Delta-rational chord: Difference between revisions

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=== Measuring the error of an approximation ===

==== Linear error ====

Say we want the linear error of a chord 1:''r''1:''r''2:...:''r''''n'' (in increasing order) as an approximation to a fully delta-rational chord with signature +δ1 +δ2 ... +δ''n''. Solving the linear least-squares optimization problem

Say we want the linear error of a chord 1:''r''1:''r''2:...:''r''''n'' (in increasing order) as an approximation to a fully delta-rational chord with signature +δ1 +δ2 ... +δ''n'', i.e. a chord

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

where we vary α.

Solving the linear least-squares optimization problem

<math> \displaystyle{ \min_\alpha \sum_{i=1}^n \Bigg( \alpha r_i - \alpha - \sum_{l=1}^i \delta_l \Bigg)^2 } </math>

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</math>

which can be plugged back into the error formula to obtain the least-squares error in the linear domain.

which can be plugged back into the error formula to obtain the least-squares error in the linear domain. We multiply the 1:''r''1:''r''2:...:''r''''n'' chord by α because we want to compare it to the target DR chord on the same root.

== DR and RTT ==

@@ Line 55: / Line 55: @@
 === Measuring the error of an approximation ===
 ==== Linear error ====
-Say we want the linear error of a chord 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> (in increasing order) as an approximation to a fully delta-rational chord with signature +δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>. Solving the linear least-squares optimization problem
+Say we want the linear error of a chord 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> (in increasing order) as an approximation to a fully delta-rational chord with signature +δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>, i.e. a chord
+<math> \alpha : \alpha + \delta_1 : \cdots : \alpha + \sum_{l=1}^n \delta_l </math>
+where we vary α.
+Solving the linear least-squares optimization problem
 <math> \displaystyle{ \min_\alpha \sum_{i=1}^n \Bigg( \alpha r_i - \alpha - \sum_{l=1}^i \delta_l \Bigg)^2 } </math>
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 </math>
-which can be plugged back into the error formula to obtain the least-squares error in the linear domain.
+which can be plugged back into the error formula to obtain the least-squares error in the linear domain. We multiply the 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> chord by α because we want to compare it to the target DR chord on the same root.
 == DR and RTT ==