Temperament merging: Difference between revisions

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== 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 {{nowrap|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 | ("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 {{nowrap|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 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>.

The ampersand 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 ==

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 {{nowrap|12 & 19 {{=}} meantone}}.

== 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 {{nowrap|22 & 34d & 37}} which gives [[ares]].

== Non-uniqueness ==

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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 column-rank-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:

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 column-rank-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:

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=== 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 include—for example on [http://x31eq.com/temper/ Graham Breed's temperament finding tool]. This allows for things like {{nowrap|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 ==

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== 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: [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT#Temperament merging]].

== Cross-domain temperament merging ==

@@ Line 82: / Line 82: @@
 == 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 &amp; ("ampersand") symbol is used (for example, on [http://x31eq.com/temper/ Graham Breed's temperament finding tool]) to notate map-merging, as in {{nowrap|12 &amp; 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 | ("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 {{nowrap|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 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>.
+The ampersand 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 ==
-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 {{nowrap|12 &amp; 19 {{=}} meantone}}.
 == 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 {{nowrap|22 &amp; 34d &amp; 37}} which gives [[ares]].
 == Non-uniqueness ==
@@ Line 142: / Line 142: @@
-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 column-rank-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:
+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 column-rank-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:
@@ Line 207: / Line 207: @@
 === 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 &amp; operator, the [[defactoring]] part of canonicalization is not include—for example on [http://x31eq.com/temper/ Graham Breed's temperament finding tool]. This allows for things like {{nowrap|5 &amp; 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 ==
@@ Line 229: / Line 229: @@
 == 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: [[Dave Keenan &amp; Douglas Blumeyer's guide to EA for RTT#Temperament merging]].
 == Cross-domain temperament merging ==