Temperament merging: Difference between revisions
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== Merging == | == Merging == | ||
"Merging" in this context refers to concatenating the matrices in question and then [[Temperament merging#Canonicalization|canonicalizing]] them. | "Merging" in this context refers to concatenating the matrices in question and then [[Temperament merging#Canonicalization|canonicalizing]] them. | ||
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== Application == | == Application == | ||
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. | 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. | ||
For discussions of temperament merging in context, see: | For discussions of temperament merging in context, see: | ||
* Dave Keenan & Douglas Blumeyer's guide to RTT | * [[Dave Keenan & Douglas Blumeyer's guide to RTT]] | ||
** [[Dave Keenan & Douglas Blumeyer's guide to RTT: mappings#Mappings|mappings]] | ** [[Dave Keenan & Douglas Blumeyer's guide to RTT: mappings#Mappings|mappings]] | ||
** [[Dave Keenan & Douglas Blumeyer's guide to RTT: mappings#Multiple forms|multiple mapping forms]] | ** [[Dave Keenan & Douglas Blumeyer's guide to RTT: mappings#Multiple forms|multiple mapping forms]] | ||
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== Notation == | == Notation == | ||
The & ("ampersand") symbol is used (for example, on [http://x31eq.com/temper/ Graham Breed's temperament finding tool]) to notate map-merging, as in 12&19 = meantone; we can read this as "12-ET and 19-ET is meantone" or "12-ET map-merge 19-ET is meantone". Here, 12 and 19 are [[wart notation]] for 12-ET and 19-ET. | The & ("ampersand") symbol is used (for example, on [http://x31eq.com/temper/ Graham Breed's temperament finding tool]) to notate map-merging, as in 12&19 = meantone; we can read this as "12-ET and 19-ET is meantone" or "12-ET map-merge 19-ET is meantone". Here, 12 and 19 are [[wart notation]] for 12-ET and 19-ET. | ||
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== Cross-breeding == | == Cross-breeding == | ||
Perhaps the most basic example of temperament merging is map-merging [[equal temperament]]s (ETs), which is sometimes called "cross-breeding". And so meantone could be said to be a cross-breed of 12-ET and 19-ET, because 12&19 = meantone. | Perhaps the most basic example of temperament merging is map-merging [[equal temperament]]s (ETs), which is sometimes called "cross-breeding". And so meantone could be said to be a cross-breed of 12-ET and 19-ET, because 12&19 = meantone. | ||
== Multiple temperament merging == | == Multiple temperament merging == | ||
More than two temperaments may be merged at a time, such as 22&34d&37 to give [[ares]]. | More than two temperaments may be merged at a time, such as 22&34d&37 to give [[ares]]. | ||
== Non-uniqueness == | == Non-uniqueness == | ||
Note that while a given temperament merging expression unambiguously refers to a single temperament, a given temperament can be expressed by many possible different temperament merging expressions. | Note that while a given temperament merging expression unambiguously refers to a single temperament, a given temperament can be expressed by many possible different temperament merging expressions. | ||
== Canonicalization == | == Canonicalization == | ||
The canonicalization step is important for eliminating any redundancies that may have been introduced by merging related temperaments, such as [[rank-deficient|rank-deficiencies]] or [[enfactoring]]. | The canonicalization step is important for eliminating any redundancies that may have been introduced by merging related temperaments, such as [[rank-deficient|rank-deficiencies]] or [[enfactoring]]. | ||
=== Rank-deficiencies === | === Rank-deficiencies === | ||
Sometimes when temperaments are merged, rank-deficiencies may occur. For example, comma-merging septimal meantone and miracle temperaments: | Sometimes when temperaments are merged, rank-deficiencies may occur. For example, comma-merging septimal meantone and miracle temperaments: | ||
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\left[ \begin{array} {r|r|r|r} | \left[ \begin{array} {r|r|r|r} | ||
\style{background-color:#F2B2B4;padding:5px}{0} & -49 & -45 & -36 \\ | \style{background-color:#F2B2B4;padding:5px}{0} & -49 & -45 & -36 \\ | ||
\style{background-color:#F2B2B4;padding:5px}{0} & 31 & 27 & 21 \\ | \style{background-color:#F2B2B4;padding:5px}{0} & 31 & 27 & 21 \\ | ||
\style{background-color:#F2B2B4;padding:5px}{0} & 0 & 1 & 0 \\ | \style{background-color:#F2B2B4;padding:5px}{0} & 0 & 1 & 0 \\ | ||
