Tenney–Euclidean temperament measures: Difference between revisions

Given a [[Wedgies_and_Multivals|wedgie]] M, that is a canonically reduced ''r''-val correspondng to a temperament of rank ''r'', the norm ||M|| is a measure of the ''complexity'' of M; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. This complexity and related measures have been [http://x31eq.com/temper/primerr.pdf extensively studied] by [[Graham_Breed|Graham Breed]], and we may call it Tenney-Euclidean complexity since it can be defined in terms of the [[Tenney-Euclidean_metrics|Tenney-Euclidean norm]].

In fact, while TE complexity is easily computed if a routine for computing multivectors is available, such a routine is not required, as it can also be computed using the [http://en.wikipedia.org/wiki/Gramian_matrix Gramian]. This is the determinant of the square matrix, called the Gram matrix, defined from a list of ''r'' vectors, whose (''i'', ''j'')-th entry is v_''i''·v_''j'', the [http://en.wikipedia.org/wiki/Dot_product dot product] of the ''i''-th vector with the ''j''-th vector. The square of the ordinary Euclidean norm of a multivector is the Gramian of the vectors wedged together to define it, and hence in terms of the TE norm we have

||M|| = ||v_1 \wedge v_2 \wedge ... \wedge v_r|| = \sqrt{\frac{det([v_i \cdot v_j])}{C(n, r)}}</math>

where C(''n'', ''r'') is the number of combinations of ''n'' things taken ''r'' at a time, and v''_i''·v''_j'' is the TE [http://en.wikipedia.org/wiki/Symmetric_bilinear_form symmetric form] on vals, which in weighted coordinates is simply the ordinary dot product. Here ''n'' is the number of primes up to the prime limit ''p'', and ''r'' is the rank of the temperament, which equals the number of vals wedged together to compute the wedgie.

If V is a matrix whose rows are the weighted vals ''v_i'', we may write det([''v_i''·''v_j'']) as det(VV^T), where V^T denotes the transpose. If W is [http://en.wikipedia.org/wiki/Diagonal_matrix diagonal matrix] with 1, 1/log₂3, …, 1/log₂''p'' along the diagonal and A is the matrix corresponding to V with unweighted vals as rows, then V = AW and det(VV^T) = det(AW²A^T). This may be related to the [[Tenney-Euclidean_metrics|TE tuning projection matrix]] P, which is V^T(VV^T)^-1V, and the corresponding matrix for unweighted monzos '''P''' = A^T(AW²A^T)^-1A.

If J = &lt;1 1 ... 1| is the JI point in weighted coordinates, then the simple badness of M, which we may also call the relative error of M, is defined as ||J∧M||. This may considered to be a sort of badness which heavily favors complex temperaments, or it may be considered error relativized to the complexity of the temperament: relative error is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step. Once again, if we have a list of vectors we may use a Gramian to compute relative error. First we note that a_''i'' = J·v_''i''/''n'' is the mean value of the entries of v_''i''. Then note that J∧(v₁ - a₁J)∧(v₂ - a₂J)∧...∧(v_''r'' - a_''r''J) = J∧v₁∧v₂∧...∧v_''r'', since wedge products with more than one term J are zero. The Gram matrix of the vectors J and v₁ - a_''i''J will have ''n'' as the (1,1) entry, and 0s in the rest of the first row and column. Hence we obtain: