Delta-rational chord: Difference between revisions

Line 59:

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

We can vary x and ask, "By at least how much does the approximating chord 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?" When a specific x achieves this minimum, the resulting chord with delta signature {{nowrap|+δ1 +δ2 ... +δ''n''

}} is taken to be the DR chord that is being approximated.

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 59: / Line 59: @@
 <math> x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l.</math>
-We can vary x and ask, "By at least how much does the approximating chord 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?" When a specific x achieves this minimum, the resulting chord with delta signature {{nowrap|+&delta;<sub>1</sub> +&delta;<sub>2</sub> ... +&delta;<sub>''n''</sub>
+}} is taken to be the DR chord that is being approximated.
 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