Generator embedding optimization: Difference between revisions
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As discussed in [[Dave Keenan & Douglas Blumeyer's guide to RTT | As discussed in [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning fundamentals#Absolute errors]], these vertical bars mean to take the absolute value of each entry of this vector, not to take its magnitude. | ||
As discussed elsewhere, we can simplify this to: | As discussed elsewhere, we can simplify this to: | ||
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But sometimes we have a tie between tunings for least average damage, though. For example, if we had we done a unity-weight tuning, in which case <math>W = I</math>, and included the interval <math>\frac85</math> in our set, we would have found that quarter-comma meantone tied with another tuning, one with generators of <math>\sqrt[5]{\frac{2560}{81}}</math> and <math>\sqrt[5]{\frac{200}{27}}</math>, which are approximately 1195.7 ¢ and 693.352 ¢. | But sometimes we have a tie between tunings for least average damage, though. For example, if we had we done a unity-weight tuning, in which case <math>W = I</math>, and included the interval <math>\frac85</math> in our set, we would have found that quarter-comma meantone tied with another tuning, one with generators of <math>\sqrt[5]{\frac{2560}{81}}</math> and <math>\sqrt[5]{\frac{200}{27}}</math>, which are approximately 1195.7 ¢ and 693.352 ¢. | ||
In this case, we fall back to our general method, which is equipped to find the true optimum somewhere in between these two extreme ends of goodness, albeit as an approximate solution.<ref group="note">The article for the minimax tuning scheme, [[Target tunings]], suggests that you fall back to the miniRMS method to tie-break between these, but that sort of misses the point of the problem. The two tied points are on extreme opposite ends of the slice of good solutions, and the optimum solution lies somewhere in between them. We don't want the tie-break to choose one or the other extreme; we want to find a better solution somewhere in between them.</ref> This method is discussed here: [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation#Tie_breaking:_power_limit_method|power limit method]]. Or, if you'd like a refresher on how to think about non-unique tunings, please see [[Dave Keenan & Douglas Blumeyer's guide to RTT | In this case, we fall back to our general method, which is equipped to find the true optimum somewhere in between these two extreme ends of goodness, albeit as an approximate solution.<ref group="note">The article for the minimax tuning scheme, [[Target tunings]], suggests that you fall back to the miniRMS method to tie-break between these, but that sort of misses the point of the problem. The two tied points are on extreme opposite ends of the slice of good solutions, and the optimum solution lies somewhere in between them. We don't want the tie-break to choose one or the other extreme; we want to find a better solution somewhere in between them.</ref> This method is discussed here: [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation#Tie_breaking:_power_limit_method|power limit method]]. Or, if you'd like a refresher on how to think about non-unique tunings, please see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning fundamentals#Non-unique tunings]]. | ||
We note that there may be a way to find an exact solution to a nested miniaverage, in a similar fashion to the nested minimax discussed in the coinciding-damage method section below, but it raises some conceptual issues about what a nested miniaverage even means.<ref group="note">There does not seem to be any consensus about how to identify a true optimum in the case of multiple solutions when <math>p=1</math>. See https://en.wikipedia.org/wiki/Least_absolute_deviations#Properties, https://www.researchgate.net/publication/223752233_Dealing_with_the_multiplicity_of_solutions_of_the_l1_and_l_regression_models, and https://stats.stackexchange.com/questions/275931/is-it-possible-to-force-least-absolute-deviations-lad-regression-to-return-the.</ref> We have done some pondering of this problem but it remains open; we didn't prioritize solving it, on account of the fact that nobody uses miniaverage tunings anyway. | We note that there may be a way to find an exact solution to a nested miniaverage, in a similar fashion to the nested minimax discussed in the coinciding-damage method section below, but it raises some conceptual issues about what a nested miniaverage even means.<ref group="note">There does not seem to be any consensus about how to identify a true optimum in the case of multiple solutions when <math>p=1</math>. See https://en.wikipedia.org/wiki/Least_absolute_deviations#Properties, https://www.researchgate.net/publication/223752233_Dealing_with_the_multiplicity_of_solutions_of_the_l1_and_l_regression_models, and https://stats.stackexchange.com/questions/275931/is-it-possible-to-force-least-absolute-deviations-lad-regression-to-return-the.</ref> We have done some pondering of this problem but it remains open; we didn't prioritize solving it, on account of the fact that nobody uses miniaverage tunings anyway. | ||