Xenharmonic Wiki talk:Things to do: Difference between revisions
Moving the proposal here |
→Set a semi-objective standard for classifying edos as subgroup temperaments: More moved to here |
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== Set a semi-objective standard for classifying edos as subgroup temperaments == | == Set a semi-objective standard for classifying edos as subgroup temperaments == | ||
We need a criterion that's as objective as possible for when an EDO should be said to be good for JI subgroups. Some possibilities: | |||
* absolute error | |||
:: [Inthar: arbitrary since different people accept different amounts of error] | |||
* consistent | |||
:: [Inthar: semi-objective, but breaks for small edos; actual accuracy depends too much on the size of the edo] | |||
* consistent to distance 1? | |||
:: [Inthar: see new definition in [[Consistent]] page), The justification is that some small piece of the JI subgroup lattice (maybe to distance one of the "fundamental chord") should map "consistently" in the edo, in addition to the chord itself being consistent. May be too strong for large EDOs.] | |||
:: [Inthar: Subgroup information might be considered technical data IMO.] | |||
'''Proposal''': for nEDk (meaning n equal divisions of the interval k/1, so k=2 is an octave/ditave, k=3 is a tritave, k=5 a pentave, etc.), consider the step errors, defined as err(x) = round(n*log_k(x)) - n*log_k(x), of the first L positive integers AKA of the first L harmonics. Specifically, let X be the set of the errors, meaning for all x in X, we have x in the range [-1/2, 1/2] so that |x| does not exceed 1/2. Then, to determine the error of a subgroup, pick a subset S of X (it does not have to include any powers of 2), and look at the statistical variance (AKA the square of the standard deviation) of the set of error values, however, weight the contributions of harmonics according to their expected frequency of use in factorisations of JI intervals intended to be approximated. This is taken to be the "expected error" (note that the (weighted) mean of the (signed) errors in S is the reference by which error is judged, as this provides a sort of "agnosticism" to the subgroup).<br/> | '''Proposal''': for nEDk (meaning n equal divisions of the interval k/1, so k=2 is an octave/ditave, k=3 is a tritave, k=5 a pentave, etc.), consider the step errors, defined as err(x) = round(n*log_k(x)) - n*log_k(x), of the first L positive integers AKA of the first L harmonics. Specifically, let X be the set of the errors, meaning for all x in X, we have x in the range [-1/2, 1/2] so that |x| does not exceed 1/2. Then, to determine the error of a subgroup, pick a subset S of X (it does not have to include any powers of 2), and look at the statistical variance (AKA the square of the standard deviation) of the set of error values, however, weight the contributions of harmonics according to their expected frequency of use in factorisations of JI intervals intended to be approximated. This is taken to be the "expected error" (note that the (weighted) mean of the (signed) errors in S is the reference by which error is judged, as this provides a sort of "agnosticism" to the subgroup).<br/> | ||