Delta-rational chord: Difference between revisions

Line 53:

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.

=== ~~Linear~~ error of an approximate DR chord ===

=== Measuring the error of an approximate fully DR chord ===

==== ~~Fully DR~~ ====

==== Based on the least-squares problem on delta error ====

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

Line 63:

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>

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

yields

Line 73:

which can be plugged back into the error formula to obtain the error. (We multiply the 1:''r''1:''r''2:...:''r''''n'' chord by α in order to compare it to the target DR chord on the same isodifferential series.)

This error measure is called '''least-squares delta error'''{{idiosyncratic}} (LSD error). LSD error does not depend on whether the chord whose error is being measured is 1:''r''1:''r''2:...:''r''''n'' or the same chord linearly shifted to have root α. Unfortunately, LSD error does not form a metric on the set of delta signatures with a fixed number of terms. ~~There is ongoing research seeking~~ a ~~natural and psychoacoustically useful~~ metric space structure on delta signatures.

This error measure is called '''least-squares delta error'''{{idiosyncratic}} (LSD error). LSD error does not depend on whether the chord whose error is being measured is 1:''r''1:''r''2:...:''r''''n'' or the same chord linearly shifted to have root α. Unfortunately, LSD error does not form a metric on the set of delta signatures with a fixed number of terms.

==== Naive <math>L^p</math> metrics on delta signatures ====

To impose a metric space structure on delta signatures of length <math>k</math> we can first normalize their respective cumulative forms and then compare the signatures by using the <math>L^p</math> metric.

Given normalized delta signatures <math>1 = \delta_0, \delta_1, ..., \delta_n</math> and <math>1 = \epsilon_0, \epsilon_1,..., \epsilon_n,</math> we rewrite them as <math>\mathbf{D} = (D_1, ..., D_n), \ 1 < D_1 < D_2 < \cdots < D_n</math> and <math>\mathbf{E} = (E_1, ..., E_n), \ 1 < E_1 < E_2 < \cdots < E_n</math> where <math>D_k = \sum_{i=0}^k \delta_i</math> and <math>E_k = \sum_{i=0}^k \epsilon_i.</math> Then we take the ''p''-norm of the differences:

<math>\displaystyle{\left\| \mathbf{D} - \mathbf{E} \right\|_p = \Bigg( \sum_{i=1}^n (D_i - E_i)^{p} \Bigg)^{1/p}.}</math>

<!--

==== Partially DR ====

When the DR signature has one or more free ("+?") terms, the optimization problem becomes a multivariate one: we have one variable <math>\alpha_i, i \ge 2</math> for each free term, as well as the variable <math>\alpha_1</math> for the root. However, solving it is not much more difficult than the univariate case.

Line 90:

Line 96:

\alpha_1 + \cdots + \alpha_m + \sum_{i=1}^{m-1} \sum_{l_i=1}^{n_i} \delta_{i,l_i} + \delta_{m,1}, ..., \alpha_1 + \cdots + \alpha_m + \sum_{i=1}^m \sum_{l_i=1}^{n_i} \delta_{i,l_i}].

</math>

-->

== DR and RTT ==

@@ Line 53: / Line 53: @@
 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.
-=== Linear error of an approximate DR chord ===
+=== Measuring the error of an approximate fully DR chord ===
-==== Fully DR ====
+==== Based on the least-squares problem on delta error ====
 Say we want the error of a chord 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> (in increasing order) 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
@@ Line 63: / Line 63: @@
 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>
+<math> \displaystyle{ \min_\alpha  \sqrt{\sum_{i=1}^n \Bigg( \alpha r_i - \alpha - \sum_{l=1}^i \delta_l \Bigg)^2 } }</math>
 yields
@@ Line 73: / Line 73: @@
 which can be plugged back into the error formula to obtain the error. (We multiply the 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> chord by α in order to compare it to the target DR chord on the same isodifferential series.)
-This error measure is called '''least-squares delta error'''{{idiosyncratic}} (LSD error). LSD error does not depend on whether the chord whose error is being measured is 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> or the same chord linearly shifted to have root α. Unfortunately, LSD error does not form a metric on the set of delta signatures with a fixed number of terms. There is ongoing research seeking a natural and psychoacoustically useful metric space structure on delta signatures.
+This error measure is called '''least-squares delta error'''{{idiosyncratic}} (LSD error). LSD error does not depend on whether the chord whose error is being measured is 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> or the same chord linearly shifted to have root α. Unfortunately, LSD error does not form a metric on the set of delta signatures with a fixed number of terms.
+==== Naive <math>L^p</math> metrics on delta signatures ====
+To impose a metric space structure on delta signatures of length <math>k</math> we can first normalize their respective cumulative forms and then compare the signatures by using the <math>L^p</math> metric.
+Given normalized delta signatures <math>1 = \delta_0, \delta_1, ..., \delta_n</math> and <math>1 = \epsilon_0, \epsilon_1,..., \epsilon_n,</math> we rewrite them as <math>\mathbf{D} = (D_1, ..., D_n), \ 1 < D_1 < D_2 < \cdots < D_n</math> and <math>\mathbf{E} = (E_1, ..., E_n), \ 1 < E_1 < E_2 < \cdots < E_n</math> where <math>D_k = \sum_{i=0}^k \delta_i</math> and <math>E_k = \sum_{i=0}^k \epsilon_i.</math> Then we take the ''p''-norm of the differences:
+<math>\displaystyle{\left\| \mathbf{D} - \mathbf{E} \right\|_p = \Bigg( \sum_{i=1}^n (D_i - E_i)^{p} \Bigg)^{1/p}.}</math>
+<!--
 ==== Partially DR ====
 When the DR signature has one or more free ("+?") terms, the optimization problem becomes a multivariate one: we have one variable <math>\alpha_i, i \ge 2</math> for each free term, as well as the variable <math>\alpha_1</math> for the root. However, solving it is not much more difficult than the univariate case.
@@ Line 90: / Line 96: @@
 \alpha_1 + \cdots + \alpha_m +  \sum_{i=1}^{m-1} \sum_{l_i=1}^{n_i} \delta_{i,l_i} + \delta_{m,1}, ..., \alpha_1 + \cdots + \alpha_m + \sum_{i=1}^m \sum_{l_i=1}^{n_i} \delta_{i,l_i}].
 </math>
+-->
 == DR and RTT ==