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.

== ~~Error measures~~ ==

== Least-squares error ==

The idea motivating least-squares error ~~measures~~ on delta ~~signatures~~ is the following: Say we want the error of a chord 1:''r''1:''r''2:...:''r''''n'' (in increasing order), with {{nowrap|''n'' > 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+δ1 +δ2 ... +δ''n''

The idea motivating least-squares error on a chord as an approximation to a given delta signature is the following: Say we want the error of a chord 1:''r''1:''r''2:...:''r''''n'' (in increasing order), with {{nowrap|''n'' > 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+δ1 +δ2 ... +δ''n''

}} (where the delta signature is written based on the chord written to have root 1), i.e. a chord

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We can replace the 1 with x, vary x and ask, "By at least how much do the deltas have to be off for any x?"

~~=== Least-squares error ===~~

Rewriting a bit, if 1:''r''1:''r''2:...:''r''''n'' has delta signature +ε1 +ε2 ... +ε''n'' (where the chord is written to start on 1, i.e. 1:1+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

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This error measure is called the ''least-squares error''. This error measure does not form a metric on the set of delta signatures with a fixed number of terms, since it is not symmetric.

~~=== Sphere metric error ===~~

Project the set of delta signatures with ''n'' terms (as directions of rays in the positive orthant of <math>\mathbb{R}^n</math>) onto the sphere <math>S^{n-1}</math> and measure the distance on the sphere. This results in a metric/distance function, the ''sphere metric error''{{idiosyncratic}}.

== DR triads in small edos ==

@@ Line 52: / Line 52: @@
 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.
-== Error measures ==
+== Least-squares error ==
-The idea motivating least-squares error measures on delta signatures is the following: 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 {{nowrap|''n'' &gt; 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>
+The idea motivating least-squares error on a chord as an approximation to a given delta signature is the following: 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 {{nowrap|''n'' &gt; 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>
 }} (where the delta signature is written based on the chord written to have root 1), i.e. a chord
@@ Line 60: / Line 60: @@
 We can replace the 1 with x, vary x and ask, "By at least how much do the deltas have to be off for any x?"
-=== 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 written to start on 1, i.e. 1:1+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
@@ Line 76: / Line 75: @@
 This error measure is called the ''least-squares error''. This error measure does not form a metric on the set of delta signatures with a fixed number of terms, since it is not symmetric.
-=== Sphere metric error ===
-Project the set of delta signatures with ''n'' terms (as directions of rays in the positive orthant of <math>\mathbb{R}^n</math>) onto the sphere <math>S^{n-1}</math> and measure the distance on the sphere. This results in a metric/distance function, the ''sphere metric error''{{idiosyncratic}}.
 == DR triads in small edos ==