Constrained tuning: Difference between revisions

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All constrained tunings are standard temperament optimization problems. Specifically, as [[TE tuning]] can be viewed as a [[Wikipedia: Least squares|least squares problem]], CTE tuning can be viewed as an equality-constrained least squares problem.

The most common subject of constraint is the octave, which is assumed unless specified otherwise. For higher-rank temperaments, it may make sense to add multiple constraints, such as a pure-2.3 constrained tuning. For a rank-''r'' temperament, specifying ''m'' ~~eigenmonzos~~ will yield ''r'' - ''m'' [[Wikipedia: Degrees of freedom|degrees of freedom]] to be optimized.

The most common subject of constraint is the octave, which is assumed unless specified otherwise. For higher-rank temperaments, it may make sense to add multiple constraints, such as a pure-2.3 (pure-octave pure-fifth) constrained tuning. For a rank-''r'' temperament, specifying a rank-''m'' constraint list will yield ''r'' - ''m'' [[Wikipedia: Degrees of freedom|degrees of freedom]] to be optimized.

== Definition ==

Given a temperament [[mapping]] V and the [[~~JIP~~]] J, we specify a weight and a skew, represented by transformation matrices W and X, respectively, and a ''p''-norm. Suppose the tuning is constrained by the eigenmonzo list MC. The goal is to find the generator list G by

Given a temperament [[mapping]] V and the [[just tuning map]] J, we specify a weight and a skew, represented by transformation matrices W and X, respectively, and a ''p''-norm. Suppose the tuning is constrained by the eigenmonzo list MC. The goal is to find the generator list G by

Minimize

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The problem is feasible if

# rank (MC) ≤ rank (V), and

# The ~~subspaces~~ of MC and N (V) are [[Wikipedia:linear independence|linearly independent]].

# The subgroups of MC and N (V) are [[Wikipedia:linear independence|linearly independent]].

== Computation ==

As a standard optimization problem, numerous algorithms exist to solve it, such as [[Wikipedia: Sequential quadratic programming|sequential quadratic programming]], to name one. [[Flora Canou]]'s [https://github.com/FloraCanou/temperament_evaluator/blob/32a2198f1ebd9038fc64e02b10b547e011d4eaa4/te_optimizer_legacy.py tuning optimizer] is such an implementation in [https://www.python.org Python]. Note: it ~~depends on~~ [https://scipy.org/ Scipy].

As a standard optimization problem, numerous algorithms exist to solve it, such as [[Wikipedia: Sequential quadratic programming|sequential quadratic programming]], to name one. [[Flora Canou]]'s [https://github.com/FloraCanou/temperament_evaluator/blob/32a2198f1ebd9038fc64e02b10b547e011d4eaa4/te_optimizer_legacy.py tuning optimizer] is such an implementation in [https://www.python.org Python]. Note: it uses [https://scipy.org/ Scipy].

{{Databox| Code |

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<math>\langle \begin{matrix} 1200.000 & 1896.656 & 2786.625 & 3366.562 \end{matrix} ]</math>

The CTE tuning map is even sharper, with 3 tuned flat and 5 and 7 sharp.

The CTE tuning map is even sharper, with 3 tuned flat and 5 and 7 sharp:

<math>\langle \begin{matrix} 1200.000 & 1896.952 & 2787.809 & 3369.521 \end{matrix} ]</math>

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== Systematic name ==

In D&D's guide to RTT, the [[Dave Keenan & Douglas Blumeyer's guide to RTT: alternative complexities #Naming|systematic name]] for the CTE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT: all-interval tuning schemes #Held-octave minimax-.28E.29S|held-octave minimax-ES]]'', and the systematic name for the CTWE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT: tuning fundamentals #Held-intervals|held-octave]] [[~~Dave_Keenan~~ %26 Douglas Blumeyer%27s guide to RTT: alternative complexities #Tunings used in 7|minimax-E-lils-S]]''.

In D&D's guide to RTT, the [[Dave Keenan & Douglas Blumeyer's guide to RTT: alternative complexities #Naming|systematic name]] for the CTE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT: all-interval tuning schemes #Held-octave minimax-.28E.29S|held-octave minimax-ES]]'', and the systematic name for the CTWE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT: tuning fundamentals #Held-intervals|held-octave]] [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT: alternative complexities #Tunings used in 7|minimax-E-lils-S]]''.

