Tp tuning: Difference between revisions

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== Dual norm ==

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''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 [[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''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).

== Applying the Hahn-Banach theorem ==

Suppose T = T''p'' (S) is an T''p'' tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings of S since T is the T''p'' tuning. By the [[Wikipedia: Hahn–Banach theorem|Hahn–Banach theorem], Ɛ can be extended to an element Ƹ in the space of full p-limit tuning maps with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a [http://www.math.unl.edu/%7Es-bbockel1/928/node25.html corollary of Hahn-Banach], the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.

Suppose T = T''p'' (S) is an T''p'' tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings of S since T is the T''p'' tuning. By the [[Wikipedia: Hahn–Banach theorem|Hahn–Banach theorem]], Ɛ can be extended to an element Ƹ in the space of full p-limit tuning maps with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a [http://www.math.unl.edu/%7Es-bbockel1/928/node25.html corollary of Hahn-Banach], the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.

||Ƹ||, the norm of the full p-limit error map, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full ''p''-limit [[JIP]], must equal the T''p'' tuning for S*. Thus to find the T''p'' tuning of S for the group G, we may first find the T''p'' tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.

@@ Line 15: / Line 15: @@
 == Dual norm ==
-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''p'' (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''p'' (S).
 == Applying the Hahn-Banach theorem ==
-Suppose T = T''p'' (S) is an T''p'' tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings of S since T is the T''p'' tuning. By the [[Wikipedia: Hahn–Banach theorem|Hahn–Banach theorem], Ɛ can be extended to an element Ƹ in the space of full p-limit tuning maps with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a [http://www.math.unl.edu/%7Es-bbockel1/928/node25.html corollary of Hahn-Banach], the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.
+Suppose T = T''p'' (S) is an T''p'' tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings of S since T is the T''p'' tuning. By the [[Wikipedia: Hahn–Banach theorem|Hahn–Banach theorem]], Ɛ can be extended to an element Ƹ in the space of full p-limit tuning maps with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a [http://www.math.unl.edu/%7Es-bbockel1/928/node25.html corollary of Hahn-Banach], the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.
 ||Ƹ||, the norm of the full p-limit error map, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full ''p''-limit [[JIP]], must equal the T''p'' tuning for S*. Thus to find the T''p'' tuning of S for the group G, we may first find the T''p'' tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.