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|(''V''12)~~~~''W''~~{{mpp~~}} ~ {{!(}}{{monzo| 0.027706 0.027677 0.027842 }}{{)!}}}}. Then {{nowrap|''P'' {{=}} (''V''12)~~~~''W''~~{{mpp~~}}(''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

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which we may also write as [{{monzo| -1.81029 -6.68250 4.83496 3.67652 }}, {{monzo| 1.00052 3.66285 -2.63063 -1.99757 }}]

{{nowrap|''P'' {{=}} ''V''~~~~''W''~~{{mpp~~}}''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.

{{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'':

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 : ''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|(''V''<sub>12</sub>)<sub>''W''</sub>{{mpp}} ~ {{!(}}{{monzo| 0.027706 0.027677 0.027842 }}{{)!}}}}. Then {{nowrap|''P'' {{=}} (''V''<sub>12</sub>)<sub>''W''</sub>{{mpp}}(''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
@@ Line 121: / Line 121: @@
 which we may also write as [{{monzo| -1.81029 -6.68250 4.83496 3.67652 }}, {{monzo| 1.00052 3.66285 -2.63063 -1.99757 }}]
-{{nowrap|''P'' {{=}} ''V''<sub>''W''</sub>{{mpp}}''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.
 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>: