Tp tuning: Difference between revisions
→T2 tuning: distinguish between the temperament and its optimal tuning |
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We can extend the T''p'' norm on monzos to a [[Wikipedia: Normed vector space|vector space norm]] on [[Monzos and interval space|interval space]], thereby defining the real normed interval space T''p''. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full ''p''-limit will be the whole of T''p'' but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a [[Wikipedia: Dual norm|dual norm]]. If ''r''<sub>1</sub>, ''r''<sub>2</sub>, … , ''r''<sub>''n''</sub> are a set of generators for G, which in particular could be a normal list and so define [[Smonzos and Svals|smonzos]] for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [''r''<sub>1</sub> ''r''<sub>2</sub> … ''r''<sub>''n''</sub>] is the normal G generator list, then {{val| cents (''r''<sub>1</sub>) cents (''r''<sub>2</sub>) … cents (''r''<sub>''n''</sub>) }} is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the '''L''p'' tuning''' L<sub>''p''</sub> (S). | We can extend the T''p'' norm on monzos to a [[Wikipedia: Normed vector space|vector space norm]] on [[Monzos and interval space|interval space]], thereby defining the real normed interval space T''p''. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full ''p''-limit will be the whole of T''p'' but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a [[Wikipedia: Dual norm|dual norm]]. If ''r''<sub>1</sub>, ''r''<sub>2</sub>, … , ''r''<sub>''n''</sub> are a set of generators for G, which in particular could be a normal list and so define [[Smonzos and Svals|smonzos]] for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [''r''<sub>1</sub> ''r''<sub>2</sub> … ''r''<sub>''n''</sub>] is the normal G generator list, then {{val| cents (''r''<sub>1</sub>) cents (''r''<sub>2</sub>) … cents (''r''<sub>''n''</sub>) }} is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the '''L''p'' tuning''' L<sub>''p''</sub> (S). | ||
In the special case where ''p'' = 2, this becomes L2 tuning. This is called '''inharmonic TE''' in Graham Breed's temperament finder, because the basis entries are treated as if they were primes, reminiscent of some inharmonic timbres. Inharmonic TE depends on the basis used for | In the special case where ''p'' = 2, this becomes L2 tuning. This is called '''inharmonic TE''' in Graham Breed's temperament finder, because the basis entries are treated as if they were primes, reminiscent of some inharmonic timbres. Inharmonic TE depends on the basis used for the subgroup.<!-- It may be preferable when optimizing a specific voicing of a tempered JI chord. --> | ||
== Applying the Hahn-Banach theorem == | == Applying the Hahn-Banach theorem == | ||