the one where the weighting on the primes is given by <math>p^s</math>, and the weighting for an arbitrary monzo <math>m = |a\, b\, c\, ...\rangle</math> is given by
 <math>\text{sopfr}^s(m) = 2^s|a| + 3^s|b| + 5^s|c| + ...</math>