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

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?

Utonal approximations of 11edt

Based off Werckmeister's Septenarius tuning, I have derived something similar to it in principle, but for 11edt.

This 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]\displaystyle{ \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. 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 lengths is:

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

By making each of the steps equivalent to 3^(1/11), we obtain this temperament. This has a very interesting structure if you consider the mapping

[ 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]

The top row describes 11edt steps, the most macro level of tuning. The second row describes the number of semitone-like accidentals (which can be described as 3^(20/11)/7) 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 (7 and 111, and 22 and 176 repectively).

Of these three tunings:

• 7edo makes the semitone and fine tuning increment 0 steps.
• 22edt makes the semitone 1 step and fine tuning increment 0 steps.
• 111edo/176edt makes the semitone 8 steps and fine tuning increment 1 step.

Subsets

7&111

7&111 maps the semitone to eight of the fine tuning increments. The 7edo-based mapping is:

[ 7 11 16 20 24 26  29 30  32  35 39 49]
[-1  0  2 -8  0 -5 -10 -8 -10 -10 -7 -8]

b22&b176

This maps the semitone 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. The 22edt-based mapping is:

[​14 22​ 32 ​39​ 48​ 51 ​57​ 59 ​63​ 69​ 77 ​97]
[-1  0  2  0  0  3 -2  0 -2 -2  1  0]

7&b22

Also note that if one removes the fine tunings, one obtains 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. The 7edo-based mapping is:

[ 7 11 16 20 24 26 29 30 32 35 39 49]
[ 0  0  0 -1  0 -1 -1 -1 -1 -1 -1 -1]

Restrictions

No-23s

If one removes 23, we get the three-patent-val-EDO-defined temperament 7&111&236:

[ 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]

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:

[ 7 11 16 20 24 26 30]
[ 0  0  0 -1  0 -1 -1]
[-1  0  2  0  0  3  0]

This is also supported by 229edo.