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.

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

== Error measures ==

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 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''

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

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It satisfies SLSE({D_i}, {E_i}) = SLSE({E_i}, {D_i}) and is an upper bound for the NLSE. However, it is invariant under scaling neither of the arguments.

=== Round metric ===

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 the ''round metric''{{idiosyncratic}}.

=== Open problems ===

Find a version of the symmetric least-squares error that forms a metric on the set of delta signatures with n terms. This should ideally itself be a solution to a least-squares optimization problem.

~~== Other error measures ==~~

~~=== Round metric ===~~

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 the ''round metric''{{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.
-== Least-squares error measures ==
+== 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 {{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 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>
 }} (where the delta signature is written based on the chord written to have root 1), i.e. a chord
@@ Line 100: / Line 100: @@
 It satisfies SLSE({D_i}, {E_i}) = SLSE({E_i}, {D_i}) and is an upper bound for the NLSE. However, it is invariant under scaling neither of the arguments.
+=== Round metric ===
+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 the ''round metric''{{idiosyncratic}}.
 === Open problems ===
 Find a version of the symmetric least-squares error that forms a metric on the set of delta signatures with n terms. This should ideally itself be a solution to a least-squares optimization problem.
-== Other error measures ==
-=== Round metric ===
-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 the ''round metric''{{idiosyncratic}}.
 == DR triads in small edos ==