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FloraC

Joined 30 March 2020
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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,  


''t''<sub>2</sub>/''v''<sub>2</sub> = ''t''<sub>1</sub>/''v''<sub>1</sub>
[math]t_2/v_2 = t_1/v_1[/math]


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 ''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>
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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


''e'' = ||t - j||<sub>RMS</sub> = sqrt (((''t''<sub>1</sub> - 1)<sup>2</sup> + (''t''<sub>2</sub> - 1)<sup>2</sup>)/2)
[math]e = ||\vec t - \vec j||_\text {RMS} = \sqrt {\frac {(t_1 - 1)^2 + (t_2 - 1)^2)}{2}}[/math]


Since (''t''<sub>1</sub> - 1)<sup>2</sup> + (''t''<sub>2</sub> - 1)<sup>2</sup>
Since  


= ''t''<sub>1</sub><sup>2</sup> - 2''t''<sub>1</sub> + 1 + ''c''<sup>2</sup> ''t''<sub>1</sub><sup>2</sup> - 2''ct''<sub>1</sub> + 1
[math]
(t_1 - 1)^2 + (t_2 - 1)^2 \\
= t_1^2 - 2t_1 + 1 + c^2 t_1^2 - 2c t_1 + 1 \\
= (c^2 + 1)t_1^2 - 2(c + 1)t_1 + 2
[/math]


= (''c''<sup>2</sup> + 1)''t''<sub>1</sub><sup>2</sup> - 2(''c'' + 1)''t''<sub>1</sub> + 2
has minimum at


has minimum at ''t''<sub>1</sub> = (''c'' + 1)/(''c''<sup>2</sup> + 1) = ''v''<sub>1</sub>(''v''<sub>1</sub> + ''v''<sub>2</sub>) / (''v''<sub>1</sub><sup>2</sup> + ''v''<sub>2</sub><sup>2</sup>)
[math]t_1 = \frac{c + 1}{c^2 + 1} = \frac {v_1 (v_1 + v_2)}{v_1^2 + v_2^2}[/math]


and ''f'' (''x'') = sqrt (''x''/2) is a monotonously increasing function
and ''f'' (''x'') = sqrt (''x''/2) is a monotonously increasing function
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Now substitute ''t''<sub>2</sub>/''c'' for ''t''<sub>1</sub>,  
Now substitute ''t''<sub>2</sub>/''c'' for ''t''<sub>1</sub>,  


''t''<sub>''i''</sub> = ''v''<sub>''i''</sub>(''v''<sub>1</sub> + ''v''<sub>2</sub>)/(''v''<sub>1</sub><sup>2</sup> + ''v''<sub>2</sub><sup>2</sup>), ''i'' = 1, 2
[math]
 
t_i = \frac {v_i (v_1 + v_2)}{v_1^2 + v_2^2}, i = 1, 2 \\
''e'' = |''v''<sub>1</sub> - ''v''<sub>2</sub>|/sqrt (2(''v''<sub>1</sub><sup>2</sup> + ''v''<sub>2</sub><sup>2</sup>))
e = \frac {|v_1 - v_2|}{\sqrt {2(v_1^2 + v_2^2)}}
[/math]


=== 3-limit TOP tuning of ets ===
=== 3-limit TOP tuning of ets ===
This part is deduced from Paul Erlich's ''Middle Path''.  
This part is deduced from Paul Erlich's ''Middle Path''.  


''t''<sub>''i''</sub> = 2''v''<sub>''i''</sub>/(''v''<sub>1</sub> + ''v''<sub>2</sub>), ''i'' = 1, 2
[math]
 
t_i = \frac {2v_i}{v_1 + v_2}, i = 1, 2 \\
''e'' = |''v''<sub>1</sub> - ''v''<sub>2</sub>|/(''v''<sub>1</sub> + ''v''<sub>2</sub>)
e = \frac {|v_1 - v_2|}{v_1 + v_2}
[/math]


This ''e'' is also the amount to stretch or compress each prime.  
This ''e'' is also the amount to stretch or compress each prime.  


=== General TE tuning of ets ===
=== General TE tuning of ets ===
This time we have a sequence c = {''c''<sub>''n''</sub>}, where ''c''<sub>''i''</sub> = ''v''<sub>''i''</sub>/''v''<sub>1</sub>, ''i'' = 1, 2, 3, …, ''n''
This time we have a sequence c = {''c''<sub>''n''</sub>}, where  
 
[math]c_i = v_i/v_1, i = 1, 2, \ldots, n[/math]


And just proceed as before,  
And just proceed as before,  


''t''<sub>1</sub> = (&Sigma;c + 1)/(cc<sup>T</sup> + 1) = v<sub>1</sub>&Sigma;v/vv<sup>T</sup>
[math]t_1 = \frac {\sum \vec c}{\vec c \cdot \vec c} = \frac {v_1 \sum \vec v}{\vec v \cdot \vec v}[/math]


Substitute ''t''<sub>''i''</sub>/''c''<sub>''i''</sub> for ''t''<sub>1</sub>,  
Substitute ''t''<sub>''i''</sub>/''c''<sub>''i''</sub> for ''t''<sub>1</sub>,  


''t''<sub>''i''</sub> = ''v''<sub>''i''</sub>&Sigma;v/vv<sup>T</sup>
[math]
 
t_i = \frac {v_i \sum \vec v}{\vec v \cdot \vec v}, i = 1, 2, \ldots, n \\
''e'' = sqrt (1 - (&Sigma;v)<sup>2</sup>/(''n''vv<sup>T</sup>))
e = \sqrt {1 - \frac {(\sum \vec v)^2}{n \vec v \cdot \vec v}}
[/math]


=== Notes ===
=== Notes ===
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