User:FloraC: Difference between revisions

Line 28:

== Quick reference ==

=== ~~To quickly obtain TOP~~ tuning ===

* For any ~~temperament tempering out {{monzo|~~ ''m''1 ''m''2 … ''m''''n'' }}, ~~each prime~~ ''pi'' ~~is tuned to log~~2 (''pi'')~~(Σ~~''i'' = 1''n'' ''m''''i'' ~~log~~2 (''pi''))/~~(Σ~~''i'' = 1''n'' |''m''''i''| ~~log~~2 (''pi'')) (~~in 8ves~~).

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

* For ets, 3-~~limit TOP tuning and TE tuning are identical~~ (~~needs further study in higher limits~~).

* For any et tempering out {{monzo| ''n'' ''m'' }}, ~~stretch the octave by~~ ''&~~delta~~;'' = (''m'' log2~~3 +~~ ''n'')/(|''m''| log2~~3 + |~~''n''|) ~~to obtain 3-limit~~ TOP/TE tuning ~~(which is my preferred tuning for most ets)~~.

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,

''t''2/''v''2 = ''t''1/''v''1

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

''e'' = ||t - j||RMS = sqrt (((''t''1 - 1)2 + (''t''2 - 1)2)/2)

Since (''t''1 - 1)2 + (''t''2 - 1)2

= ''t''12 - 2''t''1 + 1 + ''c''2 ''t''12 - 2''ct''1 + 1

= (''c''2 + 1)''t''12 - 2(''c'' + 1)''t''1 + 2

has minimum at ''t''1 = (''c'' + 1)/(''c''2 + 1) = ''v''1(''v''1 + ''v''2) / (''v''12 + ''v''22)

and ''f'' (''x'') = sqrt (''x''/2) is a monotonously increasing function

''e'' has the same minimum point.

Now substitute ''t''2/''c'' for ''t''1,

''t''''i'' = ''v''''i''(''v''1 + ''v''2)/(''v''12 + ''v''22), ''i'' = 1, 2

''e'' = |''v''1 - ''v''2|/sqrt (2(''v''12 + ''v''22))

=== 3-limit TOP tuning of ets ===

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

''t''''i'' = 2''v''''i''/(''v''1 + ''v''2), ''i'' = 1, 2

''e'' = |''v''1 - ''v''2|/(''v''1 + ''v''2)

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

=== General TE tuning of ets ===

This time we have a sequence c = {''c''''n''}, where ''c''''i'' = ''v''''i''/''v''1, ''i'' = 1, 2, 3, …, ''n''

And just proceed as before,

''t''1 = (Σc + 1)/(ccT + 1) = v1Σv/vvT

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

''t''''i'' = ''v''''i''Σv/vvT

''e'' = sqrt (1 - (Σv)2/(''n''vvT))

=== Notes ===

* For any temperament tempering out {{monzo| ''m''1 ''m''2 … ''m''''n'' }}, each prime ''pi'' is tuned to log2 (''pi'')(Σ''i'' = 1''n'' ''m''''i'' log2 (''pi''))/(Σ''i'' = 1''n'' |''m''''i''| log2 (''pi'')).

* For ets, TOP tuning and TE tuning are close but not identical.

------

@@ Line 28: / Line 28: @@
 == Quick reference ==
-=== To quickly obtain TOP tuning ===
-* For any temperament tempering out {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>''n''</sub> }}, each prime ''p<sub>i</sub>'' is tuned to log<sub>2</sub> (''p<sub>i</sub>'')(&Sigma;<sub>''i'' = 1</sub><sup>''n''</sup> ''m''<sub>''i''</sub> log<sub>2</sub> (''p<sub>i</sub>''))/(&Sigma;<sub>''i'' = 1</sub><sup>''n''</sup> |''m''<sub>''i''</sub>| log<sub>2</sub> (''p<sub>i</sub>'')) (in 8ves).
+I call equal temperaments in Tenney-Euclidean tuning "ette".
-* For ets, 3-limit TOP tuning and TE tuning are identical (needs further study in higher limits).
-* For any et tempering out {{monzo| ''n'' ''m'' }}, stretch the octave by ''&delta;'' = (''m'' log<sub>2</sub>3 + ''n'')/(|''m''| log<sub>2</sub>3 + |''n''|) to obtain 3-limit TOP/TE tuning (which is my preferred tuning for most ets).
+-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,
+''t''<sub>2</sub>/''v''<sub>2</sub> = ''t''<sub>1</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>
+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)
+Since (''t''<sub>1</sub> - 1)<sup>2</sup> + (''t''<sub>2</sub> - 1)<sup>2</sup>
+= ''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
+= (''c''<sup>2</sup> + 1)''t''<sub>1</sub><sup>2</sup> - 2(''c'' + 1)''t''<sub>1</sub> + 2
+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>)
+and ''f'' (''x'') = sqrt (''x''/2) is a monotonously increasing function
+''e'' has the same minimum point.
+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
+''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>))
+=== 3-limit TOP tuning of ets ===
+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
+''e'' = |''v''<sub>1</sub> - ''v''<sub>2</sub>|/(''v''<sub>1</sub> + ''v''<sub>2</sub>)
+This ''e'' is also the amount to stretch or compress each prime.
+=== 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''
+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>
+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>
+''e'' = sqrt (1 - (&Sigma;v)<sup>2</sup>/(''n''vv<sup>T</sup>))
+=== Notes ===
+* For any temperament tempering out {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>''n''</sub> }}, each prime ''p<sub>i</sub>'' is tuned to log<sub>2</sub> (''p<sub>i</sub>'')(&Sigma;<sub>''i'' = 1</sub><sup>''n''</sup> ''m''<sub>''i''</sub> log<sub>2</sub> (''p<sub>i</sub>''))/(&Sigma;<sub>''i'' = 1</sub><sup>''n''</sup> |''m''<sub>''i''</sub>| log<sub>2</sub> (''p<sub>i</sub>'')).
+* For ets, TOP tuning and TE tuning are close but not identical.
 ------