Temperament merging: Difference between revisions

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'''Temperament merging''' is a way to find new [[regular temperaments]] by merging others. There are two ways to merge temperaments: '''map-merge~~'''~~, which works by merging the temperaments' [[mapping]]s, and '''comma-merge''', which works by merging the temperaments' [[comma basis|comma bases]].

'''Temperament merging''' is a way to find new [[regular temperaments]] by merging others. There are two ways to merge temperaments: '''joining''' (or map-merge), which works by merging the temperaments' [[mapping]]s, and '''comma-merge''', which works by merging the temperaments' [[comma basis|comma bases]].

== ~~Merging~~ ==

These are multiple ways in which a temperament can be defined in terms of the properties of another temperament.

~~"Merging" in~~ this ~~context refers to concatenating~~ the ~~matrices~~ in ~~question~~ and then [[Temperament merging#Canonicalization|~~canonicalizing~~]] them.

'''Joining''' two temperaments ''a'' and ''b'' (notated a & b) results in a higher-rank temperament which tempers out only the commas that both ''a'' and ''b'' temper out. Usually, this is done with two [[Equal temperament|ETs]] ([[vals]], usually written in wart notation) to receive a rank-2 temperament (sometimes called cross-breeding), and indeed, all possible rank-2 temperaments can be written as a combination of two ETs. The resulting rank-2 essentially captures the similarities between the two ETs: [[15edo|15]] & [[22edo|22]] is [[porcupine]], because both ETs have an [[11/10]] that doubles to [[6/5]] and triples to [[4/3]]. Similarly, [[19edo|19]] & [[26edo|26]] is [[flattone]], because in the diatonic scale of both edos, the [[Major third (interval region)|major third]] is 5/4 and the [[Major sixth#As a diatonic interval category|diminished seventh]] is 7/4. Higher-rank temperaments can also be joined; [[garibaldi]] & [[rodan]] is [[hemifamity]], because both garibaldi and rodan conflate [[81/80]] and [[64/63]] into a single comma-sized interval.

'''Comma-merging''' two temperaments ''a'' and ''b'' (notated a | b) results in a lower-rank temperament which tempers out all of the commas that either ''a'' or ''b'' temper out. This can be done with two rank-2 temperaments to find the equal temperament which [[Support|supports]] them both. For example, [[meantone]] | [[Augmented (temperament)|augmented]] is [[12edo|12-ET]], since 12-ET both has 5/4 as its diatonic major third and has that 5/4 equal to [[3edo|1\3]] of the [[Octave|octave.]]

More than two temperaments may be merged at once. For example, joining three ETs results in a [[rank-3 temperament]] (e.g. 22 & 34d & 37 is [[ares]]).

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.

== With mappings ==

To perform the join with mappings, we vertically concatenate the matrices. In this form, the mapping does represent the temperament (and is the form used in [[Diatonic, chromatic, enharmonic, and subchromatic steps|diatonic, chromatic, enharmonic, and subchromatic]] theory), but to get a more conventional mapping, we can then [[Temperament merging#Canonicalization|canonicalize]] it.

Similarly, to perform the join with comma bases, we horizontally concatenate them, and then canonicalize the result.

~~For mappings, the concatenation is vertical, while for comma-bases, the concatenation is horizontal:~~

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</math>

== ~~Application~~ ==

== With multivals ==

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.

Joining is equivalent to the [[wedge product]], and can be calculated in that manner. Wedging two vals results in the same temperament (in [[wedgie]] form) as joining them does.

~~For discussions of temperament merging in context, see:~~

* [[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#Multiple forms|Multiple mapping forms]]

** [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Temperament merging|Temperament merging]]

* [[Diatonic, chromatic, enharmonic, subchromatic#Chromatic and diatonic interval classes]]

~~== 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 {{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. This is in fact a notation for the [[wedge product]].

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

~~== Multiple temperament merging ==~~

~~More than two temperaments may be merged at a time, such~~ as ~~{{nowrap|22 & 34d & 37}} which gives [[ares]].~~

~~== 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~~.

