Tp tuning: Difference between revisions

If ''p'' ≥ 1, define the T_''p'' norm, which we may also call the T_''p'' complexity, of any monzo in weighted coordinates m as

<math>||\ |m_2 \ m_3 \ \ldots \ m_k> ||_p = (|m_2|^p + |m_3|^p + \ldots + |m_k|^p)^{1/p}</math>

<math>||\ |b_2 \ b_3 \ \ldots \ b_k> ||_p = (|b_2 \log_2 2 |^p + |b_3 \log_2 3|^p + \ldots + |b_k \log_2 k |^p)^{1/p}</math>

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 T (''c'') = 0 for any comma ''c'' of the temperament. We define the error of the tuning on ''q'', Err (''q''), as |T (''q'') - cents (''q'')|, and if ''q'' ≠ 1, the ''T_p proportional error'' is PE_''p'' (''q'') = Err (''q'')/||''q''||_''p''. For any tuning T of the temperament, the set of PE_''p'' (''q'') for all ''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 sup (PE_''p'' (T)) = E_''p'' (S), is an T_''p'' tuning. Usually this tuning is unique, but in the case ''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 coresponding pure-octaves tuning (POL_''p'' tuning) by dividing by the tuning of 2: 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 ''p'' = 2, POL₂ tuning generalizes POTE tuning.

Suppose 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 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 ||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, 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 ''p'' = 1 or ''p'' = Infinity, so that we may get a unique L_''p'' tuning (called the "TIPTOP" tuning for ''p'' = Infinity) by simply taking the limit as ''p'' approaches our value.

In the special case where ''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 ||(T - J)/''n'' ||₂ = ||T - J||_RMS.

{{todo|review}}