Tp tuning: Difference between revisions

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 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.

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.

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}}.