Tp tuning: Difference between revisions

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== Dual norm ==

We can extend the T''p'' norm on monzos to a {{w|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 {{w|dual norm}}. If ''r''1, ''r''2, … , ''r''''n'' 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''1 ''r''2 … ''r''''n''] is the normal ''G'' generator list, then {{val| cents (''r''1) cents (''r''2) … cents (''r''''n'') }} 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''''p''(''S'').

We can extend the T''p'' norm on monzos to a {{w|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 {{w|dual norm}}. If ''r''1, ''r''2, … , ''r''''n'' 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''1 ''r''2 … ''r''''n''] is the normal ''G'' generator list, then {{val| cents (''r''1) cents (''r''2) … cents (''r''''n'') }} 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''''p''(''S'').

In the special case where {{nowrap|''p'' {{=}} 2}}, this becomes L2 tuning. This is called ''inharmonic TE'' in Graham Breed's temperament finder.

== Applying the Hahn-Banach theorem ==

@@ Line 24: / Line 24: @@
 == Dual norm ==
-We can extend the T<sub>''p''</sub> norm on monzos to a {{w|normed vector space|vector space norm}} on [[monzos and interval space|interval space]], thereby defining the real normed interval space T<sub>''p''</sub>. 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<sub>''p''</sub> 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 {{w|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<sub>''p''</sub> norm on monzos to a {{w|normed vector space|vector space norm}} on [[monzos and interval space|interval space]], thereby defining the real normed interval space T<sub>''p''</sub>. 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<sub>''p''</sub> 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 {{w|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<sub>''p''</sub> tuning''', ''L''<sub>''p''</sub>(''S'').
 In the special case where {{nowrap|''p'' {{=}} 2}}, this becomes L<sub>2</sub> tuning. This is called ''inharmonic TE'' in Graham Breed's temperament finder.
 == Applying the Hahn-Banach theorem ==