Optimization: Difference between revisions
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<math>\displaystyle W = \operatorname {diag} (1/\log_2 (Q)) </math> | <math>\displaystyle W = \operatorname {diag} (1/\log_2 (Q)) </math> | ||
indicates that the prime harmonic ''q'' in Q has the importance of 1/log<sub>2</sub>(''q''). Since the tuning space and the interval space are [[dual]] to each other, such a rating of importance in the tuning space has the dual effect in the interval space: the prime harmonic ''q'' has the complexity log<sub>2</sub>(''q''). The more complex it is, the more error will be allowed for it. | indicates that the prime harmonic ''q'' in Q = {{val| 2 3 5 … }} has the importance of 1/log<sub>2</sub>(''q''). Since the tuning space and the interval space are [[dual]] to each other, such a rating of importance in the tuning space has the dual effect in the interval space: the prime harmonic ''q'' has the complexity log<sub>2</sub>(''q''). The more complex it is, the more error will be allowed for it. | ||
=== Skew === | === Skew === | ||
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The '''Chebyshevian norm''' aka ''L''<sup>inifinity</sup> norm is the opposite of the Minkowsky norm – it is the maximum number of steps in any direction, so a diagonal movement is the same as a horizontal or vertical one. | The '''Chebyshevian norm''' aka ''L''<sup>inifinity</sup> norm is the opposite of the Minkowsky norm – it is the maximum number of steps in any direction, so a diagonal movement is the same as a horizontal or vertical one. | ||
It should be noted that the dual norm of ''L''<sup>1</sup> is ''L''<sup>infinity</sup>, and vice versa. Thus, the Minkowskian norm corresponds to the ''L''<sup>infinity</sup> tuning space, and the Chebyshevian norm corresponds to the ''L''<sup>1</sup> tuning space. The dual of ''L''<sup>2</sup> norm is itself, so the Euclidean norm corresponds to Euclidean tuning as one may expect. | It should be noted that the dual norm of ''L''<sup>1</sup> is ''L''<sup>infinity</sup>, and vice versa. Thus, the Minkowskian norm corresponds to the ''L''<sup>infinity</sup> tuning space, and the Chebyshevian norm corresponds to the ''L''<sup>1</sup> tuning space. The dual of ''L''<sup>2</sup> norm is itself, so the Euclidean norm corresponds to Euclidean tuning as one may expect. | ||
== Enforcement == | == Enforcement == | ||