Tenney–Euclidean tuning: Difference between revisions

{{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'':

Multiplying {{val| 12 19 28 }} by ''P''_''F'' gives {{val| 12 19 28 }} again. Multiplying the monzos for 81/80, 128/125, 648/625 etc. gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, {{val| 19 30 44 }}, by P_F gives {{val| 24360 38570 56840 }}/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is {{monzo| -1 1 0 }}; times F gives the fractional monzo corresponding to (2⁸⁴ 3¹³³ 5¹⁹⁶)^1/1289, which equates to 698.121 cents, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of P_F, which is to say by {{monzo| 1 0 0 }}P_F, of 1196.778 cents.

We can do the same thing with a matrix ''V'' with rows consisting of the vals for 7-limit 12 and 22 equal; then ''V''{{+}}''V'', the Frobenius projection matrix for pajara, is

P_\text{F} = \frac{1}{305}