Tp tuning: Difference between revisions

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For a subgroup temperament over a general [[JI subgroup]], and for a given choice of ''p'' (most commonly {{nowrap|''p'' {{=}} 2}}), there are two notions of T''p'' tuning:

* The first is called '''inharmonic T''p''''', ~~because~~ the basis entries are treated as if they were primes, reminiscent of ~~some~~ inharmonic ~~timbres~~. Inharmonic T''p'' depends on the basis used for the subgroup. In non-octave temperaments, inharmonic T''p'' could be used when optimizing a specific voicing of a tempered JI chord. For example in 3/2.7/4.5/2 semiwolf temperament which tempers out 245/243, the 3/2.7/4.5/2 inharmonic T''p'' optimizes the 4:6:7:10 chord.

* The first is called '''inharmonic T''p''''', or '''L''p''''' in earlier materials. This is where the basis entries are treated as if they were primes, reminiscent of using an inharmonic timbre. Inharmonic T''p'' depends on the basis used for the subgroup. In non-octave temperaments, inharmonic T''p'' could be used when optimizing a specific voicing of a tempered JI chord. For example in 3/2.7/4.5/2 semiwolf temperament which tempers out 245/243, the 3/2.7/4.5/2 inharmonic T''p'' optimizes the 4:6:7:10 chord.

* The second is called '''subgroup T''p''''', because it treats the temperament as a restriction of a full prime-limit temperament to a subgroup of the prime-limit. Subgroup T''p'' does not depend on the basis used for the subgroup, and as stated, extends naturally to the T''p'' tuning of the full prime-limit temperament.

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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 ==

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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 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.~~

== See also ==

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 For a subgroup temperament over a general [[JI subgroup]], and for a given choice of ''p'' (most commonly {{nowrap|''p'' {{=}} 2}}), there are two notions of T<sub>''p''</sub> tuning:
-* The first is called '''inharmonic T<sub>''p''</sub>''', because the basis entries are treated as if they were primes, reminiscent of some inharmonic timbres. Inharmonic T<sub>''p''</sub> depends on the basis used for the subgroup. In non-octave temperaments, inharmonic T<sub>''p''</sub> could be used when optimizing a specific voicing of a tempered JI chord. For example in 3/2.7/4.5/2 semiwolf temperament which tempers out 245/243, the 3/2.7/4.5/2 inharmonic T<sub>''p''</sub> optimizes the 4:6:7:10 chord.
+* The first is called '''inharmonic T<sub>''p''</sub>''', or '''L<sub>''p''</sub>''' in earlier materials. This is where the basis entries are treated as if they were primes, reminiscent of using an inharmonic timbre. Inharmonic T<sub>''p''</sub> depends on the basis used for the subgroup. In non-octave temperaments, inharmonic T<sub>''p''</sub> could be used when optimizing a specific voicing of a tempered JI chord. For example in 3/2.7/4.5/2 semiwolf temperament which tempers out 245/243, the 3/2.7/4.5/2 inharmonic T<sub>''p''</sub> optimizes the 4:6:7:10 chord.
 * The second is called '''subgroup T<sub>''p''</sub>''', because it treats the temperament as a restriction of a full prime-limit temperament to a subgroup of the prime-limit. Subgroup T<sub>''p''</sub> does not depend on the basis used for the subgroup, and as stated, extends naturally to the T<sub>''p''</sub> tuning of the full prime-limit temperament.
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 == 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<sub>''p''</sub> 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 ==
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 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.
-This is called ''subgroup TE'' in Graham Breed's temperament finder.
 == See also ==