TOP tuning: Difference between revisions

Line 46:

Given this restriction, the proof is easy: for any rational number, any "inconsistent" tuning must have a weighted error that is no worse than the "consistent" tuning, which in turn is no worse than the worst weighted prime error. However, there is no such thing as an "inconsistent prime mapping" - the mapping of a prime must always be consistent with itself! As a result, the worst-weighted error of the entire temperament cannot be changed by improving the errors of individual composite rationals - it will always be found at the worst-weighted prime, which will never change in this way.

As a result, the tuning that minimizes the max Tenney-weighted error on the primes is the same tuning that minimizes the max Tenney-weighted error on all rationals, even if ~~those~~ rationals ~~are inconsistently adjusted to get a~~ better ~~tuning~~ than the consistent ones.

As a result, the tuning that minimizes the max Tenney-weighted error on the primes is the same tuning that minimizes the max Tenney-weighted error on all rationals, even if there exist rationals with extra inconsistent mappings that have better tunings than the consistent ones.

Note that the above proof is only for full-prime limits: for arbitrary subgroups, some care is needed to extend the above argument, as it is possible (for instance) to work in the 2.5.9 subgroup without mapping 3 at all. In this situation, it is no longer the case that we have an extra mapping for 9/1 that is "inconsistent" with the mapping for 3/1, because there is no mapping for 3/1 at all, so 9/1 needs to be treated as thought it were a "prime." While a more thorough treatment of inconsistent mappings on arbitrary subgroups is needed, it is easy to see that for subgroups with a basis consisting only of prime powers, the same argument is easily shown to hold, with the worst weighted error being found at a prime power rather than a prime.

Most importantly, the above result holds without any hitches for prime-limit subgroups as the limit tends to infinity, and in particular for infinite-limit generalized patent vals, where the TOP tuning minimizes the error on all rationals regardless of whether those rationals are mapped "consistently" given the mapping on the primes, or "inconsistently" given their direct rounding to the nearest EDO-step.

@@ Line 46: / Line 46: @@
 Given this restriction, the proof is easy: for any rational number, any "inconsistent" tuning must have a weighted error that is no worse than the "consistent" tuning, which in turn is no worse than the worst weighted prime error. However, there is no such thing as an "inconsistent prime mapping" - the mapping of a prime must always be consistent with itself! As a result, the worst-weighted error of the entire temperament cannot be changed by improving the errors of individual composite rationals - it will always be found at the worst-weighted prime, which will never change in this way.
-As a result, the tuning that minimizes the max Tenney-weighted error on the primes is the same tuning that minimizes the max Tenney-weighted error on all rationals, even if those rationals are inconsistently adjusted to get a better tuning than the consistent ones.
+As a result, the tuning that minimizes the max Tenney-weighted error on the primes is the same tuning that minimizes the max Tenney-weighted error on all rationals, even if there exist rationals with extra inconsistent mappings that have better tunings than the consistent ones.
 Note that the above proof is only for full-prime limits: for arbitrary subgroups, some care is needed to extend the above argument, as it is possible (for instance) to work in the 2.5.9 subgroup without mapping 3 at all. In this situation, it is no longer the case that we have an extra mapping for 9/1 that is "inconsistent" with the mapping for 3/1, because there is no mapping for 3/1 at all, so 9/1 needs to be treated as thought it were a "prime." While a more thorough treatment of inconsistent mappings on arbitrary subgroups is needed, it is easy to see that for subgroups with a basis consisting only of prime powers, the same argument is easily shown to hold, with the worst weighted error being found at a prime power rather than a prime.
+Most importantly, the above result holds without any hitches for prime-limit subgroups as the limit tends to infinity, and in particular for infinite-limit generalized patent vals, where the TOP tuning minimizes the error on all rationals regardless of whether those rationals are mapped "consistently" given the mapping on the primes, or "inconsistently" given their direct rounding to the nearest EDO-step.