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 an [[interval size unit]] proportional to [[cent]]s, such that {{nowrap| ''T''(''c'') {{=}} 0 }} for any comma ''c'' of the temperament. Let the just value of ''q'' be ''J''(''q''), we define the error of the tuning on ''q'', ''Ɛ''(''q''), as {{nowrap| ''T''(''q'') − ''J''(''q'') }}, and if {{nowrap| ''q'' ≠ 1 }}, the T_''p'' proportional error, or [[damage]], is {{nowrap|''D''_''p''(''q'') {{=}} {{!}}''Ɛ''(''q''){{!}}/‖''q''‖_''p''}}. For any tuning ''T'' of the temperament, the set of ''D''_''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 (''D''_''p''(''T'')). The set of values sup (''D''_''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'' in the same unit as we have defined above. Any tuning achieving this minimum, so that {{nowrap| sup (''D''_''p''(''T'')) {{=}} ''E''_''p''(''S'') }}, is a 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''.