Delta-rational chord: Difference between revisions
| Line 75: | Line 75: | ||
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. | 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. | 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 space structure on delta signatures. | ||
==== Partially DR ==== | ==== 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. | 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. | ||