Constrained tuning
The CTE tuning (constrained Tenney-Euclidean tuning) is TE tuning under the constraints of some purely tuned intervals (i.e. eigenmonzos). While the TE tuning can be viewed as a least squares problem, the CTE tuning can be viewed as an equality-constrained least squares problem. For a rank-r temperament, specifying m eigenmonzos will yield r - m degrees of freedom to be optimized.
The most significant form of CTE tuning is pure-octave constrained. For higher-rank temperaments, it may make sense to add multiple constraints, such as the pure-{2, 3} CTE tuning.
Definition
Given a temperament mapping A and the JIP J0, denote the Tenney-weighted temperament mapping by V = AW, and the Tenney-weighted JIP by J = J0W. If the tuning is contrained by the eigenmonzo list B, the CTE tuning is equivalent to the following optimization problem:
Minimize
[math]\displaystyle{ \lVert GV - J \rVert }[/math]
subject to
[math]\displaystyle{ (GA - J_0)B = O }[/math]
where G is the generator list, and O the zero matrix.
The problem is feasible if
- rank (B) ≤ rank (A), and
- Each column in B and N (A) are linearly independent.
Computation
The tuning can be solved in the method of Lagrange multiplier. The solution is given by
[math]\displaystyle{ \begin{bmatrix} G^{\mathsf T} \\ \Lambda^{\mathsf T} \end{bmatrix} = \begin{bmatrix} VV^{\mathsf T} & AB \\ (AB)^{\mathsf T} & O \end{bmatrix}^{-1} \begin{bmatrix} VJ^{\mathsf T}\\ (J_0 B)^{\mathsf T} \end{bmatrix} }[/math]
which is almost an analytical solution. 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.
Otherwise, as a standard optimization problem, numerous algorithms exist to solve it, such as sequential quadratic programming, to name one.
The following Python code is an excerpt from Flora Canou's tuning optimizer. Note: it depends on Scipy.
# © 2020-2021 Flora Canou | Version 0.8
# This work is licensed under the GNU General Public License version 3.
import numpy as np
from scipy import optimize, linalg
np.set_printoptions (suppress = True, linewidth = 256)
PRIME_LIST = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61]
SCALAR = 1200 #could be in octave, but for precision reason
def weighted (matrix, subgroup, type = "tenney"):
if not type in {"tenney", "frobenius"}:
type = "tenney"
if type == "tenney":
weighter = np.diag (1/np.log2 (subgroup))
elif type == "frobenius":
weighter = np.eye (len (subgroup))
return matrix @ weighter
def error (gen, map, jip, order = 2):
return linalg.norm (gen @ map - jip, ord = order)
def optimizer_main (map, subgroup = [], order = 2, weighter = "tenney", cons_monzo_list = np.array ([]), stretch_monzo = np.array ([]), show = True):
if len (subgroup) == 0:
subgroup = PRIME_LIST[:map.shape[1]]
jip = np.log2 (subgroup)*SCALAR
map_w = weighted (map, subgroup, type = weighter)
jip_w = weighted (jip, subgroup, type = weighter)
if order == 2 and not cons_monzo_list.size: #te with no constraints, simply use lstsq for better performance
res = linalg.lstsq (map_w.T, jip_w)
gen = res[0]
print ("L2 tuning without constraints, solved using lstsq. ")
else:
gen0 = [SCALAR]*map.shape[0] #initial guess
cons = {'type': 'eq', 'fun': lambda gen: (gen @ map - jip) @ cons_monzo_list} if cons_monzo_list.size else ()
res = optimize.minimize (error, gen0, args = (map_w, jip_w, order), method = "SLSQP", constraints = cons)
print (res.message)
if res.success:
gen = res.x
if stretch_monzo.size:
gen *= (jip @ stretch_monzo)/(gen @ map @ stretch_monzo)
if show:
print (f"Generators: {gen} (¢)", f"Tuning map: {gen @ map} (¢)", sep = "\n")
return gen
Constraints can be added from the parameter cons_monzo_list of the optimizer_main function.
Versus POTE tuning
The pure-octave CTE tuning can be very different from POTE tuning. Take 7-limit meantone as an example, the POTE tuning map:
⟨1200.000 1896.495 2785.980 3364.949]
This is a little bit flatter than quarter-comma meantone, with all the primes tuned flat.
The pure-octave CTE tuning map:
⟨1200.000 1896.952 2787.809 3369.521]
This is a little bit sharper than quarter-comma meantone, with prime 3 tuned flat and 5 and 7 sharp.
It can be speculated that POTE tends to result in biased tunings whereas CTE less so.
Special constraint
The special eigenmonzo Wj removes the weighted tuning bias, where j is the all-ones monzo:
[math]\displaystyle{ W \vec j = [ \begin{matrix} 1 & 1/\log_2 (3) & \ldots & 1/\log_2 (p) \end{matrix} \rangle }[/math]
The monzo introduces weight to the constraint:
[math]\displaystyle{ (GV - J)\vec j = 0 }[/math]
It can be regarded as a distinct optimum, the TOCTE tuning (Tenney ones constrained Tenney-Euclidean tuning).
For equal temperaments
Since the one degree of freedom of equal temperaments is determined by the constraint, optimization is not involved. It is thus reduced to TOC tuning (Tenney ones constrained tuning). This constrained tuning demonstrates interesting properties.
The step size g can be found by
[math]\displaystyle{ g = 1/\operatorname {mean} (V) }[/math]
The edo number n can be found by
[math]\displaystyle{ n = 1/g = \operatorname {mean} (V) }[/math]
Unlike TE or TOP, the optimal edo number space in TOC is linear with respect to A. That is, if A = A1 + A2, then
[math]\displaystyle{ n = \frac {AW \vec j}{J \vec j} \\ = \frac {(A_1 + A_2) W \vec j}{J \vec j} \\ = \frac {A_1 W \vec j}{J \vec j} + \frac {A_2 W \vec j}{J \vec j} \\ = n_1 + n_2 }[/math]
As a result, the relative error space is also linear with respect to A.
For example, the relative errors of 12ettoc5 (12et in 5-limit TOC) is
[math]\displaystyle{ E_\text {r, 12} = \langle \begin{matrix} -1.55\% & -4.42\% & +10.08\% \end{matrix} ] }[/math]
That of 19ettoc5 is
[math]\displaystyle{ E_\text {r, 19} = \langle \begin{matrix} +4.08\% & -4.97\% & -2.19\% \end{matrix} ] }[/math]
As 31 = 12 + 19, the relative errors of 31ettoc5 is
[math]\displaystyle{ E_\text {r, 31} = \langle \begin{matrix} +2.52\% & -9.38\% & +7.88\% \end{matrix} ] }[/math]