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| == Badness == | | == Badness == |
| We take all of the fractions greater than 1 within the temperament subgroup and map them to orthogonal Kronecker vectors in an infinite-dimensional vector space (because there are infinitely many fractions within the temperament subgroup). We plot all of our infinitely many commas on this vector space (for example, if our comma basis contains elements that generate 80/81, the vector [-4 1] in the subgroup with first coordinate corresponding to 3/2 and second coordinate 5/1 is plotted here because (3/2)^-4 * 5/1 = 80/81). We then stretch each axis (which is a linear transformation where the eigenvectors are the kronecker vectors) to have length equal to (min_axis_length + the square root of the cent error) * the sum of the numerator and denominator of the basis element, where min_axis_length is a constant and the cent error is the difference between the tempered cent value (using the tempered generators) and the real cent value. Then the score of a comma is score_persistence to the power of its 1-norm (taxicab norm), where score_persistence < 1. | | We say that is the score of a "step" is equal to 1/(min_cents + sqrt(the motion's error in cents)) * badness_penalty**(the complexity of the motion). We then say that the score of a "path" is equal to the product of the scores of the steps. The badness of a rank-2 temperament is equal to the sum of the scores of all paths that reach the original interval. (This works for irregular and regular temperaments.) |
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| The score for a temperament is the sum of the scores of all the commas. | |