Constrained tuning: Difference between revisions

'''Constrained tunings''' are tuning [[optimization]] techniques using the constraint of some purely tuned intervals (i.e. [[eigenmonzo|unit eigenmonzos, or unchanged-intervals]]).

It can also be solved analytically using the method of {{w|Lagrange multipliers}}. The solution is given by:

\begin{bmatrix}

G^{\mathsf T}  \\

(VM)^{\mathsf T} & O

\end{bmatrix}^{-1}

\begin{bmatrix}

V_X J_X^{\mathsf T}\\

(JM)^{\mathsf T}

\end{bmatrix}

Notice we introduced the vector of lagrange multipliers ''Λ'', with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be discarded.

=== Simple fast closed-form algorithm ===

Another way to compute the CTE and CWE tunings, and the CTWE tuning in general, is to use the pseudoinverse.

Take 7-limit meantone as an example. The POTE [[tuning map]] is a little bit flatter than [[quarter-comma meantone]], with all the primes tuned flat:  

The CWE tuning map is a little bit sharper than quarter-comma meantone, with 5 tuned sharp and 3 and 7 flat:  

The CTE tuning map is even sharper, with 3 tuned flat and 5 and 7 sharp:  

==== Blackwood ====

Note that the POTE tuning still has prime 5 tuned sharp, even though it could be tuned pure:

The CWE gives similar results, although tunes it a few cents flatter:

The CTE tuning, on the other hand, tunes prime 5 pure:

Since prime 5 is not involved in the comma to begin with, it is understandable that it is tuned pure as in 5-limit JI. This, as mentioned above, leads to very lopsided behavior for compact chords like 1–5/4–3/2. Note that the tunings for KE and POTE distribute the error between 5/4 and 6/5 relatively evenly; both are very close to the delta-rational 0–397–720. The CTE tuning, on the other hand, has that chord tuned to 0–386–720, so that all of the error is on the 6/5 at about 18 cents sharp.

The step size ''g'' can be found by

The edo number ''n'' can be found by

Unlike TE or TOP, the optimal edo number space in TOC is linear with respect to ''V''. That is, if {{nowrap|''V'' {{=}} ''αV''₁ + ''βV''₂}}, then

\begin{align}

n &= \operatorname {mean} (VX) \\

&= \alpha n_1 + \beta n_2

\end{align}

As a result, the [[Relative interval error #Linearity|relative error space]] is also linear with respect to ''V''.  

For example, the relative errors of 12ettoc5 (12et in 5-limit TOC) is

That of 19ettoc5 is

As 31 = 12 + 19, the relative errors of 31ettoc5 is

\begin{align}

\mathcal{E}_\text {r}(31) &= \mathcal{E}_\text {r}(12) + \mathcal{E}_\text {r}(19) \\

&= \val{+2.52\% & -9.38\% & +7.88\% }

\end{align}

== Systematic name ==