Normal forms: Difference between revisions

Line 190:

=== Defactored Hermite form ===

Given a matrix of ''p''-limit intervals, we can find its defactored Hermite form in much the same way as a mapping. The only difference is that ~~the matrix~~ must be antitransposed once at the beginning ~~of the operation~~ and ~~once~~ again at the end. For an example, ~~see~~: [[~~defactoring#Canonical_comma_bases~~]]

Given a matrix of ''p''-limit intervals, we can find its defactored Hermite form in much the same way as a mapping. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed<ref>See a discussion of the antitranspose here: [[Douglas_Blumeyer%27s_RTT_How-To#Null-space]]</ref>once at the beginning, and then antitransposed again at the end.

For example, suppose we have the comma basis for septimal meantone:

<math>

\left[ \begin{array} {rrr}

-4 & 1 \\

4 & 2 \\

-1 & -3 \\

0 & 1 \\

\end{array} \right]

</math>

Note that the interval vectors are columns when put into matrix form like this.

So now we antitranspose, or in other words, transpose the matrix but instead of across its main diagonal (top-left to bottom-right) as with the traditional transpose, across its antidiagonal (top-right to bottom-left).

<math>

\begin{array} {l}

\left[ \begin{array} {rrr}

\colorbox{skyblue}{-4} & \colorbox{yellow}{1} \\

\colorbox{yellow}{4} & \colorbox{pink}{2} \\

\colorbox{springgreen}{-1} & \colorbox{pink}{-3} \\

\colorbox{springgreen}{0} & \colorbox{pink}{1} \\

\end{array} \right]

& → &

\left[ \begin{array} {rrr}

\colorbox{pink}{1} & \colorbox{pink}{-3} & \colorbox{pink}{2} & \colorbox{yellow}{1} \\

\colorbox{springgreen}{0} & \colorbox{springgreen}{-1} & \colorbox{yellow}{4} & \colorbox{skyblue}{-4} \\

\end{array} \right]

\end{array}

</math>

This has the effect of both reversing the entries within each interval, as well as reversing the order of the intervals themselves. The purpose of these reversals is so that when the HNF tries to put all the zeros in the bottom-left corner, it gravitates them toward where we want them: the higher primes, and commas that will be earlier in the list after the second antitranspose<ref>Because these are going to be put into HNF soon, the reversing of the order of the intervals themselves at the beginning is irrelevant. But it is important that the order of the intervals themselves reverses on the way out, in the second antitranspose. And so for simplicity of explanation's sake, we simply say to do an antitranspose at both the beginning and end of the operation.</ref>.

Now we can defactor and HNF this as if it were a mapping.

<math>

\left[ \begin{array} {rrr}

1 & 0 & -10 & -13 \\

0 & 1 & -4 & 4 \\

\end{array} \right]

</math>

Finally we antitranspose again:

<math>

\begin{array} {l}

\left[ \begin{array} {rrr}

\colorbox{pink}{1} & \colorbox{pink}{0} & \colorbox{pink}{-10} & \colorbox{yellow}{-13} \\

\colorbox{springgreen}{0} & \colorbox{springgreen}{1} & \colorbox{yellow}{-4} & \colorbox{skyblue}{4} \\

\end{array} \right]

& → &

\left[ \begin{array} {rrr}

\colorbox{skyblue}{4} & \colorbox{yellow}{13} \\

\colorbox{yellow}{-4} & \colorbox{pink}{-10} \\

\colorbox{springgreen}{1} & \colorbox{pink}{0} \\

\colorbox{springgreen}{0} & \colorbox{pink}{1} \\

\end{array} \right]

\end{array}

</math>

And there's our canonical comma basis.

The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any ''p''-limit group it lives inside. The normalized list contains a minimal set of ratios, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [80/81, 57344/59049]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because [[regular temperament]]s, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal comma list of septimal meantone.

@@ Line 190: / Line 190: @@
 === Defactored Hermite form ===
-Given a matrix of ''p''-limit intervals, we can find its defactored Hermite form in much the same way as a mapping. The only difference is that the matrix must be antitransposed once at the beginning of the operation and once again at the end. For an example, see: [[defactoring#Canonical_comma_bases]]
+Given a matrix of ''p''-limit intervals, we can find its defactored Hermite form in much the same way as a mapping. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed<ref>See a discussion of the antitranspose here: [[Douglas_Blumeyer%27s_RTT_How-To#Null-space]]</ref>once at the beginning, and then antitransposed again at the end.
+For example, suppose we have the comma basis for septimal meantone:
+<math>
+\left[ \begin{array} {rrr}
+-4 & 1 \\
+& 2 \\
+-1 & -3 \\
+& 1 \\
+\end{array} \right]
+</math>
+Note that the interval vectors are columns when put into matrix form like this.
+So now we antitranspose, or in other words, transpose the matrix but instead of across its main diagonal (top-left to bottom-right) as with the traditional transpose, across its antidiagonal (top-right to bottom-left).
+<math>
+\begin{array} {l}
+\left[ \begin{array} {rrr}
+\colorbox{skyblue}{-4} & \colorbox{yellow}{1} \\
+\colorbox{yellow}{4} & \colorbox{pink}{2} \\
+\colorbox{springgreen}{-1} & \colorbox{pink}{-3} \\
+\colorbox{springgreen}{0} & \colorbox{pink}{1} \\
+\end{array} \right]
+& → &
+\left[ \begin{array} {rrr}
+\colorbox{pink}{1} & \colorbox{pink}{-3} & \colorbox{pink}{2} & \colorbox{yellow}{1} \\
+\colorbox{springgreen}{0} & \colorbox{springgreen}{-1} & \colorbox{yellow}{4} & \colorbox{skyblue}{-4} \\
+\end{array} \right]
+\end{array}
+</math>
+This has the effect of both reversing the entries within each interval, as well as reversing the order of the intervals themselves. The purpose of these reversals is so that when the HNF tries to put all the zeros in the bottom-left corner, it gravitates them toward where we want them: the higher primes, and commas that will be earlier in the list after the second antitranspose<ref>Because these are going to be put into HNF soon, the reversing of the order of the intervals themselves at the beginning is irrelevant. But it is important that the order of the intervals themselves reverses on the way out, in the second antitranspose. And so for simplicity of explanation's sake, we simply say to do an antitranspose at both the beginning and end of the operation.</ref>.
+Now we can defactor and HNF this as if it were a mapping.
+<math>
+\left[ \begin{array} {rrr}
+& 0 & -10 & -13 \\
+& 1 & -4 & 4 \\
+\end{array} \right]
+</math>
+Finally we antitranspose again:
+<math>
+\begin{array} {l}
+\left[ \begin{array} {rrr}
+\colorbox{pink}{1} & \colorbox{pink}{0} & \colorbox{pink}{-10} & \colorbox{yellow}{-13} \\
+\colorbox{springgreen}{0} & \colorbox{springgreen}{1} & \colorbox{yellow}{-4} & \colorbox{skyblue}{4} \\
+\end{array} \right]
+& → &
+\left[ \begin{array} {rrr}
+\colorbox{skyblue}{4} & \colorbox{yellow}{13} \\
+\colorbox{yellow}{-4} & \colorbox{pink}{-10} \\
+\colorbox{springgreen}{1} & \colorbox{pink}{0} \\
+\colorbox{springgreen}{0} & \colorbox{pink}{1} \\
+\end{array} \right]
+\end{array}
+</math>
+And there's our canonical comma basis.
 The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any ''p''-limit group it lives inside. The normalized list contains a minimal set of ratios, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [80/81, 57344/59049]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because [[regular temperament]]s, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal comma list of septimal meantone.