Temperament merging: Difference between revisions
Cmloegcmluin (talk | contribs) GCF back to GCD, for better distention from "greatest factor", and it's more popular anyway |
Cmloegcmluin (talk | contribs) →Grade-deficiencies: add highlighting for extra clarity |
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\left[ \begin{array} {rrr} | \left[ \begin{array} {rrr} | ||
0 & -24 & -5 & -23 \\ | \colorbox{pink}0 & -24 & -5 & -23 \\ | ||
0 & 29 & 4 & 26 \\ | \colorbox{pink}0 & 29 & 4 & 26 \\ | ||
0 & 0 & 3 & 2 \\ | \colorbox{pink}0 & 0 & 3 & 2 \\ | ||
0 & 0 & 0 & 1 \\ | \colorbox{pink}0 & 0 & 0 & 1 \\ | ||
\end{array} \right] | \end{array} \right] | ||
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Note that we've only ''normalized'' here so far, that is, put it into [[Hermite normal form]]; we've done this to illustrate that one of the vectors is now entirely zeros. This means that the matrix was nullity-deficient, or in layperson's terms, contained redundant commas. In other words, these two temperaments tempered out some of the same commas, 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. | Note that we've only ''normalized'' here so far, that is, 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 tempered out some of the same commas, 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. | ||
Once we fully [[canonical form|canonicalize]], though, the all- | Once we fully [[canonical form|canonicalize]], though, the all-zero row(s) are removed, and we end up with: | ||