Tp tuning: Difference between revisions

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== Applying the Hahn-Banach theorem ==

Suppose {{nowrap|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 {{nowrap|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.

Suppose {{nowrap|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 {{nowrap|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.

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 L''p'' tuning and the extension of Ɛ to Ƹ is in that case unique. It is also easy to see that this can only be non-unique if {{nowrap|''p'' {{=}} 1}} or {{nowrap|''p'' {{=}} ∞}}, so that we may get a unique L''p'' tuning (called the "TIPTOP" tuning for {{nowrap|''p'' {{=}} ∞}}) by simply taking the limit as ''p'' approaches our value.

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 == Applying the Hahn-Banach theorem ==
-Suppose {{nowrap|T {{=}} T<sub>''p''</sub>(S)}} is an T<sub>''p''</sub> 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 {{nowrap|&#400; {{=}} T - J}} is also. The norm &#8741;&#400;&#8741; of &#400; is minimal among all error maps for tunings of S since T is the T<sub>''p''</sub> tuning. By the [[Wikipedia: Hahn–Banach theorem|Hahn–Banach theorem]], &#400; can be extended to an element &#440; in the space of full ''p''-limit tuning maps with the same norm; that is, so that &#8741;&#400;&#8741; = &#8741;&#440;&#8741;. 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. &#8741;&#440;&#8741;, the norm of the full ''p''-limit error map, must also be minimal among all valid error maps for S*, or the restriction of &#440; to G would improve on &#400;. Hence, as &#8741;&#440;&#8741; is minimal, J* + &#440;, where J* is the full ''p''-limit [[JIP]], must equal the T<sub>''p''</sub> tuning for S*. Thus to find the T<sub>''p''</sub> tuning of S for the group G, we may first find the T<sub>''p''</sub> tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.
+Suppose {{nowrap|T {{=}} T<sub>''p''</sub>(S)}} is an T<sub>''p''</sub> 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 {{nowrap|&#400; {{=}} T &minus; J}} is also. The norm &#8741;&#400;&#8741; of &#400; is minimal among all error maps for tunings of S since T is the T<sub>''p''</sub> tuning. By the [[Wikipedia: Hahn–Banach theorem|Hahn–Banach theorem]], &#400; can be extended to an element &#440; in the space of full ''p''-limit tuning maps with the same norm; that is, so that &#8741;&#400;&#8741; = &#8741;&#440;&#8741;. 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. &#8741;&#440;&#8741;, the norm of the full ''p''-limit error map, must also be minimal among all valid error maps for S*, or the restriction of &#440; to G would improve on &#400;. Hence, as &#8741;&#440;&#8741; is minimal, J* + &#440;, where J* is the full ''p''-limit [[JIP]], must equal the T<sub>''p''</sub> tuning for S*. Thus to find the T<sub>''p''</sub> tuning of S for the group G, we may first find the T<sub>''p''</sub> 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 L<sub>''p''</sub> tuning and the extension of &#400; to &#440; is in that case unique. It is also easy to see that this can only be non-unique if {{nowrap|''p'' {{=}} 1}} or {{nowrap|''p'' {{=}} &infin;}}, so that we may get a unique L<sub>''p''</sub> tuning (called the "TIPTOP" tuning for {{nowrap|''p'' {{=}} &infin;}}) by simply taking the limit as ''p'' approaches our value.