Tenney–Euclidean temperament measures: Difference between revisions

Given a [[Wedgies_and_Multivals|multival]] or multimonzo which is a [http://en.wikipedia.org/wiki/Exterior_algebra wedge product] of weighted vals or monzos, we may define a norm by means of the usual Euclidean norm. We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS ([http://en.wikipedia.org/wiki/Root_mean_square root mean square]) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different prime limits can be meaningfully compared. If M is a multivector, we denote the RMS norm as ||M||.

There have been several ways to define TE complexity, and each will be discussed below. All of them can be easily computed either from the multivector or from the mapping matrix, using the [http://en.wikipedia.org/wiki/Gramian_matrix Gramian].  

It is clear that the definitions differ from each other by a factor of rank and limit. For the same rank and limit though, any of them will provide meaningful comparison.

If J = {{val|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:

<math>\displaystyle ||J \wedge M|| = \sqrt{\frac{n}{C(n,r+1)}} det([v_i \cdot v_j - na_ia_j])</math>

TE simple badness can also be called and considered to be error relative to complexity. By dividing it by TE complexity, we get an error measurement, TE error. Multiplying this by 1200, we get a figure we can consider to be a weighted average with values in cents.  

is an adjusted error which makes the error of a rank ''r'' temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament. ψ can be related to TE error as it appears on Graham Breed's [http://x31eq.com/temper/ temperament finder], which we will call RMS error; if G denotes the RMS error for a temperament, then

<math>\displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi</math>

RMS error and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the RMS normalization has the further advantage that in the rank one case,  G = sin θ, where θ is the angle between J and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin θ, TE error as it appears on the temperament finder pages. RMS error may also be found using the TE tuning map; if T is the tuning map, then

<math>\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}} = || T-J ||</math>

where the dot represents the ordinary dot product. If T is denominated in cents, then J should be also, so that J = &lt;1200 1200 ... 1200|.