Constrained tuning: Difference between revisions
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tone + all of the optimizations on this page are linear |
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=== Simple fast closed-form algorithm === | === Simple fast closed-form algorithm === | ||
Another way to compute the CTE and CWE tunings, and the CTWE tuning in general, is to use the pseudoinverse. | |||
The basic idea is that the set of all pure-octave tuning maps of some temperament will be the intersection of a linear subspace and a shifted hyperplane, and thus will be a shifted subspace. This means that any pure-octave tuning map can be expressed as the sum of some arbitrary "reference" pure-octave tuning map for the temperament, plus some other one also in the temperament whose octave-coordinate is 0. The set of all such tuning maps of the latter category form a linear subspace. | The basic idea is that the set of all pure-octave tuning maps of some temperament will be the intersection of a linear subspace and a shifted hyperplane, and thus will be a shifted subspace. This means that any pure-octave tuning map can be expressed as the sum of some arbitrary "reference" pure-octave tuning map for the temperament, plus some other one also in the temperament whose octave-coordinate is 0. The set of all such tuning maps of the latter category form a linear subspace. | ||