Error measures for DR chords: Difference between revisions

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with root real-valued harmonic ''x''. Let <math>D_0 = 0, D_i = \sum_{k=1}^i \delta_k</math> be the delta signature {{nowrap|+δ1 +δ2 ... +δ''n''}} written cumulatively.

We want to measure the error ''without having to fix any dyad'' (as one might naively fix a dyad and measure errors in the other deltas in relation to the fixed dyad). To do this we solve a least-squares optimization problem: use a root-sum-square objective function and optimize ''x'' (and any free deltas) to minimize that function.

We want to measure the error ''without having to fix any dyad in the target chord'' (as one might naively fix a dyad and measure errors in the other deltas in relation to the fixed dyad). To do this we solve a least-squares optimization problem: use a root-sum-square objective function and optimize ''x'' (and any free deltas) to minimize that function.

== Domain and error model ==

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{| class="wikitable"

|+ style="font-size: 105%;" | ~~objective~~ function for various modes and error models

|+ style="font-size: 105%;" | Objective function for various modes and error models

|-

! Domain

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Setting the derivative to 0 gives us the closed-form solution

which can be plugged back into

to obtain the least-squares linear error.

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== External links ==

* [http://~~inthar-raven~~.github.io/delta/ Inthar's DR chord explorer (includes least-squares error calculation in both domains and multiple error models)]

* [http://turbofishcrow.github.io/delta/ Inthar's DR chord explorer (includes least-squares error calculation in both domains and multiple error models)]

[[Category:Math]]

@@ Line 11: / Line 11: @@
 with root real-valued harmonic ''x''. Let <math>D_0 = 0, D_i = \sum_{k=1}^i \delta_k</math> be the delta signature {{nowrap|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>}} written cumulatively.
-We want to measure the error ''without having to fix any dyad'' (as one might naively fix a dyad and measure errors in the other deltas in relation to the fixed dyad). To do this we solve a least-squares optimization problem: use a root-sum-square objective function and optimize ''x'' (and any free deltas) to minimize that function.
+We want to measure the error ''without having to fix any dyad in the target chord'' (as one might naively fix a dyad and measure errors in the other deltas in relation to the fixed dyad). To do this we solve a least-squares optimization problem: use a root-sum-square objective function and optimize ''x'' (and any free deltas) to minimize that function.
 == Domain and error model ==
@@ Line 28: / Line 28: @@
 {| class="wikitable"
-|+ style="font-size: 105%;" | objective function for various modes and error models
+|+ style="font-size: 105%;" | Objective function for various modes and error models
 |-
 ! Domain
@@ Line 64: / Line 64: @@
 Setting the derivative to 0 gives us the closed-form solution
-<math>x = \frac{\sum_{i=1}^n D_i }{-n + \sum_{i=1}^n r_i},</math>
+<math>x = \frac{\sum_{i=1}^n D_i^2 }{\sum_{i=1}^n D_i(r_i-1)},</math>
 which can be plugged back into
-<math>\sqrt{\sum_{1=1}^n \Bigg( 1 + \frac{D_i}{x} - r_i \Bigg)^2 }</math>
+<math>\sqrt{\sum_{i=1}^n \Bigg( 1 + \frac{D_i}{x} - r_i \Bigg)^2 }</math>
 to obtain the least-squares linear error.
@@ Line 415: / Line 415: @@
 == External links ==
-* [http://inthar-raven.github.io/delta/ Inthar's DR chord explorer (includes least-squares error calculation in both domains and multiple error models)]
+* [http://turbofishcrow.github.io/delta/ Inthar's DR chord explorer (includes least-squares error calculation in both domains and multiple error models)]
 [[Category:Math]]