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Tenney–Euclidean tuning: Difference between revisions

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Move properties of pseudoinverse to the pseudoinverse page. Set straight the weightedness of projection matrices
→Examples: this section should be mainly about tuning a temp, rather than projection matrices
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: ''For practical help, see [[POTE tuning]].''  
: ''For practical help, see [[POTE tuning]].''  


The val for 5-limit 12et is {{nowrap|''V''<sub>12</sub> {{=}} {{val| 12 19 28 }}}}. In weighted coordinates, that becomes {{nowrap|(''V''<sub>12</sub>)<sub>''W''</sub> {{=}} {{val| 12 19/log<sub>2</sub>3 28/log<sub>2</sub>5 }}}} ~&nbsp;{{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 {{nowrap|{{subsup|(''V''<sub>12</sub>)|''W''|+}} ~ [{{monzo| 0.027706 0.027677 0.027842 }}]}}. Then {{nowrap|''P'' {{=}} {{subsup|(''V''<sub>12</sub>)|''W''|+}}(''V''<sub>12</sub>)<sub>''W''</sub>}} is a projection matrix that maps onto the one-dimensional subspace whose single basis vector is (''V''<sub>12</sub>)<sub>''W''</sub>. We find that {{nowrap|(''V''<sub>12</sub>)<sub>''W''</sub>''P'' {{=}} (''V''<sub>12</sub>)<sub>''W''</sub>}}; 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 4log<sub>2</sub>3 -log<sub>2</sub>5 }} and multiply (either side, the matrix is symmetric) by ''P'', we get the zero vector, corresponding to the unison.
=== Optimizing a temperament ===
Consider [[pajara]], the 7-limit temperament tempering out 50/49 and 64/63. Its mapping ''V'' is


Now consider [[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 2×4 matrix with rows equal to the vals for 12, and 22; in weighted coordinates this would be
<math>\displaystyle
V = \begin{bmatrix}
2 & 0 & 11 & 12 \\
0 & 1 & -2 & -2
\end{bmatrix}
</math>
 
This can be found by taking two equal temperaments supporting pajara (e.g. [[12edo]] and [[22edo]]), defining a 2×4 matrix formed by the vals and then canonicalizing it. In weighted coordinates this would be


<math>\displaystyle
<math>\displaystyle
V_W =
V_W =  
\begin{bmatrix}
\begin{bmatrix}
12.000 & 11.988 & 12.059 & 12.111 \\
2.000000 & 0.000000 & 4.737442 & 4.274486 \\
22.000 & 22.083 & 21.965 & 22.085
0.000000 & 0.630930 & -0.861353 & -0.712414
\end{bmatrix}
\end{bmatrix}
</math>
</math>
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V_W^+ =
V_W^+ =
\begin{bmatrix}
\begin{bmatrix}
-1.81029 & 1.00052 \\
0.143942 & 0.622569 \\
-6.68250 & 3.66285 \\
0.196399 & 1.232405 \\
4.83496 & -2.63063 \\
0.072833 & -0.207801 \\
3.67652 & -1.99757
0.085876 & -0.060988
\end{bmatrix}
\end{bmatrix}
</math>
</math>


which we may also write as [{{monzo| -1.81029 -6.68250 4.83496 3.67652 }}, {{monzo| 1.00052 3.66285 -2.63063 -1.99757 }}]
so we obtain the generator tuning map {{nowrap|''G'' {{=}} ''J''<sub>''W''</sub>{{subsup|''V''|''W''|+}} }}:
 
<math>\displaystyle
G = \begin{bmatrix} 0.499049 & 1.586185 \end{bmatrix}
</math>
 
and the tuning map {{nowrap| ''T'' {{=}} ''GV'' }} which shows the value each prime harmonic is tuned to:
 
<math>\displaystyle
T = \begin{bmatrix} 0.998099 & 1.586185 & 2.317174 & 2.816223 \end{bmatrix}
</math>
 
The results are in octaves. Multiply them by 1200 to obtain the tuning in cents.  


=== Using a Frobenius projection matrix ===
{{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.
{{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.


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