Tenney–Euclidean temperament measures: Difference between revisions
Document my implementation of Sintel's error. Request proof for logflat badness. - editorial notes |
→TE complexity: deleted confusing/incorrect sentence |
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== TE complexity == | == TE complexity == | ||
TE complexity is the | TE complexity is the same thing as the average of the coefficients of the [[wedgie]] for that temperament. | ||
Let us define the val weighting matrix ''W'' to be the {{w|diagonal matrix}} with values 1, 1/log<sub>2</sub>3, 1/log<sub>2</sub>5 … 1/log<sub>2</sub>''p'' along the diagonal. For the prime basis {{nowrap|''Q'' {{=}} {{val| 2 3 5 … ''p'' }} }}, | Let us define the val weighting matrix ''W'' to be the {{w|diagonal matrix}} with values 1, 1/log<sub>2</sub>3, 1/log<sub>2</sub>5 … 1/log<sub>2</sub>''p'' along the diagonal. For the prime basis {{nowrap|''Q'' {{=}} {{val| 2 3 5 … ''p'' }} }}, | ||
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We can consider TE error to be a weighted average of the error of each [[prime harmonic]]s in [[TE tuning]], that is, a weighted average of the [[error map]] in the tuning where it is minimized. In this regard, TE error may be expressed in any logarithmic [[interval size unit]]s such as [[cent]]s or [[octave]]s. | We can consider TE error to be a weighted average of the error of each [[prime harmonic]]s in [[TE tuning]], that is, a weighted average of the [[error map]] in the tuning where it is minimized. In this regard, TE error may be expressed in any logarithmic [[interval size unit]]s such as [[cent]]s or [[octave]]s. | ||
As with complexity, we may simply define the TE error as the ''L''<sup>2</sup> norm of the weighted TE error map. If {{nowrap| ''T''<sub>''W''</sub> {{=}} ''TW'' }} is the weighted TE tuning map and {{nowrap| ''J''<sub>''W''</sub> {{=}} ''JW'' {{=}} {{val| 1 1 … 1 }} }} is the weighted just tuning map, then the TE error ''E'' is given by | As with complexity, we may simply define the TE error as the ''L''<sup>2</sup> norm of the weighted TE error map. If {{nowrap| ''T''<sub>''W''</sub> {{=}} ''TW'' }} is the weighted TE tuning map and {{nowrap| ''J''<sub>''W''</sub> {{=}} ''JW'' {{=}} {{val| 1 1 … 1 }} }} (or ''J''<sub>''W''</sub> {{=}} {{val| 1200 1200 … 1200 }} if using cents) is the weighted just tuning map, then the TE error ''E'' is given by | ||
$$ | $$ | ||
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$$ | $$ | ||
where <sup>+</sup> denotes the [[pseudoinverse]]. | where <sup>+</sup> denotes the [[pseudoinverse]]. The term <math>J_W(V_W^+ V_W - I)</math> immediately provides the least squares error of the primes from a given val. This least squares error in turn then optimizes the maximum tenney euclidian error over all intervals. | ||
Often, it is desirable to know the average of errors instead of the sum, which corresponds to Graham Breed's definition<ref name="primerr"/>. This error figure, ''G'', can be found by | Often, it is desirable to know the average of errors instead of the sum, which corresponds to Graham Breed's definition<ref name="primerr"/>. This error figure, ''G'', can be found by | ||