== Canonicalization ==

@@ Line 1: / Line 1: @@
 {{Beginner|Meet and join}}
-'''Temperament merging''' is a way to find new [[regular temperaments]] by merging others. There are two ways to merge temperaments: '''map-merge''', which works by merging the temperaments' [[mapping]]s, and '''comma-merge''', which works by merging the temperaments' [[comma basis|comma bases]].
+'''Temperament merging''' is a way to find new [[regular temperaments]] by merging others. There are two ways to merge temperaments: '''joining''' (or map-merge), which works by merging the temperaments' [[mapping]]s, and '''comma-merge''', which works by merging the temperaments' [[comma basis|comma bases]].
-== Merging ==
+These are multiple ways in which a temperament can be defined in terms of the properties of another temperament.
-"Merging" in this context refers to concatenating the matrices in question and then [[Temperament merging#Canonicalization|canonicalizing]] them.
+'''Joining''' two temperaments ''a'' and ''b'' (notated a & b) results in a higher-rank temperament which tempers out only the commas that both ''a'' and ''b'' temper out. Usually, this is done with two [[Equal temperament|ETs]] ([[vals]], usually written in wart notation) to receive a rank-2 temperament (sometimes called cross-breeding), and indeed, all possible rank-2 temperaments can be written as a combination of two ETs. The resulting rank-2 essentially captures the similarities between the two ETs: [[15edo|15]] & [[22edo|22]] is [[porcupine]], because both ETs have an [[11/10]] that doubles to [[6/5]] and triples to [[4/3]]. Similarly, [[19edo|19]] & [[26edo|26]] is [[flattone]], because in the diatonic scale of both edos, the [[Major third (interval region)|major third]] is 5/4 and the [[Major sixth#As a diatonic interval category|diminished seventh]] is 7/4. Higher-rank temperaments can also be joined; [[garibaldi]] & [[rodan]] is [[hemifamity]], because both garibaldi and rodan conflate [[81/80]] and [[64/63]] into a single comma-sized interval.
+'''Comma-merging''' two temperaments ''a'' and ''b'' (notated a | b) results in a lower-rank temperament which tempers out all of the commas that either ''a'' or ''b'' temper out. This can be done with two rank-2 temperaments to find the equal temperament which [[Support|supports]] them both. For example, [[meantone]] | [[Augmented (temperament)|augmented]] is [[12edo|12-ET]], since 12-ET both has 5/4 as its diatonic major third and has that 5/4 equal to [[3edo|1\3]] of the [[Octave|octave.]]
+More than two temperaments may be merged at once. For example, joining three ETs results in a [[rank-3 temperament]] (e.g. 22 & 34d & 37 is [[ares]]).
+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.
+== With mappings ==
+To perform the join with mappings, we vertically concatenate the matrices. In this form, the mapping does represent the temperament (and is the form used in [[Diatonic, chromatic, enharmonic, and subchromatic steps|diatonic, chromatic, enharmonic, and subchromatic]] theory), but to get a more conventional mapping, we can then [[Temperament merging#Canonicalization|canonicalize]] it.
+Similarly, to perform the join with comma bases, we horizontally concatenate them, and then canonicalize the result.
-For mappings, the concatenation is vertical, while for comma-bases, the concatenation is horizontal:
@@ Line 71: / Line 82: @@
 </math>
-== Application ==
+== With multivals ==
-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.
+Joining is equivalent to the [[wedge product]], and can be calculated in that manner. Wedging two vals results in the same temperament (in [[wedgie]] form) as joining them does.
-For discussions of temperament merging in context, see:
-* [[Dave Keenan &amp; Douglas Blumeyer's guide to RTT]]
-** [[Dave Keenan &amp; Douglas Blumeyer's guide to RTT/Mappings#Mappings|Mappings]]
-** [[Dave Keenan &amp; Douglas Blumeyer's guide to RTT/Mappings#Multiple forms|Multiple mapping forms]]
-** [[Dave Keenan &amp; Douglas Blumeyer's guide to RTT/Exploring temperaments#Temperament merging|Temperament merging]]
-* [[Diatonic, chromatic, enharmonic, subchromatic#Chromatic and diatonic interval classes]]
-== Notation ==
-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. This is in fact a notation for the [[wedge product]].
-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 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 {{nowrap|12 &amp; 19 {{=}} meantone}}.
-== Multiple temperament merging ==
-More than two temperaments may be merged at a time, such as {{nowrap|22 &amp; 34d &amp; 37}} which gives [[ares]].
-== 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.
 == Canonicalization ==