Tp tuning: Difference between revisions

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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'').

== Applying the ~~Hahn-Banach~~ theorem ==

== Applying the Hahn–Banach theorem ==

Suppose {{nowrap|''T'' {{=}} ''T''''p''(''S'')}} is a 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 {{w|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 {{nowrap|‖''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, {{nowrap|''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 a 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 {{w|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 {{nowrap|‖''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, {{nowrap|''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.

Note that while the Hahn–Banach theorem is usually proven using {{w|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.

== T2 tuning ==

In the special case where {{nowrap|''p'' {{=}} 2}}, the T''p'' norm for the full prime limit becomes the T2 norm, which when divided by the square root of the number ''n'' of primes in the prime limit, is the [[Tenney-Euclidean metrics|Tenney-Euclidean norm]], giving TE complexity. Associated to this norm is T2 tuning extended to arbitrary JI groups, and [[Tenney-Euclidean temperament measures #TE error|RMS error]], which for a tuning map ''T'' is {{nowrap|‖(''T'' − ''J'')/''n''‖2 {{=}} ‖''T'' − ''J''‖RMS}}.

~~For an example, consider~~ [[~~Subgroup~~ temperaments #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 ~~T~~2~~ (~~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.

== Example ==

Consider the [[subgroup temperaments #Indium|indium]] temperament, with group 2.5/3.7/3.11/3 and [[comma basis]] 3025/3024 and 3125/3087. To find the inharmonic TE tuning, start by constructing the weight matrix with the inverses of the sizes of each subgroup basis element. In this case our weight matrix is given by diag([1/cents(2), 1/cents(5/3), 1/cents(7/3), 1/cents(11/3)]). Next, we apply this weight to the sval mapping {{mapping| 1 0 0 2 | 0 6 10 -1 }} as well as the just tuning map {{val| cents(2) cents(5/3) cents(7/3) cents(11/3) }}, which turns the weighted just tuning map to all-ones. Using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the inharmonic TE generator tuning map is {{val| 1199.043 147.042 }}, and the tuning map is {{val| 1199.043 882.252 1470.420 2251.044 }}. This tuning may be regarded as optimized for the 3:5:6:7:11 chord in the temperament.

To find the subgroup TE tuning instead, we need to first find the TE tuning of the corresponding full 11-limit temperament. This temperament is of rank 3, with mapping {{mapping| 1 0 0 0 2 | 0 1 1 1 1 | 0 0 6 10 -1 }}. Here we find that the subgroup 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, so that the generator tuning map is {{val| 1199.552 146.995 }}. 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.

== See also ==

@@ Line 26: / Line 26: @@
 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'').
-== Applying the Hahn-Banach theorem ==
+== Applying the Hahn–Banach theorem ==
-Suppose {{nowrap|''T'' {{=}} ''T''<sub>''p''</sub>(''S'')}} is a 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|''Ɛ'' {{=}} ''T'' &minus; ''J''}} is also. The norm ‖''Ɛ''‖ of ''Ɛ'' is minimal among all error maps for tunings of ''S'' since ''T'' is the T<sub>''p''</sub> tuning. By the {{w|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 {{nowrap|‖''Ɛ''‖ {{=}} ‖''Ƹ''‖}}. Additionally, due to a [http://www.math.unl.edu/%7Es-bbockel1/928/node25.html corollary of Hahn&ndash;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, {{nowrap|''J''* + ''Ƹ''}}, 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 a 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|''Ɛ'' {{=}} ''T'' &minus; ''J''}} is also. The norm ‖''Ɛ''‖ of ''Ɛ'' is minimal among all error maps for tunings of ''S'' since ''T'' is the T<sub>''p''</sub> tuning. By the {{w|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 {{nowrap|‖''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, {{nowrap|''J''* + ''Ƹ''}}, 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 ''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'' {{=}} &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.
+Note that while the Hahn–Banach theorem is usually proven using {{w|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 ''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'' {{=}} &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.
 == T<sub>2</sub> tuning ==
 In the special case where {{nowrap|''p'' {{=}} 2}}, the T<sub>''p''</sub> norm for the full prime limit becomes the T<sub>2</sub> norm, which when divided by the square root of the number ''n'' of primes in the prime limit, is the [[Tenney-Euclidean metrics|Tenney-Euclidean norm]], giving TE complexity. Associated to this norm is T<sub>2</sub> tuning extended to arbitrary JI groups, and [[Tenney-Euclidean temperament measures #TE error|RMS error]], which for a tuning map ''T'' is {{nowrap|‖(''T'' &minus; ''J'')/''n''‖<sub>2</sub> {{=}} ‖''T'' &minus; ''J''‖<sub>RMS</sub>}}.
-For an example, consider [[Subgroup temperaments #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 T<sub>2</sub> (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 POT<sub>2</sub> 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.
+== Example ==
+Consider the [[subgroup temperaments #Indium|indium]] temperament, with group 2.5/3.7/3.11/3 and [[comma basis]] 3025/3024 and 3125/3087. To find the inharmonic TE tuning, start by constructing the weight matrix with the inverses of the sizes of each subgroup basis element. In this case our weight matrix is given by diag([1/cents(2), 1/cents(5/3), 1/cents(7/3), 1/cents(11/3)]). Next, we apply this weight to the sval mapping {{mapping| 1 0 0 2 | 0 6 10 -1 }} as well as the just tuning map {{val| cents(2) cents(5/3) cents(7/3) cents(11/3) }}, which turns the weighted just tuning map to all-ones. Using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the inharmonic TE generator tuning map is {{val| 1199.043 147.042 }}, and the tuning map is {{val| 1199.043 882.252 1470.420 2251.044 }}. This tuning may be regarded as optimized for the 3:5:6:7:11 chord in the temperament.
+To find the subgroup TE tuning instead, we need to first find the TE tuning of the corresponding full 11-limit temperament. This temperament is of rank 3, with mapping {{mapping| 1 0 0 0 2 | 0 1 1 1 1 | 0 0 6 10 -1 }}. Here we find that the subgroup 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, so that the generator tuning map is {{val| 1199.552 146.995 }}. Multiplying that by 1200.000/1199.552 gives a POT<sub>2</sub> 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.
 == See also ==