User:FloraC: Difference between revisions

Line 37:

=== 3-limit TE tuning of ets ===

Given a val ~~"a"~~, we have Tenney-weighted val v = aW, where W is the Tenney-weighting matrix.

Given a val A, we have Tenney-weighted val V = AW, where W is the Tenney-weighting matrix.

If t is the Tenney-weighted tuning map, then for any et, for obvious reasons,

If T is the Tenney-weighted tuning map, then for any et, for obvious reasons,

[math]t_2/v_2 = t_1/v_1[/math]

Line 45:

Let ''c'' be the coefficient of TE-weighted tuning map ''c'' = ''t''2/''t''1 = ''v''2/''v''1

Let ''e'' be the [[TE error]] in Breed's RMS, and j be the [[JIP]], then

Let ''e'' be the [[TE error]] in Breed's RMS, and J be the [[JIP]], then

[math]e = ||~~\vec t~~ - ~~\vec j~~||_\text {RMS} = \sqrt {\frac {(t_1 - 1)^2 + (t_2 - 1)^2)}{2}}[/math]

[math]e = ||T - J||_\text {RMS} = \sqrt {\frac {(t_1 - 1)^2 + (t_2 - 1)^2)}{2}}[/math]

Since

Line 89:

And just proceed as before,

[math]t_1 = \frac {\sum \vec c}{\vec c \~~cdot~~ \vec c} = \frac {v_1 \sum ~~\vec v~~}{\~~vec v \cdot \vec v~~}[/math]

[math]t_1 = \frac {\sum \vec c}{\vec c^\mathsf T \vec c} = \frac {v_1 \sum V}{VV^\mathsf T}[/math]

Substitute ''t''''i''/''c''''i'' for ''t''1,

[math]

t_i = \frac {v_i \sum ~~\vec v~~}{\~~vec v \cdot \vec v~~}, i = 1, 2, \ldots, n \\

t_i = \frac {v_i \sum V}{VV^\mathsf T}, i = 1, 2, \ldots, n \\

e = \sqrt {1 - \frac {(\sum ~~\vec v~~)^2}{n \~~vec v \cdot \vec v~~}}

e = \sqrt {1 - \frac {(\sum V)^2}{n VV^\mathsf T}}

[/math]

@@ Line 37: / Line 37: @@
 === 3-limit TE tuning of ets ===
-Given a val "a", we have Tenney-weighted val v = aW, where W is the Tenney-weighting matrix.
+Given a val A, we have Tenney-weighted val V = AW, where W is the Tenney-weighting matrix.
-If t is the Tenney-weighted tuning map, then for any et, for obvious reasons,
+If T is the Tenney-weighted tuning map, then for any et, for obvious reasons,
 [math]t_2/v_2 = t_1/v_1[/math]
@@ Line 45: / Line 45: @@
 Let ''c'' be the coefficient of TE-weighted tuning map ''c'' = ''t''<sub>2</sub>/''t''<sub>1</sub> = ''v''<sub>2</sub>/''v''<sub>1</sub>
-Let ''e'' be the [[TE error]] in Breed's RMS, and j be the [[JIP]], then
+Let ''e'' be the [[TE error]] in Breed's RMS, and J be the [[JIP]], then
-[math]e = ||\vec t - \vec j||_\text {RMS} = \sqrt {\frac {(t_1 - 1)^2 + (t_2 - 1)^2)}{2}}[/math]
+[math]e = ||T - J||_\text {RMS} = \sqrt {\frac {(t_1 - 1)^2 + (t_2 - 1)^2)}{2}}[/math]
 Since
@@ Line 89: / Line 89: @@
 And just proceed as before,
-[math]t_1 = \frac {\sum \vec c}{\vec c \cdot \vec c} = \frac {v_1 \sum \vec v}{\vec v \cdot \vec v}[/math]
+[math]t_1 = \frac {\sum \vec c}{\vec c^\mathsf T \vec c} = \frac {v_1 \sum V}{VV^\mathsf T}[/math]
 Substitute ''t''<sub>''i''</sub>/''c''<sub>''i''</sub> for ''t''<sub>1</sub>,
 [math]
-t_i = \frac {v_i \sum \vec v}{\vec v \cdot \vec v}, i = 1, 2, \ldots, n \\
+t_i = \frac {v_i \sum V}{VV^\mathsf T}, i = 1, 2, \ldots, n \\
-e = \sqrt {1 - \frac {(\sum \vec v)^2}{n \vec v \cdot \vec v}}
+e = \sqrt {1 - \frac {(\sum V)^2}{n VV^\mathsf T}}
 [/math]