Delta-rational chord: Difference between revisions
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The least-squares linear error measure does not form a metric on the set of delta signatures with a fixed number of terms, since it is not symmetric. | The least-squares linear error measure does not form a metric on the set of delta signatures with a fixed number of terms, since it is not symmetric. | ||
=== Partially DR === | === Partially DR (one free variable) === | ||
Suppose we wish to approximate a target delta signature of the form <math>+\delta_1 +? +\delta_3</math> with the chord <math>1:f_1:f_2:f_3</math> (where the +? is free to vary). By a derivation similar to the above, the least-squares problem is | Suppose we wish to approximate a target delta signature of the form <math>+\delta_1 +? +\delta_3</math> with the chord <math>1:f_1:f_2:f_3</math> (where the +? is free to vary). By a derivation similar to the above, the least-squares problem is | ||
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=== Partially DR (arbitrary) === | |||
We similarly include a free variable to be optimized for every additional +?, after coalescing strings of consecutive +?'s and omitting the middle notes, and after trimming leading and trailing +?'s. If two variables are related to each other but not to the integer deltas in the signatures, they have a common variable. | We similarly include a free variable to be optimized for every additional +?, after coalescing strings of consecutive +?'s and omitting the middle notes, and after trimming leading and trailing +?'s. If two variables are related to each other but not to the integer deltas in the signatures, they have a common variable. | ||