Constrained tuning/Analytical solution to constrained Euclidean tunings: Difference between revisions

First, it manifests itself as a form of [[tuning map]]. Its columns represent tunings of [[formal prime]]s in terms of [[monzo]]s. The tuning map in the logarithmic scale can be obtained by multiplying the projection map by the [[JIP]] on the left.  

where T is the tuning map, J the JIP, and P the projection map.  

The projection map multipled by a [[Temperament mapping matrices|temperament map]] on the left yields its [[Tmonzos and tvals|tempered monzos]]. In particular, if V is the temperament map of P, then

Second, the projection map multipled by a monzo list on the right yields the tunings of the list in terms of monzos. In particular, if M is the [[comma list]] of P, then

For any Euclidean aka ''L''² tuning without constraints, the weight-skewed projection map is

<math>\displaystyle P_{WX} = V_{WX}^+ V_{WX}</math>

where ⁺ is the [[Wikipedia:Moore–Penrose inverse|pseudoinverse]], and V_WX = VWX is the weight-skewed val list of the temperament. Removing the transformation, it is

<math>\displaystyle P = WXV_{WX}^+ V_{WX}(WX)^+ = WX(VWX)^+V</math>

Let us start with CEE tuning (constrained equilateral-Euclidean tuning): both weight and skew are identity matrices, which will be omitted below, and the constraint is the octave.  

Denote the constraint by M_C. In the case of CEE, it is the octave:  

<math>\displaystyle M_{\rm C} = [ \begin{matrix} 1 & 0 & \ldots & 0 \end{matrix} \rangle</math>

but it works as long as it is the first ''r'' elements of the [[Subgroup basis matrices|subgroup basis]].  

We will denote the projection map by P. The goal is to work out the constrained projection map P_C, which also satisfies

<math>\displaystyle  

VP_{\rm C} = V \\

P_{\rm C}M = O

<math>\displaystyle P_{\rm C} M_{\rm C} = M_{\rm C}</math>

Since P is characteristic of the temperament and is independent of the specific tuning, notice

<math>\displaystyle P = P_{\rm C}^+P_{\rm C}</math>

P_{\rm C}^+ M_{\rm C}

= P_{\rm C}^+P_{\rm C} M_{\rm C}

= P M_{\rm C}

Both P_C⁺M_C and PM_C are the same slice of the first ''r'' columns of P.

With the first ''r'' rows and columns removed, the remaining part in the mapping will be dubbed the minor matrix, denoted V_M. The minor matrix of the projection map

forms an orthogonal projection map filling the bottom-right section of P_C⁺.  

In general, if M_C is the first ''r'' elements of the subgroup basis, then P_C is of the form

V^+VM_{\rm C} & \begin{matrix} O \\ V_{\rm M}^+V_{\rm M} \end{matrix}

If there is a weight W and/or a skew X, such as CTWE tuning, the weight-skew should be applied to the map and the constraint first:  

V_{WX} &= VWX \\

(M_{\rm C})_{WX} &= (WX)^+ M_{\rm C}

Working from here, we find the weighted projection map (P_C)_WX:  

(P_{\rm C})_{WX} =  

V_{WX}^+ V_{WX} (M_{\rm C})_{WX} & \begin{matrix} O \\ (V_{WX})_{\rm M}^+ (V_{WX})_{\rm M} \end{matrix}

<math>\displaystyle P_{\rm C} = WX (P_{\rm C})_{WX} (WX)^+</math>

What if the constraint is something more complex, especially when it is not the first ''r'' elements of the subgroup basis? It turns out we can always transform the subgroup basis to encapsulate the constraint. Such a subgroup S is formed by the constraint and its orthonormal complement.  

<math>\displaystyle S = [\begin{matrix} M_{\rm C} & M_{\rm C}^\perp \end{matrix}] </math>

M_{\rm C} =  

M_{\rm C}^\perp =  

We should apply this S to the map and the constraint to convert them into the working basis:  

(M_{\rm C})_S &= S^{-1}M_{\rm C}

Similarly, if there is a weight W and/or a skew X, it should be applied to the map and the constraint first:  

V_{WX} &= VWX \\

(M_{\rm C})_{WX} &= (WX)^+ M_{\rm C}

and then the basis transformation matrix should be found out in this weight-skewed space:  

S = [\begin{matrix} M_{\rm C} & M_{\rm C}^\perp \end{matrix}]

We should apply this S to the weight-skewed map and the weight-skewed constraint to convert them into the working basis:  

V_{WXS} &= VWXS \\

(M_{\rm C})_{WXS} &= (WXS)^+ M_{\rm C}

Proceed as before. The projection map found this way will be weight-skewed and in the working basis. To reconstruct the original projection map, apply

<math>\displaystyle P_{\rm C} = WXS (P_{\rm C})_{WXS} (WXS)^+</math>

The tuning map T_C is

[[category:Math]]