Temperament merging: Difference between revisions
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== Application == | == Application == | ||
Map-merging produces a temperament that | Map-merging produces a temperament that ''only'' makes to [[vanish]] those commas that are made to vanish by ''all'' of the input temperaments. Conversely, comma-merging produces a temperament that makes to vanish ''every'' comma made to vanish by ''any'' of the input temperaments. | ||
== Notation == | == Notation == | ||
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The | ("pipe") symbol may be used to notate comma-merging, as in meantone|porcupine = 7. We could read this as "meantone or porcupine" or "meantone comma-merge porcupine is 7-ET". As a mnemonic, because commas are represented by vectors, which are vertical columns, when they merge together into matrices, the pipe resembles the seam between them as they merge. | The | ("pipe") symbol may be used to notate comma-merging, as in meantone|porcupine = 7. We could read this as "meantone or porcupine" or "meantone comma-merge porcupine is 7-ET". As a mnemonic, because commas are represented by vectors, which are vertical columns, when they merge together into matrices, the pipe resembles the seam between them as they merge. | ||
The & symbol is associated with the word "and", and in many programming languages, the | symbol is associated with the word "or". So a further mnemonic can be used to remember this pair of symbols: <math>𝓣_1 \& 𝓣_2</math> is the merge that results in the temperament that | The & symbol is associated with the word "and", and in many programming languages, the | symbol is associated with the word "or". So a further mnemonic can be used to remember this pair of symbols: <math>𝓣_1 \& 𝓣_2</math> is the merge that results in the temperament that makes the commas vanish which are made to vanish by ''both'' <math>𝓣_1</math> ''and'' <math>𝓣_2</math>, and <math>𝓣_1 | 𝓣_2</math> is the merge that results in the temperament that makes the commas vanish which are made to vanish by ''either'' <math>𝓣_1</math> ''or'' <math>𝓣_2</math>. | ||
== Cross-breeding == | == Cross-breeding == | ||
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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 | 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. | ||
Once we fully [[canonical form|canonicalize]], though, the all-zero row(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: | ||