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).

In the special case where ''p'' = 2, this becomes L2 tuning. This is called ''inharmonic TE'' in Graham Breed's temperament finder, in that the basis entries are treated as if they were primes, reminiscent of some inharmonic timbres.

== 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.

||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.

Note that while the Hahn-Banach theorem is usually proven using Zorn's lemma and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.

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For an example, consider [[Chromatic pairs #Indium|indium temperament]], with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the T2 (TE) tuning map is {{val| 1199.552 1901.846 2783.579 3371.401 4153.996 }}. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.000/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting {{val| 1200 1200 1200 1200 1200 }} gives {{val| -0.4475 -0.0685 -1.1778 0.9172 0.7741 }}. The ordinary Euclidean norm of this, i.e. the square root of the dot product, is 1.7414, and dividing by sqrt (5) gives the RMS error, 0.77879 cents.

This is called ''subgroup TE'' in Graham Breed's temperament finder.

[[Category:Math]]

@@ 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<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, in that the basis entries are treated as if they were primes, reminiscent of some inharmonic timbres.
 == 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.
+||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.
 Note that while the Hahn-Banach theorem is usually proven using Zorn's lemma and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.
@@ Line 27: / Line 29: @@
 For an example, consider [[Chromatic pairs #Indium|indium temperament]], with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the T2 (TE) tuning map is {{val| 1199.552 1901.846 2783.579 3371.401 4153.996 }}. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.000/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting {{val| 1200 1200 1200 1200 1200 }} gives {{val| -0.4475 -0.0685 -1.1778 0.9172 0.7741 }}. The ordinary Euclidean norm of this, i.e. the square root of the dot product, is 1.7414, and dividing by sqrt (5) gives the RMS error, 0.77879 cents.
+This is called ''subgroup TE'' in Graham Breed's temperament finder.
 [[Category:Math]]