BOP tuning: Difference between revisions

It so happens that we can use the same recipe to define a closely related tuning that instead minimizes the max <math>1/(nd)^s</math> weighting on all rationals, where s is a free parameter > 1. We will call this the "BOP" tuning for [[Benedetti height|Benedetti Optimal]].

Similarly to TOP, we will show that this is easily obtained by minimizing the max <math>1/p^s</math> weighting on just the primes.

For now, we show that this result holds for all full-limits, and can be easily extended to certain "nice" subgroups. More work is needed to extend it to *all* subgroups, as well as to define an analogue to "TIPTOP".

= Proof of Benedetti-Optimality On All Rationals =

First, we note that for any tuning map, call it <math>T</math>, there is an associated vector called a '''signed error map'''. This is given by <math>E = T-J</math>, where <math>J</math> is the JIP. If we assume that we are in a prime-limit, then the signed error map will contain the signed error on each prime. We assume that all errors are unweighted so far.

For any such error map, we can then evaluate the maximum ''unsigned'' weighted error on the primes. This quantity is equal to the maximum of the absolute values of the coefficients of the signed weighted error map, or equivalently the [https://en.wikipedia.org/wiki/Norm_(mathematics)#Maximum_norm_(special_case_of:_infinity_norm,_uniform_norm,_or_supremum_norm) L-inf] norm of the signed error map in the weighted coordinate system.

The Linf norm has the property of being the [https://en.wikipedia.org/wiki/Dual_norm dual norm] of the L1 norm on the dual space, which is the space of monzos. This means that it has the following property:

Technically, the supremum above also includes all vectors with arbitrary real coordinates, whereas we want to restrict to only those that have integer coordinates in the unweighted basis. However, since we divide by the norm, we note the supremum with all integer monzos is the same as the supremum with all monzos with rational coordinates, which is dense in the set of arbitrary real-valued vectors, so that the supremum will be the same - in particular, it is always found at a prime, which is an integer-coordinate anyway.

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>

Now, we note that this unusually weighted L1 norm does not provide the weighting we want on all rationals, which is <math>(nd)^s</math>. Rather, when <math>s=1</math>, this is called the [[Wilson Height|Wilson norm]], and is equivalent to the [http://mathworld.wolfram.com/SumofPrimeFactors.html sum of prime factors with repetition] of the ratio <math>n/d</math> in question, often written <math>\text{sopfr}(n/d)</math>. For arbitrary <math>s\geq 1</math>, this is the sum of the <math>s</math>'th powers of the prime factors with repetition of the ratio, which we will denote <math>\text{sopfr}^s(n/d)</math>. However, we note that this strange weighting ''does'' equal the weighting we want on the primes, where we have <math>\text{sopfr}^s(n/d) = (n/d)^s</math> - it only diverges on the composite numbers.  

 

<math>\text{sopfr}^s(n/d) \leq (n/d)^s</math>

<math>\frac{\text{err}_{n/d}}{\text{sopfr}^s(n/d)} \geq \frac{\text{err}_{n/d}}{(n/d)^s}</math>

If we can show the above, this means we have shown that after re-weighting, the weighted error on all rationals goes ''down'' -- except at the primes, where the error (and in particular the worst error) remains the same. This would give us our desired result.

 

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

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

<math>2^s|a| + 3^s|b| + 5^s|c| + ... \leq 2^{s|a|} \cdot 3^{s|b|} \cdot 5^{s|c|} \cdot ...</math>

== Extension to "Good" Subgroups ==

The above proof was only for prime limits. However, the above usually holds as long as you are working in a "nice" subgroup where the Tenney height looks like a scaled Lp norm. Examples include "prime power" subgroups such as 2.9.7.11. More work is needed to derive a corresponding result for non-nice subgroups.

= Proof of Optimality Even With Extra "Inconsistent" Mappings =

In the page on [[TOP_tuning#TOP_with_.22Inconsistent.22_Rational_Tuning_Extensions|TOP tuning]], it is shown that for full-limits (and certain "nice" subgroups), the TOP tuning remains optimal even if composite rationals are given extra "inconsistent" mappings, as long as the tuning on those mappings is no worse than the consistent one. Tuning maps with such a property are called '''admissible'''.

This property is particularly important for infinite-limit generalized patent vals, where it can be shown that regardless of whether ratios are mapped "consistently" via the prime mapping, or "inconsistently" to the nearest EDO-step, the same BOP tuning is optimal for all rational numbers.

As a simpler example, this guarantees that the BOP tuning for 2.3.5.9 16-EDO, with the inconsistent mapping of 57 steps on the 9/1, is the same as the 2.3.5 tuning for 16-EDO.