Tenney–Euclidean temperament measures: Difference between revisions
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# Taking an RMS | # Taking an RMS | ||
# Taking an RMS and also normalizing for the temperament rank | # Taking an RMS and also normalizing for the temperament rank | ||
# Any of the above and also dividing by the norm of the just | # Any of the above and also dividing by the norm of the [[JIP]] (just initonation points). | ||
Graham Breed's original definitions from his ''primerr.pdf'' paper tend to use the third definition, as do parts of his [http://x31eq.com/temper/ temperament finder], although other scaling and normalization methods are sometimes used as well. | Graham Breed's original definitions from his ''primerr.pdf'' paper tend to use the third definition, as do parts of his [http://x31eq.com/temper/ temperament finder], although other scaling and normalization methods are sometimes used as well. | ||
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The '''TE simple badness''' of M, which we may also call the '''relative error''' of M, may be considered error relativized to the complexity of the temperament. It is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step. This may considered to be a sort of badness which heavily favors complex temperaments. | The '''TE simple badness''' of M, which we may also call the '''relative error''' of M, may be considered error relativized to the complexity of the temperament. It is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step. This may considered to be a sort of badness which heavily favors complex temperaments. | ||
Gene Ward Smith defines the simple badness of M as ||J∧M||<sub>RMS</sub>, where J = {{val|1 1 ... 1}} is the | Gene Ward Smith defines the simple badness of M as ||J∧M||<sub>RMS</sub>, where J = {{val|1 1 ... 1}} is the JIP in weighted coordinates. Once again, if we have a list of vectors we may use a Gramian to compute it. First we note that a<sub>''i''</sub> = J·v<sub>''i''</sub>/''n'' is the mean value of the entries of v<sub>''i''</sub>. Then note that J∧(v<sub>1</sub> - a<sub>1</sub>J)∧(v<sub>2</sub> - a<sub>2</sub>J)∧...∧(v<sub>''r''</sub> - a<sub>''r''</sub>J) = J∧v<sub>1</sub>∧v<sub>2</sub>∧...∧v<sub>''r''</sub>, since wedge products with more than one term J are zero. The Gram matrix of the vectors J and v<sub>1</sub> - a<sub>''i''</sub>J will have ''n'' as the (1,1) entry, and 0s in the rest of the first row and column. Hence we obtain: | ||
<math>\displaystyle | <math>\displaystyle | ||