User:2^67-1/Derivation of some temperaments: Difference between revisions

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</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.

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 ==

The utonal division approximating 11edt which divides the string into 231 ~~steps~~ is:

The utonal division approximating 11edt which divides the string into 231 lengths is:

<pre>

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</pre>

The top row describes 11EDT steps, the most macro level of tuning. The second row describes the number of quartertones to reach the harmonics. As one can see, the nonzero values are all -1. The third row describes the fine-tuning increments needed.

The top row describes 11EDT steps, the most macro level of tuning. The second row describes the number of quartertones to reach the harmonics. As one can see, the nonzero values are all -1. The third row describes the fine-tuning increments needed. This temperament is also an extension of undetrita.

Note that there are less than three EDOs or EDTs which support this temperament ([[7edo|7]] and [[111edo|111]], and [[22edt|22]] and [[176edt|176]] repectively). 7&111 is very complex, so only b22&b176 will be considered. This maps the quartertone to exactly half of an [[11edt]] step, or one [[22edt]] step. This allows the 3.7.11.19.127 subgroup to be approximated solely using [[22edt]].

Also note that if one removes the fine tunings below, one obtains [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.17.19.23.31.47.127&reduce=on&weights=weil&target=&edos=7+14%5B-5%2C-13%2C-47%2C-127%5D&submit_edo=submit&commas= this temperament], which in comparison is an exotemperament: however, if one is fine with [[11edt]]'s approximations of 2, 3, 5, and 11 this temperament should be usable.

@@ Line 15: / Line 15: @@
 </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.
+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 ==
-The utonal division approximating 11edt which divides the string into 231 steps is:
+The utonal division approximating 11edt which divides the string into 231 lengths is:
 <pre>
@@ Line 44: / Line 44: @@
 </pre>
-The top row describes 11EDT steps, the most macro level of tuning. The second row describes the number of quartertones to reach the harmonics. As one can see, the nonzero values are all -1. The third row describes the fine-tuning increments needed.
+The top row describes 11EDT steps, the most macro level of tuning. The second row describes the number of quartertones to reach the harmonics. As one can see, the nonzero values are all -1. The third row describes the fine-tuning increments needed. This temperament is also an extension of undetrita.
+Note that there are less than three EDOs or EDTs which support this temperament ([[7edo|7]] and [[111edo|111]], and [[22edt|22]] and [[176edt|176]] repectively). 7&111 is very complex, so only b22&b176 will be considered. This maps the quartertone to exactly half of an [[11edt]] step, or one [[22edt]] step. This allows the 3.7.11.19.127 subgroup to be approximated solely using [[22edt]].
+Also note that if one removes the fine tunings below, one obtains [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.17.19.23.31.47.127&reduce=on&weights=weil&target=&edos=7+14%5B-5%2C-13%2C-47%2C-127%5D&submit_edo=submit&commas= this temperament], which in comparison is an exotemperament: however, if one is fine with [[11edt]]'s approximations of 2, 3, 5, and 11 this temperament should be usable.