2^67-1: Birth (Massive WIP)

2026-09-02T11:11:33Z

Birth (Massive WIP)

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This page details the derivation of some temperaments I have created.

= Colian Ultimate (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?

== Utonal approximations of [[11edt]] ==

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:

<math>
\left|\frac{X}{3^{\frac{n}{11}}}-\operatorname{round}\left(\frac{X}{3^{\frac{n}{11}}},0\right)\right|
</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.

== Mashing the steps ==

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

<pre>
1
231/209
231/189
231/171
231/155
231/140
231/127
231/115
231/104
231/94
231/85
231/77
</pre>

By making each of the steps equivalent to 3^(1/11), we obtain [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.17.19.23.31.47.127&reduce=on&weights=weil&target=&edos=&commas=3971%2F3969%0D%0A1083%2F1085%0D%0A4805%2F4788%0D%0A3920%2F3937%0D%0A16129%2F16100%0D%0A13225%2F13208%0D%0A5408%2F5405%0D%0A2209%2F2210%0D%0A7225%2F7238%0D%0A1617%2F1615&submit_comma=submit this temperament]. This has a very interesting structure if you consider the mapping

<pre>
[ 7 11 16 20 24 26 29 30 32 35 39 49]
[ 0 0 0 -1 0 -1 -1 -1 -1 -1 -1 -1]
[-1 0 2 0 0 3 -2 0 -2 -2 1 0]
</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.

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2^67-1: Birth (Massive WIP)