Tenney–Euclidean metrics

The Tenney-Euclidean metrics are metrics defined in Tenney-Euclidean space. These consist of the TE norm, which measures the complexity of an interval in just intonation, the TE temperamental norm, which measures the complexity of an interval as mapped by a temperament, and the octave-equivalent TE seminorms of both.

TE norm

The Tenney–Euclidean norm (TE norm) or Tenney–Euclidean complexity (TE complexity) applies to vals as well as to monzos.

Let us define the val weighting matrix W to be the diagonal matrix with values 1, 1/log₂3, 1/log₂5 … 1/log₂p along the diagonal. For the p-limit prime basis Q = ⟨2 3 5 … p],

$$ W = \operatorname {diag} (1/\log_2 (Q)) $$

Right-multiplying a row vector by this matrix scales each entry by the corresponding entry of the diagonal matrix.

Given a val V expressed as a row vector, the corresponding row vector in weighted coordinates is V_W = VW, with transpose V T
W  = WV^T where ^T denotes the transpose. The dot product of a weighted val with itself, or the sum of the squares of its entries, is the squared Euclidean metric of the val, ‖V_W‖ 2
2  = V_WV T
W  = VW²V^T. Thus the Euclidean metric on the val, a measure of complexity, is ‖V_W‖₂ = sqrt(V_WV T
W ) = sqrt(v 2
1  + v 2
2 /(log₂3)² + … + v 2
n /(log₂p)²), where n = π(p) is the prime-counting function which records the number of primes to p; dividing this by sqrt(n) gives the TE norm of a val.

Similarly, if m is a monzo, then in weighted coordinates the monzo becomes m_W = W⁻¹m, and the dot product is m T
W m_W = m^TW^-2m, leading to sqrt(m T
W m_W) = sqrt(m 2
1  + (log₂3)²m 2
2  + … + (log₂p)²m 2
n ); multiplying this by sqrt(n) gives the dual RMS norm on monzos which serves as a measure of complexity.

TE temperamental norm

Suppose now V is a matrix whose rows are vals defining a p-limit regular temperament. Then the corresponding weighted matrix is V_W = VW. The TE tuning projection matrix is then P_W = V +
W V_W, where ⁺ denotes the Moore–Penrose pseudoinverse. If the rows of V_W (or equivalently, V) are linearly independent, then we have V +
W  = V T
W (V_WV T
W )⁻¹. In terms of vals, the tuning projection matrix is V +
W V_W = V T
W (V_WV T
W )⁻¹V_W = WV^T(VW²V^T)⁻¹VW. P_W is a positive semidefinite matrix, so it defines a positive semidefinite bilinear form. In terms of weighted monzos (m_W)₁ and (m_W)₂, (m_W) T
1 P_W(m_W)₂ defines the semidefinite form on weighted monzos, and hence m T
1 W⁻¹P_WW⁻¹m₂ defines a semidefinite form on unweighted monzos, in terms of the matrix P = W⁻¹P_WW⁻¹ = V^T(VW²V^T)⁻¹V. From the semidefinite form we obtain an associated semidefinite quadratic form m^TPm and from this the seminorm sqrt(m^TPm).

It may be noted that (V_WV T
W )⁻¹ = (VW²V^T)⁻¹ is the inverse of the Gramian matrix used to compute TE complexity, and hence is the corresponding Gram matrix for the dual space. Hence P represents a change of basis defined by the mapping given by the vals combined with an inner product on the result. Given a monzo m, Vm represents the tempered interval corresponding to m in a basis defined by the mapping V, and P_T = (VW²V^T)⁻¹ defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by V.

Denoting the temperament-defined, or temperamental, seminorm by T(x), the subspace of interval space such that T(x) = 0 contains a lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. The quotient space of the full vector space by the commatic subspace such that T(x) = 0 is now a normed vector space with norm given by T, in which the intervals of the regular temperament define a lattice. The norm T on these lattice points is the TE temperamental norm or TE temperamental complexity of the intervals of the regular temperament; in terms of the basis defined by V, it is sqrt(t^TP_Tt) where t is the image of a monzo m by t = Vm.

Octave-equivalent TE seminorm

Instead of starting from a matrix of vals, we may start from a matrix of monzos. If M is a matrix with columns of monzos spanning the commas of a regular temperament, then M_W = W⁻¹M is the corresponding weighted matrix. Q_W = M_WM +
W  is a projection matrix dual to P_W = I − Q_W, where I is the identity matrix, and P_W is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefore linearly independent, then P_W = I − M_W(M T
W M_W)⁻¹M T
W  = I − W⁻¹M(M^TW⁻²M)⁻¹M^TW⁻¹, and m T
W P_Wm_W = m^TW⁻¹P_WW⁻¹m, or m^T(W⁻² − W⁻²M(M^TW⁻²M)⁻¹M^TW⁻²)m, so that the terms inside the parenthesis define a formula for P in terms of the matrix of monzos M.

To define the octave-equivalent Tenney–Euclidean seminorm, or OETES, we simply add a column [1 0 0 … 0⟩ representing 2 to the matrix M. An alternative procedure is to find the normal val list, and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given p-limit rational interval in terms of the p-limit regular temperament given by V.

