Delta-rational chord: Difference between revisions

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=== Fully DR ===

The idea motivating least-squares error on a chord as an approximation to a given delta signature is the following: Say we want the error of a chord 1:''r''1:''r''2:...:''r''''n'' (in increasing order), with {{nowrap|''n'' > 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+δ1 +δ2 ... +δ''n''

}} ~~(where the delta signature is written based on the chord written to have root 1)~~, i.e. a chord

}}, i.e. a chord

<math> 1 : 1 + \delta_1 : \cdots : 1 + \sum_{l=1}^n \delta_l.</math>

<math> x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l.</math>

We can ~~replace the 1 with x,~~ vary x and ask, "By at least how much do the ~~deltas~~ have to be off for any x?"

We can vary x and ask, "By at least how much does the approximating chord have to be off for any x?"

Rewriting a bit, if 1:''r''1:''r''2:...:''r''''n'' has delta signature {{nowrap|+ε1 +ε2 ... +ε''n''}} (where the chord is written to start on 1, i.e. 1:{{nowrap|1 + ε1}}:...), let <math>D_i = \sum_{k=1}^i \delta_i</math> (the ''target'' delta signature) and <math>E_i = \sum_{k=1}^i \epsilon_i</math> (the ''approximating'' delta signature). Then the resulting linear least-squares optimization problem is

@@ Line 55: / Line 55: @@
 === Fully DR ===
 The idea motivating least-squares error on a chord as an approximation to a given delta signature is the following: Say we want the error of a chord 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> (in increasing order), with {{nowrap|''n'' &gt; 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+&delta;<sub>1</sub> +&delta;<sub>2</sub> ... +&delta;<sub>''n''</sub>
-}} (where the delta signature is written based on the chord written to have root 1), i.e. a chord
+}}, i.e. a chord
-<math> 1 : 1 + \delta_1 : \cdots : 1 + \sum_{l=1}^n \delta_l.</math>
+<math> x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l.</math>
-We can replace the 1 with x, vary x and ask, "By at least how much do the deltas have to be off for any x?"
+We can vary x and ask, "By at least how much does the approximating chord have to be off for any x?"
 Rewriting a bit, if 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> has delta signature {{nowrap|+ε<sub>1</sub> +ε<sub>2</sub> ... +ε<sub>''n''</sub>}} (where the chord is written to start on 1, i.e. 1:{{nowrap|1 + ε<sub>1</sub>}}:...), let <math>D_i = \sum_{k=1}^i \delta_i</math> (the ''target'' delta signature) and <math>E_i = \sum_{k=1}^i \epsilon_i</math> (the ''approximating'' delta signature). Then the resulting linear least-squares optimization problem is