Delta-rational chord: Difference between revisions

Line 62:

}} 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

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 (after some computation) the resulting linear least-squares optimization problem is

<math>

\displaystyle{ \underset{x}{\text{minimize}} \sqrt{\sum_{i=1}^n \Bigg( ~~D_ix~~ - ~~E_i~~ \Bigg)^2 } }

\displaystyle{ \underset{x}{\text{minimize}} \sqrt{\sum_{i=1}^n \Bigg( D_i - E_ix \Bigg)^2 } }

</math>

Line 71:

<math>

x = \displaystyle{\frac{\sum_{i=1}^n D_i E_i}{\sum_{i=1}^n ~~D_i~~^2},}

x = \displaystyle{\frac{\sum_{i=1}^n D_i E_i}{\sum_{i=1}^n E_i^2},}

</math>

@@ Line 62: / Line 62: @@
 }} 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
+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 (after some computation) the resulting linear least-squares optimization problem is
 <math>
-  \displaystyle{ \underset{x}{\text{minimize}}  \sqrt{\sum_{i=1}^n \Bigg( D_ix - E_i \Bigg)^2 } }
+  \displaystyle{ \underset{x}{\text{minimize}}  \sqrt{\sum_{i=1}^n \Bigg( D_i - E_ix \Bigg)^2 } }
 </math>
@@ Line 71: / Line 71: @@
 <math>
-x = \displaystyle{\frac{\sum_{i=1}^n D_i E_i}{\sum_{i=1}^n D_i^2},}
+x = \displaystyle{\frac{\sum_{i=1}^n D_i E_i}{\sum_{i=1}^n E_i^2},}
 </math>