[[Category:Terms]]

[[Category:Math]]

[[Category:Regular temperament tuning]]

@@ Line 3: / Line 3: @@
 All constrained tunings are standard temperament optimization problems. Specifically, as [[TE tuning]] can be viewed as a [[Wikipedia: Least squares|least squares problem]], CTE tuning can be viewed as an equality-constrained least squares problem.
-The most common subject of constraint is the octave, which is assumed unless specified otherwise. For higher-rank temperaments, it may make sense to add multiple constraints, such as a pure-2.3 constrained tuning. For a rank-''r'' temperament, specifying ''m'' eigenmonzos will yield ''r'' - ''m'' [[Wikipedia: Degrees of freedom|degrees of freedom]] to be optimized.
+The most common subject of constraint is the octave, which is assumed unless specified otherwise. For higher-rank temperaments, it may make sense to add multiple constraints, such as a pure-2.3 (pure-octave pure-fifth) constrained tuning. For a rank-''r'' temperament, specifying a rank-''m'' constraint list will yield ''r'' - ''m'' [[Wikipedia: Degrees of freedom|degrees of freedom]] to be optimized.
 == Definition ==
-Given a temperament [[mapping]] V and the [[JIP]] J, we specify a weight and a skew, represented by transformation matrices W and X, respectively, and a ''p''-norm. Suppose the tuning is constrained by the eigenmonzo list M<sub>C</sub>. The goal is to find the generator list G by
+Given a temperament [[mapping]] V and the [[just tuning map]] J, we specify a weight and a skew, represented by transformation matrices W and X, respectively, and a ''p''-norm. Suppose the tuning is constrained by the eigenmonzo list M<sub>C</sub>. The goal is to find the generator list G by
 Minimize
@@ Line 27: / Line 27: @@
 The problem is feasible if
 # rank (M<sub>C</sub>) ≤ rank (V), and
-# The subspaces of M<sub>C</sub> and N (V) are [[Wikipedia:linear independence|linearly independent]].
+# The subgroups of M<sub>C</sub> and N (V) are [[Wikipedia:linear independence|linearly independent]].
 == Computation ==
-As a standard optimization problem, numerous algorithms exist to solve it, such as [[Wikipedia: Sequential quadratic programming|sequential quadratic programming]], to name one. [[Flora Canou]]'s [https://github.com/FloraCanou/temperament_evaluator/blob/32a2198f1ebd9038fc64e02b10b547e011d4eaa4/te_optimizer_legacy.py tuning optimizer] is such an implementation in [https://www.python.org Python]. Note: it depends on [https://scipy.org/ Scipy].
+As a standard optimization problem, numerous algorithms exist to solve it, such as [[Wikipedia: Sequential quadratic programming|sequential quadratic programming]], to name one. [[Flora Canou]]'s [https://github.com/FloraCanou/temperament_evaluator/blob/32a2198f1ebd9038fc64e02b10b547e011d4eaa4/te_optimizer_legacy.py tuning optimizer] is such an implementation in [https://www.python.org Python]. Note: it uses [https://scipy.org/ Scipy].
 {{Databox| Code |
@@ Line 196: / Line 196: @@
 <math>\langle \begin{matrix} 1200.000 & 1896.656 & 2786.625 & 3366.562 \end{matrix} ]</math>
-The CTE tuning map is even sharper, with 3 tuned flat and 5 and 7 sharp.
+The CTE tuning map is even sharper, with 3 tuned flat and 5 and 7 sharp:
 <math>\langle \begin{matrix} 1200.000 & 1896.952 & 2787.809 & 3369.521 \end{matrix} ]</math>
@@ Line 247: / Line 247: @@
 == Systematic name ==
-In D&D's guide to RTT, the [[Dave Keenan & Douglas Blumeyer's guide to RTT: alternative complexities #Naming|systematic name]] for the CTE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT: all-interval tuning schemes #Held-octave minimax-.28E.29S|held-octave minimax-ES]]'', and the systematic name for the CTWE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT: tuning fundamentals #Held-intervals|held-octave]] [[Dave_Keenan %26 Douglas Blumeyer%27s guide to RTT: alternative complexities #Tunings used in 7|minimax-E-lils-S]]''.
+In D&D's guide to RTT, the [[Dave Keenan & Douglas Blumeyer's guide to RTT: alternative complexities #Naming|systematic name]] for the CTE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT: all-interval tuning schemes #Held-octave minimax-.28E.29S|held-octave minimax-ES]]'', and the systematic name for the CTWE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT: tuning fundamentals #Held-intervals|held-octave]] [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT: alternative complexities #Tunings used in 7|minimax-E-lils-S]]''.
 [[Category:Terms]]
 [[Category:Math]]
 [[Category:Regular temperament tuning]]