Tenney–Euclidean tuning: Difference between revisions

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{{nowrap|''P'' {{=}} {{subsup|''V''|''W''|+}}''V''''W''}} is a 4×4 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 matrix instead, whose rows are [[fractional monzos]]. The unweighted pseudoinverse {{subsup|V|12|+}} of the 5-limit val ''V''12 for 12 equal is the column matrix {{subsup|''V''|12|T}}/1289; that is, the 1×3 matrix with column {{monzo| 12/1289 19/1289 28/1289 }}. Then {{subsup|V|12|+}}''V''12 is the 3×3 Frobenius projection matrix ''P''''F'':

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

<math>\displaystyle

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 {{nowrap|''P'' {{=}} {{subsup|''V''|''W''|+}}''V''<sub>''W''</sub>}} is a 4×4 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 matrix instead, whose rows are [[fractional monzos]]. The unweighted pseudoinverse {{subsup|V|12|+}} of the 5-limit val ''V''<sub>12</sub> for 12 equal is the column matrix {{subsup|''V''|12|T}}/1289; that is, the 1×3 matrix with column {{monzo| 12/1289 19/1289 28/1289 }}. Then {{subsup|V|12|+}}''V''<sub>12</sub> is the 3×3 Frobenius projection matrix ''P''<sub>''F''</sub>:
+If we use unweighted coordinates we get the Frobenius projection matrix instead, whose rows are [[fractional monzos]]. The unweighted pseudoinverse {{subsup|''V''|12|+}} of the 5-limit val ''V''<sub>12</sub> for 12 equal is the column matrix {{subsup|''V''|12|T}}/1289; that is, the 1×3 matrix with column {{monzo| 12/1289 19/1289 28/1289 }}. Then {{subsup|''V''|12|+}}''V''<sub>12</sub> is the 3×3 Frobenius projection matrix ''P''<sub>''F''</sub>:
 <math>\displaystyle