Constrained tuning: Difference between revisions
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It can be speculated that POTE tends to result in biased tunings whereas CTE less so. | It can be speculated that POTE tends to result in biased tunings whereas CTE less so. | ||
== Special constraint == | |||
The special eigenmonzo Wj removes the weighted tuning bias, where j is the all-ones monzo: | |||
<math>W \vec j = [ \begin{matrix} 1 & 1/\log_2 (3) & \ldots & 1/\log_2 (p) \end{matrix} \rangle </math> | |||
The monzo introduces weight to the constraint: | |||
<math>(GV - J)\vec j = 0</math> | |||
It can be regarded as a distinct optimum, the '''TOCTE tuning''' ('''Tenney ones constrained Tenney-Euclidean tuning'''). | |||
=== For equal temperaments === | |||
Since the one degree of freedom of equal temperaments is determined by the constraint, optimization is not involved. It is thus reduced to '''TOC tuning''' ('''Tenney ones constrained tuning'''). This constrained tuning demonstrates interesting properties. | |||
The step size ''g'' can be found by | |||
<math>g = 1/\operatorname {mean} (V)</math> | |||
The edo number ''n'' can be found by | |||
<math>n = 1/g = \operatorname {mean} (V)</math> | |||
Unlike TE or TOP, the optimal edo number space in TOC is linear with respect to A. That is, if A = A<sub>1</sub> + A<sub>2</sub>, then | |||
<math> | |||
n = \frac {AW \vec j}{J \vec j} \\ | |||
= \frac {(A_1 + A_2) W \vec j}{J \vec j} \\ | |||
= \frac {A_1 W \vec j}{J \vec j} + \frac {A_2 W \vec j}{J \vec j} \\ | |||
= n_1 + n_2 | |||
</math> | |||
As a result, the [[Relative interval error #Linearity|relative error space]] is also linear with respect to A. | |||
For example, the relative errors of 12ettoc5 (12et in 5-limit TOC) is | |||
<math>E_\text {r, 12} = \langle \begin{matrix} -1.55\% & -4.42\% & +10.08\% \end{matrix} ]</math> | |||
That of 19ettoc5 is | |||
<math>E_\text {r, 19} = \langle \begin{matrix} +4.08\% & -4.97\% & -2.19\% \end{matrix} ]</math> | |||
As 31 = 12 + 19, the relative errors of 31ettoc5 is | |||
<math>E_\text {r, 31} = \langle \begin{matrix} +2.52\% & -9.38\% & +7.88\% \end{matrix} ]</math> | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||