Optimization: Difference between revisions
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The weight, represented by a diagonal transformation matrix, determines the importance of each formal prime. Since the tuning space and the interval space are {{w|dual (mathematics)|dual}} to each other, rating of importance in the tuning space is equivalent to rating of complexity in the interval space. The Tenney weight is the most common weight: | The weight, represented by a diagonal transformation matrix, determines the importance of each formal prime. Since the tuning space and the interval space are {{w|dual (mathematics)|dual}} to each other, rating of importance in the tuning space is equivalent to rating of complexity in the interval space. The Tenney weight is the most common weight: | ||
$$ W = \operatorname {diag} (1/\log_2 (Q)) $$ | |||
which indicates that the prime harmonic ''q'' in ''Q'' = {{val| 2 3 5 … }} has the importance of 1/log<sub>2</sub>(''q''). Its dual states that ''q'' has the complexity of log<sub>2</sub>(''q''). | which indicates that the prime harmonic ''q'' in ''Q'' = {{val| 2 3 5 … }} has the importance of 1/log<sub>2</sub>(''q''). Its dual states that ''q'' has the complexity of log<sub>2</sub>(''q''). | ||
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== General formulation == | == General formulation == | ||
In general, the temperament optimization problem (except for the destretch) can be defined as follows. Given a temperament mapping ''V'' and the [[just tuning map]] ''J'', we specify a weight–skew | In general, the temperament optimization problem (except for the destretch) can be defined as follows. Given a temperament mapping matrix ''V'' and the [[just tuning map]] ''J'', we specify a weight–skew transformation matrix ''X'', a ''q''-norm, and optionally a unit eigenmonzo list ''M''. Let ''G'' denote the generator tuning map, we want to | ||
$$ | |||
\begin{align} | |||
& \text{find} && G \\ | |||
& \text{that minimizes} && \lVert GV_X - J_X \rVert_q \\ | |||
subject to | & \text{subject to} && (GV - J)M = O | ||
\end{align} | |||
$$ | |||
where (·)<sub>''X''</sub> denotes the variable in the weight–skew transformed space, found by | where (·)<sub>''X''</sub> denotes the variable in the weight–skew transformed space, found by | ||
$$ | |||
\begin{align} | \begin{align} | ||
V_X &= VX \\ | V_X &= VX \\ | ||
J_X &= JX | J_X &= JX | ||
\end{align} | \end{align} | ||
$$ | |||
== Common tunings == | == Common tunings == | ||