Tp tuning: Difference between revisions

If {{nowrap|''p'' ≥ 1}}, define the [[Generalized Tenney norms and Tp interval space|T_''p'' norm]], which we may also call the T_''p'' complexity, of any monzo in weighted coordinates m as

If ''q'' is any positive rational number, ‖''q''‖_''p'' is the T_''p'' norm defined by its monzo.

For some just intonation group G, which is to say some finitely generated group of positive rational numbers which can be either a full prime-limit group or some subgroup of such a group, a regular temperament [[tuning map|tuning]] T for an abstract temperament S is defined by a linear map from monzos belonging to G to a value in cents, such that {{nowrap|T(''c'') {{=}} 0}} for any comma ''c'' of the temperament. We define the error of the tuning on ''q'', Err (''q''), as {{nowrap|{{!}}T(''q'') − cents (''q''){{!}}}}, and if {{nowrap|''q'' ≠ 1}}, the ''T_p proportional error'' is {{nowrap|PE_''p''(''q'') {{=}} Err(''q'')/‖''q''‖_''p''}}. For any tuning T of the temperament, the set of PE_''p''(''q'') for all {{nowrap|''q'' ≠ 1}} in G is bounded, and hence has a least upper bound, the {{w|infimum and supremum|supremum}} sup(PE_''p''(T)). The set of values sup (PE_''p''(T)) is bounded below, and by continuity achieves its minimum value, which is the T_''p'' error E_''p''(S) of the abstract temperament S; if we measure in cents as we've defined above, E_''p''(S) has units of cents. Any tuning achieving this minimum, so that {{nowrap|sup(PE_''p''(T)) {{=}} E_''p''(S)}}, is an T_''p'' tuning. Usually this tuning is unique, but in the case {{nowrap|''p'' {{=}} 1}}, called the [[TOP tuning]], it may not be. In this case we can choose a TOP tuning canonically by setting it to the limit as ''p'' tends to 1 of the T_''p'' tuning, thereby defining a unique tuning T_''p''(S) for any abstract temperament S on any group G.

Given T_''p''(S) in a group G containing 2, we may define a corresponding pure-octaves tuning (POL_''p'' tuning) by dividing by the tuning of 2: {{nowrap|T_''p''{{'}}(S) {{=}} 1200 T_''p''(S)/(T_''p''(S))₁}}, where (T_''p''(S))₁ is the first entry of T_''p''(S). When {{nowrap|''p'' {{=}} 2}}, POL₂ tuning generalizes POTE tuning.

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).

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 {{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.

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 [[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.