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 {{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 have 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 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''.  

Suppose {{nowrap|''T'' {{=}} ''T''_''p''(''S'')}} is a 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 {{w|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}}.