User:FloraC: Difference between revisions

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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 ===

Revision as of 08:12, 27 February 2021

Name's Flora Canou (Fumica#5144).

English & Chinese Mandarin;

Mostly microtonal theory currently.

I contributed to the n-EDO Retuner plugin for MuseScore and made a fork which has key signatures reordered into fifths for my own use.

I found this chord – Canovian chord.

I found this temperament – Canou family.

Tools

TE Tuning & Temperament Measures Calculator – how I find all the TE tunings, badnesses, optimal patent vals, etc.

Writings

Well temperaments

I developed well temperaments on 12et and 17et which can be seen here. I also tried one on 19et but gave up for multiple reasons.

Q: Why I gave up developing a 19wt

A: First, unlike 12- and 17et with ambiguous major and minor thirds, 19et's thirds are close enough to 5-limit JI that interpreting them otherwise is like a force. In 12- and 17et, those intervals can represent different ratios in different keys, while in 19et they represent the same ratios better or worse in different keys, and I'm not fond of that. Second, the harmonics of 3, 5, 7, and 13 in 19-et are all flat, so there's not much room to operate. Third, the ambiguity of 4\19 and 15\19 is nice and I want them ambiguous in every key.

Quick reference

I call equal temperaments in Tenney-Euclidean tuning "ette".

3-limit TE tuning, which is my preferred tuning for most ets, is "ette3".

Some super easy formulae for such a tuning follows.

3-limit TE tuning of ets

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,

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

Let c be the coefficient of TE-weighted tuning map c = t2/t1 = v2/v1

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]

Since

[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]

has minimum at

[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

e has the same minimum point.

Now substitute t2/c for t1,

[math] t_i = \frac {v_i (v_1 + v_2)}{v_1^2 + v_2^2}, i = 1, 2 \\ e = \frac {|v_1 - v_2|}{\sqrt {2(v_1^2 + v_2^2)}} [/math]

3-limit TOP tuning of ets

This part is deduced from Paul Erlich's Middle Path.

[math] t_i = \frac {2v_i}{v_1 + v_2}, i = 1, 2 \\ e = \frac {|v_1 - v_2|}{v_1 + v_2} [/math]

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

General TE tuning of ets

This time we have a sequence c = {cn}, where

[math]c_i = v_i/v_1, i = 1, 2, \ldots, n[/math]

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]

Substitute ti/ci for t1,

[math] t_i = \frac {v_i \sum \vec v}{\vec v \cdot \vec v}, i = 1, 2, \ldots, n \\ e = \sqrt {1 - \frac {(\sum \vec v)^2}{n \vec v \cdot \vec v}} [/math]

Notes

  • For any temperament tempering out [m1 m2 … mn⟩, each prime pi is tuned to log2 (pi)(Σi = 1n mi log2 (pi))/(Σi = 1n |mi| log2 (pi)).
  • For ets, TOP tuning and TE tuning are close but not identical.
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