View source for User:Sintel/CTE tuning

The '''CTE tuning''' ('''constrained Tenney-Euclidean tuning''') is [[TE tuning]] under the constraints of some purely tuned intervals (i.e. [[eigenmonzo]]s). While the TE tuning can be viewed as a [[Wikipedia: Least squares|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'' [[Wikipedia: Degrees of freedom|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, the CTE tuning is equivalent to the following optimization problem: 

Minimize

<math>\text{minimize} \left\|   GV - J   \right\|   \\
\text{subject to} </math>

subject to

<math>(GA - J_0)B = O</math>

where G is the generator list, V = AW the Tenney-weighted temperament mapping, J = J<sub>0</sub>W the Tenney-weighted [[JIP]], and B the monzo list. 

The problem is feasible if
# rank (B) ≤ rank (A), and
# Each column in B and N (A) are [[Wikipedia:linear independence|linearly independent]].

== Computation ==
As a standard optimization problem, numerous algorithms exist to solve for this tuning, such as [[Wikipedia: Sequential quadratic programming|sequential quadratic programming]], to name one. 

The following [https://www.python.org Python] code is an excerpt from [[Flora Canou]]'s [https://github.com/FloraCanou/te_temperament_measures/blob/54c9ec58acf0ad5adc7c1d96f2deaf1d2a503702/tuning_optimizer.py tuning optimizer]. Note: it depends on [https://scipy.org/ Scipy]. 

{{Databox| Code |
<syntaxhighlight lang="python">
# © 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

</syntaxhighlight>
}}

Constraints can be added from the parameter <code>cons_monzo_list</code> of the <code>optimizer_main</code> 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]]: 

{{val| 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: 

{{val| 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.

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]
[[Category:Tuning]]
[[Category:Tuning technique]]