Tenney–Euclidean temperament measures: Difference between revisions

Line 88:

: '''Note''': that is the definition used by Graham Breed's temperament finder.

Gene Ward Smith defines the TE error as the ratio ‖''M''''W'' ∧ ''J''''W''‖/‖''M''''W''‖, derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''. From {{nowrap|‖''M''''W'' ∧ ''J''''W''‖/‖''M''''W''‖}} we can extract a coefficient {{nowrap| sqrt(''C''(''n'', ''r'' + 1)/''C''(''n'', ''r'')) {{=}} sqrt((''n'' − ''r'')/(''r'' + 1)) }}, which relates ''Ψ'' with ''E'' as follows:

Gene Ward Smith defines the TE error as the ratio {{nowrap|‖''M''''W'' ∧ ''J''''W''‖/‖''M''''W''‖}}, derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''. From {{nowrap|‖''M''''W'' ∧ ''J''''W''‖/‖''M''''W''‖}} we can extract a coefficient {{nowrap| sqrt(''C''(''n'', ''r'' + 1)/''C''(''n'', ''r'')) {{=}} sqrt((''n'' − ''r'')/(''r'' + 1)) }}, which relates ''Ψ'' with ''E'' as follows:

$$ \Psi = \sqrt{\frac{r + 1}{n - r}} E $$

Line 99:

''G'' and ''ψ'' error both have the advantage that higher-rank temperament error corresponds directly to rank-1 error, but the RMS normalization has the further advantage that in the rank-1 case, {{nowrap| ''G'' {{=}} sin ''θ'' }} octaves, where ''θ'' is the angle between ''J''''W'' and the val in question.

Sintel defines the TE error as the ratio {{nowrap|''G'' {{=}} ‖''M''''U'' ∧ ''J''''U''‖/‖''M''''U''‖}}, using ''U''-weighted norm (see the next section), and it results to the same value of Graham's definition.

== TE simple badness ==

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Line 159:

| 5.400

| 2.763

| 12.~~435×10~~−3

| 1.244×10−2

|-

| Septimal magic

| 7.195

| 2.149

| 12.~~882×10~~−3

| 1.288×10−2

|}

{| class="wikitable center-all left-1"

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Line 201:

| 2.631

| 6.441×10−3

|}

{| class="wikitable center-all left-1"

|+ style="font-size: 105%;" | Sintel's norm

|-

! Temperament

! Complexity

! Error (¢)

! Simple badness

|-

| Septimal meantone

| 17.357

| 1.382

| 1.999×10−2

|-

| Septimal magic

| 23.126

| 1.074

| 2.070×10−2

|}

@@ Line 88: / Line 88: @@
 : '''Note''': that is the definition used by Graham Breed's temperament finder.
-Gene Ward Smith defines the TE error as the ratio ‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖, derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''. From {{nowrap|‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖}} we can extract a coefficient {{nowrap| sqrt(''C''(''n'', ''r'' + 1)/''C''(''n'', ''r'')) {{=}} sqrt((''n'' − ''r'')/(''r'' + 1)) }}, which relates ''Ψ'' with ''E'' as follows:
+Gene Ward Smith defines the TE error as the ratio {{nowrap|‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖}}, derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''. From {{nowrap|‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖}} we can extract a coefficient {{nowrap| sqrt(''C''(''n'', ''r'' + 1)/''C''(''n'', ''r'')) {{=}} sqrt((''n'' − ''r'')/(''r'' + 1)) }}, which relates ''Ψ'' with ''E'' as follows:
 $$ \Psi = \sqrt{\frac{r + 1}{n - r}} E $$
@@ Line 99: / Line 99: @@
 ''G'' and ''ψ'' error both have the advantage that higher-rank temperament error corresponds directly to rank-1 error, but the RMS normalization has the further advantage that in the rank-1 case, {{nowrap| ''G'' {{=}} sin ''θ'' }} octaves, where ''θ'' is the angle between ''J''<sub>''W''</sub> and the val in question.
+Sintel defines the TE error as the ratio {{nowrap|''G'' {{=}} ‖''M''<sub>''U''</sub> ∧ ''J''<sub>''U''</sub>‖/‖''M''<sub>''U''</sub>‖}}, using ''U''-weighted norm (see the next section), and it results to the same value of Graham's definition.
 == TE simple badness ==
@@ Line 157: / Line 159: @@
 | 5.400
 | 2.763
-| 12.435×10<sup>−3</sup>
+| 1.244×10<sup>−2</sup>
 |-
 | Septimal magic
 | 7.195
 | 2.149
-| 12.882×10<sup>−3</sup>
+| 1.288×10<sup>−2</sup>
 |}
 {| class="wikitable center-all left-1"
@@ Line 199: / Line 201: @@
 | 2.631
 | 6.441×10<sup>−3</sup>
+|}
+{| class="wikitable center-all left-1"
+|+ style="font-size: 105%;" | Sintel's norm
+|-
+! Temperament
+! Complexity
+! Error (¢)
+! Simple badness
+|-
+| Septimal meantone
+| 17.357
+| 1.382
+| 1.999×10<sup>−2</sup>
+|-
+| Septimal magic
+| 23.126
+| 1.074
+| 2.070×10<sup>−2</sup>
 |}