Tp tuning: Difference between revisions

* The second is called '''subgroup TE''', because it treats the temperament as a restriction of a full prime-limit temperament to a subgroup of the prime-limit. Subgroup TE does not depend on the basis used for the subgroup, and as stated, extends naturally to the TE tuning of the full prime-limit temperament.

The two notions agree exactly when the temperament is defined on a JI subgroup with a basis consisting of rationally independent (i.e. pairwise coprime) members. That is, the subgroup has a basis where no two elements share a prime factor (examples: 2.3.5 and 2.9.5; nonexample: 2.9.5.21).

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 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 ''Tp 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.

In the special case where ''p'' = 2, this becomes L2 tuning. This is called ''inharmonic TE'' in Graham Breed's temperament finder, in that the basis entries are treated as if they were primes, reminiscent of some inharmonic timbres.  

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.

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 Lp 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=0 or p=Infinity, so that we may get a unique Lp tuning (called the "TIPTOP" tuning for p=Infinity) by simply taking the limit as p approaches our value.

== T2 tuning ==

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.

For an example, consider [[Chromatic pairs #Indium|indium temperament]], with group 2.5/3.7/3.11/3 and [[comma basis]] 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the T2 (TE) tuning map is {{val| 1199.552 1901.846 2783.579 3371.401 4153.996 }}. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.000/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting {{val| 1200 1200 1200 1200 1200 }} gives {{val| -0.4475 -0.0685 -1.1778 0.9172 0.7741 }}. The ordinary Euclidean norm of this, i.e. the square root of the dot product, is 1.7414, and dividing by sqrt (5) gives the RMS error, 0.77879 cents.