Delta-rational chord: Difference between revisions
m Clarify/strengthen wording about what "nonoverlapping" means. |
m →Mathematics of DR: Start section on error measures |
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The existence of an exact tuning for a delta signature specification is only guaranteed to hold when we only care about a ratio between ''two'' terms in the delta signature being exact. If we want to optimize an arbitrary specified delta signature (with some deltas possibly held free), we can use a least-squared-error solution instead to minimize the error. | The existence of an exact tuning for a delta signature specification is only guaranteed to hold when we only care about a ratio between ''two'' terms in the delta signature being exact. If we want to optimize an arbitrary specified delta signature (with some deltas possibly held free), we can use a least-squared-error solution instead to minimize the error. | ||
=== 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> 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 | |||
<math> \displaystyle{ \max_\alpha \sum_{i=1}^n \Bigg( \alpha + \sum_{l=1}^i \delta_l - \alpha r_i \Bigg)^2 } </math> | |||
yields | |||
<math> | |||
\alpha = \displaystyle{\frac{\sum_{i=1}^n \delta_i (r_i - 1)}{\sum_{i=1}^n (r_i - 1)^2},} | |||
</math> | |||
which can be plugged back into the error formula to obtain the least-squares error in the linear domain. | |||
== DR and RTT == | == DR and RTT == | ||