Tp tuning: Difference between revisions

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.