Tenney–Euclidean tuning: Difference between revisions

Tenney-Euclidean tuning is a variant of [[TOP tuning]] which employs the [[Tenney-Euclidean_metrics|TE norm]]. Just as the TOP tuning minimizes the maximum Tenney-weighted (L1) error of any interval, the TE tuning minimizes the maximum TE-weighted error of any interval.

In more pragmatic terms, suppose M is the known mapping of the [[Regular temperament|abstract temperament]], '''g''' the row vector of TE generators (in per-octave), '''p''' the row vector of targeted JI intervals (in per-octave), and W the weighting matrix. In this case, W_''ij'' = 1/log₂''p'' if ''i'' = ''j'', and 0 otherwise. TE tuning then defines a [[wikipedia:Least_squares|least square]] problem of the following overdetermined linear equation system:  

where V = MW, i.e. the mapping in the weighted basis, and J = '''p'''W, i.e. the JI point, in this case it is {{val|1 1 … 1}}.  

 

The system simply says that the sum of V_''ij'' steps of generator '''g'''_''i'' for all ''i'' should equal the ''j''-th targeted JI interval J_''j''.  

gives the TE tuning in the sense that it gives (not necessarily independent) generators which correspond to the rows of V. How many of each generator to take to map a rational number contained in the prime limit in question is determined by applying the val corresponding to the generator to the rational number.

We may call pure-octaves Tenney-Euclidean tuning the '''POTE tuning'''.  

 

If T = JP = '''g'''V is the TE tuning map, then a corresponding pure-octaves map can be found by [[wikipedia:Scalar_multiplication|scalar multiplication]], T/T₁, where T₁, the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.  

The val for 5-limit 12et is {{val|12 19 28}}. In weighted coordinates, that becomes '''v'''₁₂ = {{val|12 19/log₂3 28/log₂5}} ~ {{val|12.0 11.988 12.059}}. If we take this to be a 1×3 matrix and take the pseudoinverse, we get the 3×1 matrix '''v'''₁₂⁺ ~ [{{monzo|0.027706 0.027677 0.027842}}]. Then P = '''v'''₁₂⁺'''v'''₁₂ is a projection map onto the one-dimensional subspace whose single basis vector is '''v'''₁₂. We find that '''v'''₁₂P equals '''v'''₁₂; on the other hand, if we take the monzo for 81/80, which is {{monzo|-4 4 -1}}; and monzo-weight it to {{monzo|-4 4log₂3 -log₂5}} and multiply (either side, the matrix is symmetric) by P, we get the zero vector, corresponding to the unison.

Now consider [[Diaschismic_family #Pajara|pajara]], the 7-limit temperament tempering out both 50/49 and 64/63. Two possible equal temperament tunings for pajara are [[12edo]] and [[22edo]]. We may define a 2x4 matrix with rows equal to the vals for 12, and 22; in weighted coordinates this would be

Paj = V⁺V is a 4x4 symmetrical matrix which projects weighted vals in TE tuning space, or weighted monzos in TE interval space, to a subspace defined by pajara. It therefore projects the weighted monzos for 50/49, 64/63, 225/224, 2048/2025 etc. to the zero vector, whereas it leaves pajara vals such as [[10edo]] in weighted coordinates unchanged.

If we use unweighted coordinates we get the Frobenius projection map instead, whose rows are [[fractional monzos]]. The unweighted pseudoinverse '''u'''₁₂⁺ of the 5-limit val '''u'''₁₂ for 12 equal is the column matrix '''u'''₁₂^T/1289; that is, the 1×3 matrix with column {{monzo|12/1289 19/1289 28/1289}}. Then '''u'''₁₂⁺'''u'''₁₂ is the 3×3 Frobenius projection map F: