Delta-rational chord: Difference between revisions
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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-squares-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-squares-error solution instead to minimize the error. | ||
== Least-squares error measures == | |||
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 ''n'' > 1, in the linear domain 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 | 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 ''n'' > 1, in the linear domain 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 | ||
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where we vary x and ask, "By at least how much do the deltas have to be off for any x?" | where we vary x and ask, "By at least how much do the deltas have to be off for any x?" | ||
=== Naive least-squares error === | |||
Rewriting a bit, if 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> has delta signature +ε<sub>1</sub> +ε<sub>2</sub> ... +ε<sub>''n''</sub> (where the chord is 1:1+ε<sub>1</sub>:...), let <math>D_i = \sum_{k=1}^i \delta_i</math> and <math>E_i = \sum_{k=1}^i \epsilon_i.</math> 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 +ε<sub>1</sub> +ε<sub>2</sub> ... +ε<sub>''n''</sub> (where the chord is 1:1+ε<sub>1</sub>:...), let <math>D_i = \sum_{k=1}^i \delta_i</math> and <math>E_i = \sum_{k=1}^i \epsilon_i.</math> Then the resulting linear least-squares optimization problem is | ||
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This error measure was found by Inthar and groundfault. | This error measure was found by Inthar and groundfault. | ||
=== Symmetric least-squares error === | |||
'''Symmetric least-squares error''' (SLS error) is found by solving | '''Symmetric least-squares error''' (SLS error) is found by solving | ||