Constrained tuning/Analytical solution to constrained Euclidean tunings

Preliminaries

The projection map is useful in a lot of ways. We will work extensively with the projection map in the course of solving constrained tunings.

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

[math]\displaystyle{ \displaystyle T = JP }[/math]

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

The projection map multipled by a temperament map on the left yields its tempered monzos. In particular, if A is the temperament map of P, then

[math]\displaystyle{ \displaystyle AP = A }[/math]

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 B is the comma list of P, then

[math]\displaystyle{ \displaystyle PB = O }[/math]

For any Euclidean aka L₂ tunings, the weighted projection map is

[math]\displaystyle{ \displaystyle P_W = V^+V }[/math]

where V = AW is the weighted val list of the temperament. Removing the weight, it is

[math]\displaystyle{ \displaystyle P = WV^+VW^{-1} = W(AW)^+A }[/math]

CFE tuning

Let us start with CFE tuning (constrained Frobenius tuning): there is no weight or skew, and the constraint is the octave.

Denote the constraint by B_C. In the case of CFE, it is the octave:

[math]\displaystyle{ \displaystyle B_{\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.

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{ \displaystyle AP_{\rm C} = A \\ P_{\rm C}B = O }[/math]

in addition to

[math]\displaystyle{ \displaystyle P_{\rm C} B_{\rm C} = B_{\rm C} }[/math]

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

[math]\displaystyle{ \displaystyle P = P_{\rm C}^+P_{\rm C} }[/math]

where ⁺ is the pseudoinverse. That makes the pseudoinverse of P_C easier to work with than P_C itself, as

[math]\displaystyle{ \displaystyle P_{\rm C}^+ B_{\rm C} = P_{\rm C}^+P_{\rm C} B_{\rm C} = P B_{\rm C} }[/math]

Both P_C⁺B_C and PB_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, denoted A_M. The minor of the projection map

[math]\displaystyle{ \displaystyle P_{\rm M} = A_{\rm M}^+ A_{\rm M} }[/math]

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

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

[math]\displaystyle{ \displaystyle P_{\rm C} = \begin{bmatrix} A^+AB_{\rm C} & \begin{matrix} O \\ A_{\rm M}^+A_{\rm M} \end{matrix} \end{bmatrix}^+ }[/math]

Otherwise normed tuning

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:

[math]\displaystyle{ \displaystyle \begin{align} V &= AWX \\ M_{\rm C} &= (WX)^+ B_{\rm C} \end{align} }[/math]

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

[math]\displaystyle{ \displaystyle (P_{\rm C})_{WX} = \begin{bmatrix} V^+VM_{\rm C} & \begin{matrix} O \\ V_{\rm M}^+V_{\rm M} \end{matrix} \end{bmatrix}^+ }[/math]

To reconstruct the original projection map, apply

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

Nontrivially constrained tuning

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

For example, if the temperament is in the subgroup basis of 2.3.5.7, and if the constraint is 2.5/3, then

[math]\displaystyle{ \displaystyle B_{\rm C} = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}, B_{\rm C}^\perp = \begin{bmatrix} 0 & 0 \\ 1/\sqrt{2} & 0 \\ 1/\sqrt{2} & 0 \\ 0 & 1 \end{bmatrix}, S = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 1/\sqrt{2} & 0 \\ 0 & 1 & 1/\sqrt{2} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} }[/math]

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

[math]\displaystyle{ \displaystyle \begin{align} A_S &= AS \\ (B_{\rm C})_S &= S^{-1}B_{\rm C} \end{align} }[/math]

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

[math]\displaystyle{ \displaystyle P_{\rm C} = S (P_{\rm C})_S S^{-1} }[/math]

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

[math]\displaystyle{ \displaystyle \begin{align} V &= AWX \\ M_{\rm C} &= (WX)^+ B_{\rm C} \end{align} }[/math]

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

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

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

[math]\displaystyle{ \displaystyle \begin{align} V_S &= VS \\ (M_{\rm C})_S &= S^{-1} M_{\rm C} \end{align} }[/math]

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{ \displaystyle P_{\rm C} = WXS (P_{\rm C})_{WXS} S^{-1} (WX)^+ }[/math]

Example

Let us try tuning septimal meantone to CFE.

Its mapping is

[math]\displaystyle{ \displaystyle A = \begin{bmatrix} 1 & 0 & -4 & -13 \\ 0 & 1 & 4 & 10 \end{bmatrix} }[/math]

The projection map is

[math]\displaystyle{ \displaystyle \begin{align} P &= A^+A \\ &= \frac{1}{446} \begin{bmatrix} 117 & 146 & 116 & -61 \\ 146 & 186 & 160 & -38 \\ 116 & 160 & 176 & 92 \\ -61 & -38 & 92 & 413 \end{bmatrix} \end{align} }[/math]

The minor of the mapping is

[math]\displaystyle{ \displaystyle A_{\rm M} = \begin{bmatrix} 1 & 4 & 10\end{bmatrix} }[/math]

and the minor projection map is

[math]\displaystyle{ \displaystyle \begin{align} P_{\rm M} &= A_{\rm M}^+A_{\rm M} \\ &= \frac{1}{117} \begin{bmatrix} 1 & 4 & 10 \\ 4 & 16 & 40 \\ 10 & 40 & 100 \end{bmatrix} \end{align} }[/math]

In fact,

[math]\displaystyle{ \displaystyle P_{\rm C}^+ = \begin{bmatrix} 117/446 & 0 & 0 & 0 \\ 146/446 & 1/117 & 4/117 & 10/117 \\ 116/446 & 4/117 & 16/117 & 40/117 \\ -61/446 & 10/117 & 40/117 & 100/117 \end{bmatrix} }[/math]

Hence,

[math]\displaystyle{ \displaystyle P_{\rm C} = \frac{1}{117} \begin{bmatrix} 117 & 146 & 116 & -61 \\ 0 & 1 & 4 & 10 \\ 0 & 4 & 16 & 40 \\ 0 & 10 & 40 & 100 \end{bmatrix} }[/math]