Tenney–Euclidean tuning: Difference between revisions

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: ''For practical help, see [[POTE tuning]].''

The val for 5-limit 12et is {{nowrap|''V''12 {{=}} {{val| 12 19 28 }}}}. In weighted coordinates, that becomes {{nowrap|(''V''12)''W'' {{=}} {{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 {{nowrap|{{subsup|(''V''12)|''W''|+}} ~ ~~{{!(}}~~{{monzo| 0.027706 0.027677 0.027842 }}~~{{)!}}~~}}. Then {{nowrap|''P'' {{=}} {{subsup|(''V''12)|''W''|+}}(''V''12)''W''}} is a projection matrix that maps onto the one-dimensional subspace whose single basis vector is (''V''12)''W''. We find that {{nowrap|(''V''12)''W''''P'' {{=}} (''V''12)''W''}}; 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.

The val for 5-limit 12et is {{nowrap|''V''12 {{=}} {{val| 12 19 28 }}}}. In weighted coordinates, that becomes {{nowrap|(''V''12)''W'' {{=}} {{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 {{nowrap|{{subsup|(''V''12)|''W''|+}} ~ [{{monzo| 0.027706 0.027677 0.027842 }}]}}. Then {{nowrap|''P'' {{=}} {{subsup|(''V''12)|''W''|+}}(''V''12)''W''}} is a projection matrix that maps onto the one-dimensional subspace whose single basis vector is (''V''12)''W''. We find that {{nowrap|(''V''12)''W''''P'' {{=}} (''V''12)''W''}}; 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 [[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

@@ Line 95: / Line 95: @@
 : ''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.
+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.
 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