Tenney–Euclidean tuning: Difference between revisions

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We may also obtain the TE tuning from a projection matrix. {{nowrap|''P'' {{=}} ''V''''W''{{mpp}}''V''''W''}} is the orthogonal projection matrix that maps onto the space spanned by the rows of V. This space corresponds to the temperament, and so does ''P''. However, ''P'' is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether [[contorsion]] has been removed. The tuning map giving the tuning of each prime number is found by multiplying by the JIP: {{nowrap|''T'' {{=}} ''JP''}} where ''J'' is the JIP, which is the nearest point in the subspace corresponding to the temperament to ''J''.

We may find the same projection matrix starting from a list of weighted monzos rather than vals. If ''M''''W'' is a rank-''n'' matrix whose columns are weighted monzos, and ''I'' is the ''n''×''n'' identity matrix, then {{nowrap|''P'' {{=}} ''I'' − ''M''''W''{{subsup|''M''|''W''|+}} is the same projection matrix as ''V''~~~~''W''~~{{mpp~~}}''V''''W'' so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.

We may find the same projection matrix starting from a list of weighted monzos rather than vals. If ''M''''W'' is a rank-''n'' matrix whose columns are weighted monzos, and ''I'' is the ''n''×''n'' identity matrix, then {{nowrap|''P'' {{=}} ''I'' − ''M''''W''{{subsup|''M''|''W''|+}}}} is the same projection matrix as {{subsup|''V''|''W''|+}}''V''''W'' so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.

== Enforcement ==

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 We may also obtain the TE tuning from a projection matrix. {{nowrap|''P'' {{=}} ''V''<sub>''W''</sub>{{mpp}}''V''<sub>''W''</sub>}} is the orthogonal projection matrix that maps onto the space spanned by the rows of V. This space corresponds to the temperament, and so does ''P''. However, ''P'' is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether [[contorsion]] has been removed. The tuning map giving the tuning of each prime number is found by multiplying by the JIP: {{nowrap|''T'' {{=}} ''JP''}} where ''J'' is the JIP, which is the nearest point in the subspace corresponding to the temperament to ''J''.
-We may find the same projection matrix starting from a list of weighted monzos rather than vals. If ''M''<sub>''W''</sub> is a rank-''n'' matrix whose columns are weighted monzos, and ''I'' is the ''n''×''n'' identity matrix, then {{nowrap|''P'' {{=}} ''I'' − ''M''<sub>''W''</sub>{{subsup|''M''|''W''|+}} is the same projection matrix as ''V''<sub>''W''</sub>{{mpp}}''V''<sub>''W''</sub> so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.
+We may find the same projection matrix starting from a list of weighted monzos rather than vals. If ''M''<sub>''W''</sub> is a rank-''n'' matrix whose columns are weighted monzos, and ''I'' is the ''n''×''n'' identity matrix, then {{nowrap|''P'' {{=}} ''I'' − ''M''<sub>''W''</sub>{{subsup|''M''|''W''|+}}}} is the same projection matrix as {{subsup|''V''|''W''|+}}''V''<sub>''W''</sub> so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.
 == Enforcement ==