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| = Proof of Optimality Even With Extra "Inconsistent" Mappings = | | = Proof of Optimality Even With Extra "Inconsistent" Mappings = |
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| It can sometimes be useful to look not just at "indirect" prime-based mappings, but also add extra "direct" mappings for important rationals -- deliberately inconsistent with the indirect ones -- for which the indirect mapping is subpar.
| | 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'''. |
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| A good example of this is in 16-EDO, which has a perfectly good 9/8 at 225 cents, but which does not agree with the mapping of 3/2 at 675 cents. In this instance, the associated "2.3.5.9" sval would be <math>\langle 16\, 25\, 37\, 51|</math>, where it is seen that the mapping of 51 steps for 9 is "inconsistent" with the mapping of 25 steps for 3. Note that there is no mapping for 3 at all which will map 9/1 to 51 steps, since 51 is an odd number.
| | This is because the TOP tuning is the supremum of weighted error on all intervals, which is always found at a prime. Since the primes are never altered by changing composite rationals, any "improvement" will not improve on the worst weighted error, nor will it make it any worse given we only use admissible tuning maps. |
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| It so happens that for some prime-limit temperament, the BOP tuning remains optimal even if we use "inconsistent" mappings for every possible composite rational in our prime-limit, regardless of if these are viewed as "extra" direct mappings in addition to the regular "consistent" ones, or even if we discard the original "consistent" ones entirely. More importantly, this result holds ''no matter what the inconsistent ratio mappings are'', as long as you are not changing the primes!
| | The same argument holds for the BOP tuning, since we again know the worst weighted error will always be found at a prime. So likewise, adding extra mappings for rationals that have better weighted error will neither increase or decrease the supremum on the entire temperament. |
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| The one restriction we need to get this result is that we only look at tuning maps where our "inconsistent" mappings are no worse than the "consistent" ones would be. That is, if you have a tuning map with the "consistent" 9/8 tuned to 204 cents, that you don't deliberately tune the "inconsistent" 9/8 to something like 230 cents, which makes it worse for no reason.
| | As in the TOP case, some care is needed when extending this argument to arbitrary subgroup temperaments, as it is possible to use arbitrary mappings for rationals without requiring that the relevant primes even exist in the subgroup, so that there are no primes for them to be "inconsistent" with. |
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| One way to express this restriction is to write it, for all rationals n/d, as
| | 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. |
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| <math>\text{directerr}(n/d) \leq \text{indirecterr}(n/d)</math>
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| where these refer to the direct and indirect ''unweighted'' error. Since the weighting doesn't change, we also have | |
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| <math>\frac{\text{directerr}(n/d)}{\text{true}_s(n/d)} \leq \frac{\text{indirecterr}(n/d)}{\text{true}_s(n/d)}</math>
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| As a result, we know that ''adding'' extra inconsistent rational mappings can never increase the max error, because we are only adding a new set of weighted errors that are guaranteed to be no worse than things already in the original set. As a result, the supremum of weighted error remains unchanged.
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| More subtly, we also know that ''removing'' the original rational mappings, and using ''only'' the inconsistent mappings, we get the same result. This is because the supremum is always obtained at a prime. Since the prime mappings are always consistent ''by definition'', there is no way to adjust their mapping to get an "inconsistent prime".
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| As a result, the suprema are unchanged either way.
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| = Applications Of The Above = | | = Applications Of The Above = |