Tp tuning: Difference between revisions

Line 19:

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, or [[damage]], is {{nowrap|''D''''p''(''q'') {{=}} Err (''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''; if we measure in cents as we have defined above, ''E''''p''(''S'') has units of cents. 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''.

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, or [[damage]], is {{nowrap|''D''''p''(''q'') {{=}} Err(''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''; if we measure in cents as we have defined above, ''E''''p''(''S'') has units of cents. 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''.

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.

@@ Line 19: / Line 19: @@
 If ''q'' is any positive rational number, ‖''q''‖<sub>''p''</sub> is the T<sub>''p''</sub> 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'') &minus; cents (''q''){{!}}}}, and if {{nowrap|''q'' &ne; 1}}, the T<sub>''p''</sub> proportional error, or [[damage]], is {{nowrap|''D''<sub>''p''</sub>(''q'') {{=}} Err (''q'')/‖''q''‖<sub>''p''</sub>}}. For any tuning ''T'' of the temperament, the set of ''D''<sub>''p''</sub>(''q'') for all {{nowrap|''q'' &ne; 1}} in ''G'' is bounded, and hence has a least upper bound, the {{w|infimum and supremum|supremum}} sup (''D''<sub>''p''</sub>(''T'')). The set of values sup (''D''<sub>''p''</sub>(''T'')) is bounded below, and by continuity achieves its minimum value, which is the T<sub>''p''</sub> error E<sub>''p''</sub>(''S'') of the abstract temperament ''S''; if we measure in cents as we have defined above, ''E''<sub>''p''</sub>(''S'') has units of cents. Any tuning achieving this minimum, so that {{nowrap|sup (''D''<sub>''p''</sub>(''T'')) {{=}} ''E''<sub>''p''</sub>(''S'')}}, is a T<sub>''p''</sub> 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<sub>''p''</sub> tuning, thereby defining a unique tuning ''T''<sub>''p''</sub>(''S'') for any abstract temperament ''S'' on any group ''G''.
+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'') &minus; cents (''q''){{!}}}}, and if {{nowrap|''q'' &ne; 1}}, the T<sub>''p''</sub> proportional error, or [[damage]], is {{nowrap|''D''<sub>''p''</sub>(''q'') {{=}} Err(''q'')/‖''q''‖<sub>''p''</sub>}}. For any tuning ''T'' of the temperament, the set of ''D''<sub>''p''</sub>(''q'') for all {{nowrap|''q'' &ne; 1}} in ''G'' is bounded, and hence has a least upper bound, the {{w|infimum and supremum|supremum}} sup (''D''<sub>''p''</sub>(''T'')). The set of values sup (''D''<sub>''p''</sub>(''T'')) is bounded below, and by continuity achieves its minimum value, which is the T<sub>''p''</sub> error E<sub>''p''</sub>(''S'') of the abstract temperament ''S''; if we measure in cents as we have defined above, ''E''<sub>''p''</sub>(''S'') has units of cents. Any tuning achieving this minimum, so that {{nowrap|sup (''D''<sub>''p''</sub>(''T'')) {{=}} ''E''<sub>''p''</sub>(''S'')}}, is a T<sub>''p''</sub> 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<sub>''p''</sub> tuning, thereby defining a unique tuning ''T''<sub>''p''</sub>(''S'') for any abstract temperament ''S'' on any group ''G''.
 Given ''T''<sub>''p''</sub>(''S'') in a group ''G'' containing 2, we may define a corresponding pure-octaves tuning (POL<sub>''p''</sub> tuning) by dividing by the tuning of 2: {{nowrap|''T''<sub>''p''</sub>{{'}}(''S'') {{=}} 1200 ''T''<sub>''p''</sub>(''S'')/(''T''<sub>''p''</sub>(''S''))<sub>1</sub>}}, where (''T''<sub>''p''</sub>(''S''))<sub>1</sub> is the first entry of ''T''<sub>''p''</sub>(''S''). When {{nowrap|''p'' {{=}} 2}}, POL<sub>2</sub> tuning generalizes POTE tuning.