User:2^67-1/Derivation of some temperaments: Difference between revisions
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This page details the derivation of some temperaments I have created. | This page details the derivation of some temperaments I have created. | ||
= | = Grand Undetrita (rank-3, 7&b22&b176, 2.3.5.7.11.13.17.19.23.31.47.127) = | ||
Premise: [[11edt]] is a superb exotemperament in the 2.3.5.11 subgroup. What if we expanded that? | Premise: [[11edt]] is a superb exotemperament in the 2.3.5.11 subgroup. What if we expanded that? | ||
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== Utonal approximations of [[11edt]] == | == Utonal approximations of [[11edt]] == | ||
Based off Werckmeister's [[Septenarius]] tuning, I have derived something similar to it in principle, but for | Based off Werckmeister's [[Septenarius]] tuning, I have derived something similar to it in principle, but for [[11edt]]. | ||
This [https://www.desmos.com/calculator/eoix0qucng Desmos graph] shows the maximum error of the utonal divisions. For every X number of divisions of the string, the "relative error" of the nth step of [[11edt]] is: | This [https://www.desmos.com/calculator/eoix0qucng Desmos graph] shows the maximum error of the utonal divisions. For every X number of divisions of the string, the "relative error" of the nth step of [[11edt]] is: | ||
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</math> | </math> | ||
Therefore the error metric used in the graph is taken to be the maximum of this function ranging from n = 0 to n = 11. As it turns out, if one divides the string into 231 equal lengths, one has the smallest integer approximation of | Therefore the error metric used in the graph is taken to be the maximum of this function ranging from n = 0 to n = 11. As it turns out, if one divides the string into 231 equal lengths, one has the smallest integer approximation of [[11edt]]'s 11 steps with less than 25 percent relative error on all of them. | ||
== Mashing the steps == | == Mashing the steps == | ||
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[ 0 0 0 -1 0 -1 -1 -1 -1 -1 -1 -1] | [ 0 0 0 -1 0 -1 -1 -1 -1 -1 -1 -1] | ||
</pre> | </pre> | ||
== Restrictions == | |||
=== No-23s === | |||
If one removes 23, we get the three-patent-val-EDO-defined temperament 7&111&236: | |||
<pre> | |||
[ 7 11 16 20 24 26 29 30 35 39 49] | |||
[ 0 0 0 -1 0 -1 -1 -1 -1 -1 -1] | |||
[-1 0 2 0 0 3 -2 0 -2 1 0] | |||
</pre> | |||
In this tuning, [[236edo]] makes the semitone 17 steps and the fine tuning increment 2 steps. Note that the semitone is ''still'' 1/2 the [[11edt]] step. | |||
=== 2.3.5.7.11.13.19 === | |||
Since 17, 23, 31, 47, and 127 are close to 16 (or 18), 24, 32, 48, and 128, it is more difficult to use them in harmony. Restricting it further to 2.3.5.7.11.13.19 gives: | |||
<pre> | |||
[ 7 11 16 20 24 26 30] | |||
[ 0 0 0 -1 0 -1 -1] | |||
[-1 0 2 0 0 3 0] | |||
</pre> | |||
This is also supported by [[229edo]]. | |||
= Polyraider (rank-2, 7&1171, 2.3.5.11.f2.f3) = | |||
''Note: f2, f3 refer to the first 2 modes of a bar, or approximately 2.75653 and 5.40281. '' | |||
Premise: What is the next tuning that does what 11EDT does but better? Can it be possible to join it with 11EDT? | |||
== First part, solved == | |||
The next tuning that approximates the first few harmonics of a string and a bar better than 11EDT up to relative error is [[1171edo]]. It is notable for having a very accurate 2.3.5.11 subgroup. | |||
== Mapping == | |||
The mapping which I prefer to use (not Hermite normal form) is: | |||
<pre> | |||
[ 7 11 16 24 10 17] | |||
[-12 -3 15 -5 23 -23] | |||
</pre> | |||
This is because the mapping [https://sintel.pythonanywhere.com/result?subgroup=30.2.3.11.275653%2F100000.540281%2F100000&reduce=on&weights=weil&target=&edos=34+5746&submit_edo=submit&commas= Sintel's temperament finder] gives is very complex, with generator ranges reaching 297. | |||
Note that [[34ed30]] or [[3ed27/20]] are possible "spines" for this temperament. | |||