Tenney–Euclidean metrics: Difference between revisions

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== Octave equivalent TE seminorm ==

Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with ~~rows~~ of monzos spanning the commas of a regular temperament, then M = BW-1 is the corresponding weighted matrix. Q = M+M is a projection matrix dual to P = I - Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are ~~therefor~~ linearly independent, then P = I - MT~~(MM~~T)-1M = I - W-1BT~~(BW~~-2BT)-1BW-1, and ~~mPm~~T = bW-1PW-1bT~~, or b~~(W-2 - W-2BT~~(BW~~-2BT)-1BW-2)b~~T~~, so that the terms inside the parenthesis define a formula for '''P''' in terms of the matrix of monzos B.

Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with columns of monzos spanning the commas of a regular temperament, then M = W-1B is the corresponding weighted matrix. Q = MM+ is a projection matrix dual to P = I - Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefore linearly independent, then P = I - M(MTM)-1MT = I - W-1B(BTW-2B)-1BTW-1, and mTPm = bTW-1PW-1b, or bT(W-2 - W-2B(BTW-2B)-1BTW-2)b, so that the terms inside the parenthesis define a formula for '''P''' in terms of the matrix of monzos B.

To define the '''octave equivalent Tenney-Euclidean seminorm''', or '''OETES''', we simply add a ~~row~~ {{monzo|1 0 0 … 0 }} representing 2 to the matrix B. An alternative ~~proceedure~~ is to find the [[Normal lists #Normal val list|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given ''p''-limit rational interval in terms of the ''p''-limit regular temperament given by A.

To define the '''octave equivalent Tenney-Euclidean seminorm''', or '''OETES''', we simply add a column {{monzo|1 0 0 … 0 }} representing 2 to the matrix B. An alternative procedure is to find the [[Normal lists #Normal val list|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given ''p''-limit rational interval in terms of the ''p''-limit regular temperament given by A.

== TE logflat badness ==

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 == Octave equivalent TE seminorm ==
-Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with rows of monzos spanning the commas of a regular temperament, then M = BW<sup>-1</sup> is the corresponding weighted matrix. Q = M<sup>+</sup>M is a projection matrix dual to P = I - Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefor linearly independent, then P = I - M<sup>T</sup>(MM<sup>T</sup>)<sup>-1</sup>M = I - W<sup>-1</sup>B<sup>T</sup>(BW<sup>-2</sup>B<sup>T</sup>)<sup>-1</sup>BW<sup>-1</sup>, and mPm<sup>T</sup> = bW<sup>-1</sup>PW<sup>-1</sup>b<sup>T</sup>, or b(W<sup>-2</sup> - W<sup>-2</sup>B<sup>T</sup>(BW<sup>-2</sup>B<sup>T</sup>)<sup>-1</sup>BW<sup>-2</sup>)b<sup>T</sup>, so that the terms inside the parenthesis define a formula for '''P''' in terms of the matrix of monzos B.
+Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with columns of monzos spanning the commas of a regular temperament, then M = W<sup>-1</sup>B is the corresponding weighted matrix. Q = MM<sup>+</sup> is a projection matrix dual to P = I - Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefore linearly independent, then P = I - M(M<sup>T</sup>M)<sup>-1</sup>M<sup>T</sup> = I - W<sup>-1</sup>B(B<sup>T</sup>W<sup>-2</sup>B)<sup>-1</sup>B<sup>T</sup>W<sup>-1</sup>, and m<sup>T</sup>Pm = b<sup>T</sup>W<sup>-1</sup>PW<sup>-1</sup>b, or b<sup>T</sup>(W<sup>-2</sup> - W<sup>-2</sup>B(B<sup>T</sup>W<sup>-2</sup>B)<sup>-1</sup>B<sup>T</sup>W<sup>-2</sup>)b, so that the terms inside the parenthesis define a formula for '''P''' in terms of the matrix of monzos B.
-To define the '''octave equivalent Tenney-Euclidean seminorm''', or '''OETES''', we simply add a row {{monzo|1 0 0 … 0 }} representing 2 to the matrix B. An alternative proceedure is to find the [[Normal lists #Normal val list|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given ''p''-limit rational interval in terms of the ''p''-limit regular temperament given by A.
+To define the '''octave equivalent Tenney-Euclidean seminorm''', or '''OETES''', we simply add a column {{monzo|1 0 0 … 0 }} representing 2 to the matrix B. An alternative procedure is to find the [[Normal lists #Normal val list|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given ''p''-limit rational interval in terms of the ''p''-limit regular temperament given by A.
 == TE logflat badness ==