Delta-rational chord: Difference between revisions

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which can be plugged back into the error formula to obtain the error. (We multiply the 1:''r''1:''r''2:...:''r''''n'' chord by α in order to compare it to the target DR chord on the same isodifferential series.)

This error measure is called '''least-squares delta error'''{{idiosyncratic}} (LSD error). LSD error does not depend on whether the chord whose error is being measured is 1:''r''1:''r''2:...:''r''''n'' or the same chord linearly shifted to have root α~~. (TODO: check whether LSD error forms a metric)~~.

This error measure is called '''least-squares delta error'''{{idiosyncratic}} (LSD error). LSD error does not depend on whether the chord whose error is being measured is 1:''r''1:''r''2:...:''r''''n'' or the same chord linearly shifted to have root α.

Unfortunately, LSD error does not form a metric on the set of delta signatures with a fixed number of terms. There is ongoing research seeking a natural and psychoacoustically useful metric.

==== Partially DR ====

When the DR signature has one or more free ("+?") terms, the optimization problem becomes a multivariate one: we have one variable <math>\alpha_i, i \ge 2</math> for each free term, as well as the variable <math>\alpha_1</math> for the root. However, solving it is not much more difficult than the univariate case.

@@ Line 73: / Line 73: @@
 which can be plugged back into the error formula to obtain the error. (We multiply the 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> chord by α in order to compare it to the target DR chord on the same isodifferential series.)
-This error measure is called '''least-squares delta error'''{{idiosyncratic}} (LSD error). LSD error does not depend on whether the chord whose error is being measured is 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> or the same chord linearly shifted to have root α. (TODO: check whether LSD error forms a metric).
+This error measure is called '''least-squares delta error'''{{idiosyncratic}} (LSD error). LSD error does not depend on whether the chord whose error is being measured is 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> or the same chord linearly shifted to have root α.
+Unfortunately, LSD error does not form a metric on the set of delta signatures with a fixed number of terms. There is ongoing research seeking a natural and psychoacoustically useful metric.
 ==== Partially DR ====
 When the DR signature has one or more free ("+?") terms, the optimization problem becomes a multivariate one: we have one variable <math>\alpha_i, i \ge 2</math> for each free term, as well as the variable <math>\alpha_1</math> for the root. However, solving it is not much more difficult than the univariate case.