Temperament merging: Difference between revisions

Line 138:

\end{array} \right]

\text{which in ~~normal~~ form is}

\text{which in canonical form* is}

\left[ \begin{array} {r|r|r|r}

\colorbox{pink}0 & -24 & -5 & -23 \\

\colorbox{pink}0 & -49 & -45 & -36 \\

\colorbox{pink}0 & 29 & 4 & 26 \\

\colorbox{pink}0 & 31 & 27 & 21 \\

\colorbox{pink}0 & 0 & 3 & 2 \\

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

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

Line 152:

~~Note that we~~'~~ve only put it into [[Hermite normal form]]~~; we'~~ve done this to illustrate that one of~~ the ~~vectors is now entirely zeros~~ (highlighted in red). ~~This means~~ that ~~the~~ matrix was nullity-deficient, or in layperson's terms, contained redundant commas. In other words, these two temperaments make some of the same commas vanish, and so when we merged ~~them, even~~ though the input temperaments required 2 vectors each to ~~represent, their~~ merged result doesn't require all 4 vectors; it can be completely represented using only 3.

We haven't ''completely'' canonicalized yet; we didn't remove the all-zero column (highlighted in red) that was created by the [[Hermite normal form]] step. The existence of any all-zero columns like this tells us that our matrix was nullity-deficient, or in layperson's terms, that it contained redundant commas. In other words, these two temperaments make some of the same commas vanish, and so when we merged them — even though the input temperaments required 2 vectors each to represent — their merged result doesn't require all 4 vectors; it can be completely represented using only 3 vectors. So once we fully [[canonical form|canonicalize]], any all-zero column(s) are removed, and we end up with:

~~Once~~ we fully [[canonical form|canonicalize]], ~~though, the~~ all-zero ~~row~~(s) are removed, and we end up with:

@@ Line 138: / Line 138: @@
 \end{array} \right]
-\text{which in normal form is}
+\text{which in canonical form* is}
 \left[ \begin{array} {r|r|r|r}
-\colorbox{pink}0 & -24 & -5 & -23 \\
+\colorbox{pink}0 & -49 & -45 & -36 \\
-\colorbox{pink}0 & 29 & 4 & 26 \\
+\colorbox{pink}0 & 31 & 27 & 21 \\
-\colorbox{pink}0 & 0 & 3 & 2 \\
+\colorbox{pink}0 & 0 & 1 & 0 \\
 \colorbox{pink}0 & 0 & 0 & 1 \\
@@ Line 152: / Line 152: @@
-Note that we've only put it into [[Hermite normal form]]; we've done this to illustrate that one of the vectors is now entirely zeros (highlighted in red). This means that the matrix was nullity-deficient, or in layperson's terms, contained redundant commas. In other words, these two temperaments make some of the same commas vanish, and so when we merged them, even though the input temperaments required 2 vectors each to represent, their merged result doesn't require all 4 vectors; it can be completely represented using only 3.
+We haven't ''completely'' canonicalized yet; we didn't remove the all-zero column (highlighted in red) that was created by the [[Hermite normal form]] step. The existence of any all-zero columns like this tells us that our matrix was nullity-deficient, or in layperson's terms, that it contained redundant commas. In other words, these two temperaments make some of the same commas vanish, and so when we merged them — even though the input temperaments required 2 vectors each to represent — their merged result doesn't require all 4 vectors; it can be completely represented using only 3 vectors. So once we fully [[canonical form|canonicalize]], any all-zero column(s) are removed, and we end up with:
-Once we fully [[canonical form|canonicalize]], though, the all-zero row(s) are removed, and we end up with: