User:Hkm/Sandbox: Difference between revisions

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=== Bad formatting ===

2 Practically all edo pages <50 (algorithmically generated material, like GPVs and sagittal notations, should be moved to the GPV and sagittal pages, for example. The interval table gets to stay though)

== 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. Yes, this is summing over infinitely many things, so instead sum over a large finite number because the series converges.

The score for a temperament is the sum of the scores of all the commas.

@@ Line 45: / Line 45: @@
 === Bad formatting ===
 Practically all edo pages <50 (algorithmically generated material, like GPVs and sagittal notations, should be moved to the GPV and sagittal pages, for example. The interval table gets to stay though)
+== 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. Yes, this is summing over infinitely many things, so instead sum over a large finite number because the series converges.
+The score for a temperament is the sum of the scores of all the commas.