Constrained tuning/Analytical solution to constrained Euclidean tunings

< Constrained tuning

This article gives an analytical form of Euclidean-normed constrained 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 tempered tuning map in the logarithmic scale can be obtained by multiplying the projection map by the just tuning map on the left.

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

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

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

[math]\displaystyle{ \displaystyle VP = V }[/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 M is the comma list of P, then

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

The Frobenius projection map can be used to characterize a temperament. If V is the temperament mapping matrix, then its Frobenius projection map P_F is

[math]\displaystyle{ \displaystyle P_{\rm F} = V^+ V }[/math]

We can generalize that to any other Euclidean a.k.a. L² tuning without constraints, so that the weight–skew transformed projection map P_X is

[math]\displaystyle{ \displaystyle P_X = V_X^+ V_X }[/math]

where ⁺ is the pseudoinverse, and V_X = VX is the weight–skew transformed val list of the temperament. Removing the transformation, it is

[math]\displaystyle{ \displaystyle P = XV_X^+ V_X X^+ = X(VX)^+V }[/math]

CEE tuning

Let us start with CEE tuning (constrained equilateral-Euclidean tuning): the weight–skew transformation is represented by an identity matrix, which will be omitted below, and the constraint is the octave.

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

[math]\displaystyle{ \displaystyle M_I = [ \begin{matrix} 1 & 0 & \ldots & 0 \end{matrix} \rangle }[/math]

The following observations work as long as the constraint is the first r elements of the subgroup basis.

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

[math]\displaystyle{ \displaystyle \begin{align} VP_{\rm C} &= V \\ P_{\rm C}M &= O \end{align} }[/math]

in addition to

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

For an arbitrary projection map P of the same temperament, notice

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

so if we substitute P_C for P, we have

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

and

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

Since PM_I is the tuning of M_I in terms of monzos, which is just the slice of the first r columns of P in this case, it follows that P +
C  and P_F share the first r columns.

With the first r rows and columns removed, the remaining part in the mapping is another invariant of the temperament, which will be dubbed the minor matrix, denoted V_M. We observe that the minor matrix of the projection map

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

forms an orthogonal projection map filling the bottom-right section of P_C⁺, and the top-right section comprises only zeros.

Therefore, in general, if M_I is the first r elements of the subgroup basis, then P_C is of the form

[math]\displaystyle{ \displaystyle P_{\rm C} = \begin{bmatrix} V^+VM_I & \begin{matrix} O \\ V_{\rm M}^+V_{\rm M} \end{matrix} \end{bmatrix}^+ }[/math]

Otherwise normed tuning

If there is a weight–skew transformation X, such as CTWE tuning, the transformation should be applied to the map and the constraint first:

[math]\displaystyle{ \displaystyle \begin{align} V_X &= VX \\ (M_I)_X &= X^+ M_I \end{align} }[/math]

Working from here, we find the weight–skew transformed projection map (P_C)_X:

[math]\displaystyle{ \displaystyle (P_{\rm C})_X = \begin{bmatrix} V_X^+ V_X (M_I)_X & \begin{matrix} O \\ (V_X)_{\rm M}^+ (V_X)_{\rm M} \end{matrix} \end{bmatrix}^+ }[/math]

To reconstruct the original projection map, apply

[math]\displaystyle{ \displaystyle P_{\rm C} = X (P_{\rm C})_X X^+ }[/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 basis matrix S is formed by the constraint and its orthonormal complement.

[math]\displaystyle{ \displaystyle S = [\begin{matrix} M_I & M_I^\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 M_I = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}, M_I^\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} V_S &= VS \\ (M_I)_S &= S^{-1}M_I \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–skew transformation X, it should be applied to the map and the constraint first:

[math]\displaystyle{ \displaystyle \begin{align} V_X &= VX \\ (M_I)_X &= X^+ M_I \end{align} }[/math]

and then the basis transformation matrix should be found out in this weight–skew transformed space:

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

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

[math]\displaystyle{ \displaystyle \begin{align} V_{XS} &= VXS \\ (M_I)_{XS} &= (XS)^+ M_I \end{align} }[/math]

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

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

Example

Let us try tuning septimal meantone to CEE.

Its mapping is

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

The projection map is

[math]\displaystyle{ \displaystyle \begin{align} P &= V^+V \\ &= \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 matrix of the mapping is

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

and the minor matrix of the projection map is

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

The pseudoinverse of the CEE projection map can be composed as

[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]

The tuning map T_C is

[math]\displaystyle{ \displaystyle \begin{align} T_{\rm C} &= JP_{\rm C} \\ &= \langle \begin{matrix} 1200 & 1896.8843 & 2787.5374 & 3368.8435 \end{matrix} ] \end{align} }[/math]