Constrained tuning: Difference between revisions

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~~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~~.

'''Constrained tunings''' are tuning optimization techniques under the constraints of some purely tuned intervals (i.e. [[eigenmonzo]]s). The '''CTE tuning''' ('''constrained Tenney-Euclidean tuning''') is the most typical instance and will be the focus of this article. Otherwise normed tunings can be defined and computed analogously.

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.

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, which is assumed unless specified otherwise. For higher-rank temperaments, it may make sense to add multiple constraints, such as the pure-{2, 3} CTE tuning.

== Definition ==

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-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.
+'''Constrained tunings''' are tuning optimization techniques under the constraints of some purely tuned intervals (i.e. [[eigenmonzo]]s). The '''CTE tuning''' ('''constrained Tenney-Euclidean tuning''') is the most typical instance and will be the focus of this article. Otherwise normed tunings can be defined and computed analogously.
-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.
+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, which is assumed unless specified otherwise. For higher-rank temperaments, it may make sense to add multiple constraints, such as the pure-{2, 3} CTE tuning.
 == Definition ==