Examples

Consider the temperament defined by the 5-limit patent vals for 15 and 22 equal. From the vals, we may construct a 2×3 matrix V = [⟨15 24 35], ⟨22 35 51]]. From this we may obtain the matrix P as V^T(VW²V^T)⁻¹V, approximately

[math]\displaystyle{ \left[\begin{matrix} 0.9911 & 0.1118 & -0.1440 \\ 0.1118 & 1.1075 & 1.8086 \\ -0.1440 & 1.8086 & 3.0624 \\ \end{matrix}\right] }[/math]

If we want to find the temperamental seminorm T(250/243) of 250/243, we convert it into a monzo as [1 -5 3⟩. Now we may multiply P by this on the left, obtaining the zero vector. Taking the dot product of the zero vector [1 -5 3⟩ gives zero, and taking the square root of zero we get zero, the temperametal seminorm T(250/243) of 250/243. All of this is telling us that 250/243 is a comma of this temperament, which is 5-limit porcupine.

Similarly, starting from the monzo [-1 1 0⟩ for 3/2, we may multiply this by P, obtaining ⟨-0.8793 0.9957 1.9526], and taking the dot product of this with [-1 1 0⟩ gives 1.875 with square root 1.3693, which is T(3/2).

We can, however, map the monzos to elements of a rank-r abelian group (where r is the rank of the temperament) which abstractly represents the elements of the temperament without regard to tuning, the abstract regular temperament. If m is a monzo, this mapping is given by Vm. Hence we have V[1 -5 3⟩ maps to [0 0⟩ for the interval associated to 250/243, and V[-1 1 0⟩ maps to [9 13⟩ for the interval assciated to 3/2. This is the number of steps needed to get to 3/2 in 15et and 22et respectively. We now may obtain a matrix defining the temperamental norm on this abstract temperament by P_T = (VW²V^T)⁻¹, which is approximately

[math]\displaystyle{ \left[\begin{matrix} 175.3265 & -120.0291 \\ -120.0291 & 82.1730 \\ \end{matrix}\right] }[/math]

Using this, we find the temperamental norm of [9 13⟩ to be sqrt([9 13]P_T[9 13]^T) ~ sqrt(1.875), ~ 1.3693, identical to the temperamental seminorm of 3/2. Note however that while P does not depend on the choice of basis vals for the temperament, P_T does; if we choose [⟨1 2 3], ⟨0 -3 -5]] for our basis instead, then 3/2 is represented by [1 -3⟩ and P_T changes coordinates to produce the same final result of temperamental complexity.

If instead we want the OETES, we may remove the first row of [⟨1 2 3], ⟨0 -3 -5]], leaving just [⟨0 -3 -5]]. If we now call this 1×3 matrix V, then P_T = (VW²V^T)⁻¹ is a 1×1 matrix; in effect a scalar, with value [⟨0.1215588]]. Multiplying a monzo m by V on the left gives a 1×1 matrix Vm whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which m belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of generator steps.

For a more substantial example we need to consider at least a rank-3 temperament, so let us turn to 7-limit marvel, the 7-limit temperament tempering out 225/224. The 2×4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [[1 0 0 0⟩, [-5 2 2 -1⟩]. If we denote log₂ of the odd primes by p3, p5, p7, etc., then the monzo weighting of this matrix is M_W = [[1 0 0 0⟩, [-5 2p3 2p5 -p7⟩], and P_W = I − M_WM +
W  = [[1 0 0 0⟩, [0 4(p5)² + (p7)² -4(p3)(p5) 2(p3)(p7)⟩/H, [0 -4(p3)(p5) 4(p3)² + (p7)² 2(p5)(p7)⟩/H, [0 2(p3)(p7) 2(p5)(p7) 4((p3)² + (p5)²)⟩/H], where H = 4(p3)² + 4(p5)² + (p7)². On the other hand, we may start from the normal val list for the temperament, which is [⟨1 0 0 -5], ⟨0 1 0 2], ⟨0 0 1 2]]. Removing the first row gives [⟨0 1 0 2], ⟨0 0 1 2]], and val weighting this gives C_W = [⟨0 1/p3 0 2/p7], ⟨0 0 1/p5 2/p7]]. Then P_W = C_W⁺C_W is precisely the same matrix we obtained before.

Octaves are now projected to the origin as well as commas. We can as before form the quotient space with respect to the seminorm, and obtain a normed space in which octave-equivalent interval classes of the intervals of the temperament are the lattice points. This seminorm applied to monzos gives the OE complexity.

If we start from a normal val list and remove the first val, the remaining vals map to the octave classes of the notes of the temperament. If we call this reduced list of vals R, then the inner product on note classes in this basis is defined by the symmetric matrix S = (RW²R^T)⁻¹. In the case of marvel, we obtain S = [[(p3)²(4(p5)² + (p7)²) -4(p3)²(p5)²], [-4(p3)²(p5)² (p5)²(4(p3)² + (p7)²)]]/H. If k = [k₁ k₂⟩ is a note class of marvel in the coordinates defined by the truncated val list R, which in this case has a basis corresponding to tempered 3 and 5, then sqrt(k^TSk) gives the OE complexity of the note class.