\style{background-color:#F2B2B4;padding:5px}{0} & 0 & 0 & 1 \\ | \style{background-color:#F2B2B4;padding:5px}{0} & 0 & 0 & 1 \\ | ||
\end{array} \right] | \end{array} \right] | ||
</math> | </math> | ||
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<math> | <math> | ||
\left[ \begin{array} {r|r|r} | \left[ \begin{array} {r|r|r} | ||
-49 & -45 & -36 \\ | -49 & -45 & -36 \\ | ||
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0 & 0 & 1 \\ | 0 & 0 & 1 \\ | ||
\end{array} \right] | \end{array} \right] | ||
</math> | </math> | ||
=== Enfactoring === | === Enfactoring === | ||
Sometimes when temperaments are merged, enfactoring may occur. For example: | Sometimes when temperaments are merged, enfactoring may occur. For example: | ||
<math> | <math> | ||
\begin{array} {ccc} | \begin{array} {ccc} | ||
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19 & 30 & 44 \\ | 19 & 30 & 44 \\ | ||
\end{array} \right] \\ | \end{array} \right] \\ | ||
\end{array} | \end{array} | ||
</math> | </math> | ||
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<math> | <math> | ||
\left[ \begin{array} {rrr} | \left[ \begin{array} {rrr} | ||
1 & 0 & -4 \\ | 1 & 0 & -4 \\ | ||
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<math> | <math> | ||
\left[ \begin{array} {rrr} | \left[ \begin{array} {rrr} | ||
1 & 0 & -4 \\ | 1 & 0 & -4 \\ | ||
0 & 1 & 4 \\ | 0 & 1 & 4 \\ | ||
\end{array} \right] | \end{array} \right] | ||
</math> | </math> | ||
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=== Non-canonicalizing definition === | === Non-canonicalizing definition === | ||
By some definitions of the & operator, the [[defactoring]] part of canonicalization is not included — for example on [http://x31eq.com/temper/ Graham Breed's temperament finding tool]. This allows for things like 5&19 to represent 2-enfactored meantone, rather than meantone itself. Instead of a full canonicalization, then, this definition merely puts the result into Hermite normal form and removes any all-zero rows or columns resulting from rank-deficiencies. | By some definitions of the & operator, the [[defactoring]] part of canonicalization is not included — for example on [http://x31eq.com/temper/ Graham Breed's temperament finding tool]. This allows for things like 5&19 to represent 2-enfactored meantone, rather than meantone itself. Instead of a full canonicalization, then, this definition merely puts the result into Hermite normal form and removes any all-zero rows or columns resulting from rank-deficiencies. | ||
== Parallel intersections == | == Parallel intersections == | ||
Every temperament mapping has a dual comma basis, and every comma basis has a dual mapping. Because of this duality, a special parallelism exists. | Every temperament mapping has a dual comma basis, and every comma basis has a dual mapping. Because of this duality, a special parallelism exists. | ||
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== Example system of temperaments related by merging == | == Example system of temperaments related by merging == | ||
Here we have a group of temperaments that are related by merges. Moving up in this diagram corresponds with map-merges, and downward movement corresponds with comma-merges. Temperaments lower on the chart [[support]] ones higher on the chart. | Here we have a group of temperaments that are related by merges. Moving up in this diagram corresponds with map-merges, and downward movement corresponds with comma-merges. Temperaments lower on the chart [[support]] ones higher on the chart. | ||
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== Vs. the wedge product == | == Vs. the wedge product == | ||
Temperament merging is closely related to the wedge product. For more information, see: [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Temperament merging]]. | Temperament merging is closely related to the wedge product. For more information, see: [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Temperament merging]]. | ||
== Cross-domain temperament merging == | == Cross-domain temperament merging == | ||
It is possible to merge temperaments from different domains. For more information, see [[Cross-domain temperament merging]]. | It is possible to merge temperaments from different domains. For more information, see [[Cross-domain temperament merging]]. | ||
== Wolfram implementation == | == Wolfram implementation == | ||
Temperament merging has been implemented as the functions <code>mapMerge</code> and <code>commaMerge</code> in the [[RTT library in Wolfram Language]]. | Temperament merging has been implemented as the functions <code>mapMerge</code> and <code>commaMerge</code> in the [[RTT library in Wolfram Language]]. | ||