Temperament merging - Revision history

FloraC: Update the terminology. Style

2026-06-07T06:48:19Z

Update the terminology. Style

← Older revision		Revision as of 06:48, 7 June 2026
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	These are multiple ways in which a temperament can be defined in terms of the properties of another temperament.		These are multiple ways in which a temperament can be defined in terms of the properties of another temperament.

	'''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.		'''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]]s ([[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 equal temperaments. The resulting rank-2 essentially captures the similarities between the two equal temperaments: [[15edo\|15]] & [[22edo\|22]] is [[porcupine]], because both equal temperaments 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]] 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 [[aberschismic]], 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.]]		'''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]]s them both. For example, [[meantone]] \| [[augmented (temperament)\|augmented]] is [[12edo\|12et]], since 12et both has 5/4 as its diatonic major third and has that 5/4 equal to [[3edo\|1\3]] of the [[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]]).		More than two temperaments may be merged at once. For example, joining three equal temperaments 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.		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 ==		== 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.		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.		Similarly, to perform the join with comma bases, we horizontally concatenate them, and then canonicalize the result.



	<math>		<math>
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	=== 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:


	<math>		<math>
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	</math>		</math>

			We have not ''completely'' canonicalized yet; we did not 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 does not 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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	=== Enfactoring ===		=== Enfactoring ===
	Sometimes when temperaments are merged, enfactoring may occur. For example:		Sometimes when temperaments are merged, enfactoring may occur. For example:


	<math>		<math>
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	</math>		</math>

			The greatest factor of this matrix is 2, because we can produce the row {{val\| 24 38 56 }} as a coprime linear combination of its rows (that's {{val\| 5 8 12 }} + {{val\| 19 30 44 }}), and the entries of this row have a GCD of 2, so in other words this matrix is 2-enfactored. If we merely put it into Hermite normal form, we receive:
	The greatest factor of this matrix is 2, because we can produce the row {{~~map~~\|24 38 56}} as a coprime linear combination of its rows (that's {{~~map~~\|5 8 12}} + {{~~map~~\|19 30 44}}), and the entries of this row have a GCD of 2, so in other words this matrix is 2-enfactored. If we merely put it into Hermite normal form, we receive:


	<math>		<math>
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	</math>		</math>


	which is a 2-enfactored meantone mapping, and it reveals the greatest factor as the GCD of the second row. But if we fully canonicalize it (defactor, and put into normal form), then we get:		which is a 2-enfactored meantone mapping, and it reveals the greatest factor as the GCD of the second row. But if we fully canonicalize it (defactor, and put into normal form), then we get:


	<math>		<math>
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	\end{array} \right]		\end{array} \right]
	</math>		</math>


	which is simply the canonical mapping for meantone temperament.		which is simply the canonical mapping for meantone temperament.

	=== Non-canonicalizing definition ===		=== Non-canonicalizing definition ===
	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.		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 {{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 ==		== Parallel intersections ==

VectorGraphics at 19:07, 22 August 2025

2025-08-22T19:07:48Z

@@ 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.
@@ 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.
 == Canonicalization ==

VectorGraphics at 01:24, 22 August 2025

2025-08-22T01:24:11Z

← Older revision		Revision as of 01:24, 22 August 2025
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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 {{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 & ("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 \| ("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.
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	[[File:Temperament merging 7-limit example.png\|1000px\|frameless\|center]]		[[File:Temperament merging 7-limit example.png\|1000px\|frameless\|center]]

	~~== Vs. the wedge product ==~~
	~~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 ==		== Cross-domain temperament merging ==

ArrowHead294 at 13:23, 20 March 2025

2025-03-20T13:23:38Z

← Older revision		Revision as of 13:23, 20 March 2025
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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 {{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 ==		== 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 ==		== 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 ==		== 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 ===		=== 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 ==		== Parallel intersections ==
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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: [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT#Temperament merging]].

	== Cross-domain temperament merging ==		== Cross-domain temperament merging ==

ArrowHead294: /* Application */

2025-03-20T13:20:18Z

Application

← Older revision		Revision as of 13:20, 20 March 2025
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	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]]
	** [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Temperament merging\|Temperament merging]]		** [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Temperament merging\|Temperament merging]]
	* [[Diatonic, ~~Chromatic~~, ~~Enharmonic~~, ~~Subchromatic~~#Chromatic and ~~Diatonic Interval Classes~~]]		* [[Diatonic, chromatic, enharmonic, subchromatic#Chromatic and diatonic interval classes]]

	== Notation ==		== Notation ==

ArrowHead294: /* Application */

2024-10-27T01:32:55Z

Application

← Older revision		Revision as of 01:32, 27 October 2024
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	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]]
	** [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: exploring~~ temperaments#Temperament merging\|~~temperament~~ merging]]		** [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Temperament merging\|Temperament merging]]
	* [[Diatonic, Chromatic, Enharmonic, Subchromatic#Chromatic and Diatonic Interval Classes]]		* [[Diatonic, Chromatic, Enharmonic, Subchromatic#Chromatic and Diatonic Interval Classes]]

ArrowHead294 at 19:38, 30 September 2024

2024-09-30T19:38:25Z

Show changes

ArrowHead294: /* Merging */ Use matrix instead of array

2024-09-30T19:36:48Z

Merging: Use matrix instead of array

← Older revision		Revision as of 19:36, 30 September 2024
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	<math>		<math>

	\hspace{1cm}		\hspace{1cm}

	\begin{array} {ccc}		\begin{array} {ccc}
			\left[ \begin{matrix}
	\left[ \begin{~~array} {rrr~~}
	12 & 19 & 28 \\		12 & 19 & 28 \\
	\end{~~array~~} \right] \\		\end{matrix} \right] \\

	\text{map-merge} \\		\text{map-merge} \\

	\left[ \begin{~~array} {rrr~~}		\left[ \begin{matrix}
	19 & 30 & 44 \\		19 & 30 & 44 \\
	\end{~~array~~} \right] \\		\end{matrix} \right] \\

	↓ \\		↓ \\

	\left[ \begin{~~array} {rrr~~}		\left[ \begin{matrix}
	12 & 19 & 28 \\		12 & 19 & 28 \\
	19 & 30 & 44 \\		19 & 30 & 44 \\
	\end{~~array~~} \right] \\		\end{matrix} \right] \\

	\text{which canonicalizes to} \\		\text{which canonicalizes to} \\

	\left[ \begin{~~array} {rrr~~}		\left[ \begin{matrix}
	1 & 0 & -4 \\		1 & 0 & -4 \\
	0 & 1 & 4 \\		0 & 1 & 4 \\
	\end{~~array~~} \right] \\		\end{matrix} \right] \\

	\end{array}		\end{array}
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	\hspace{1cm}		\hspace{1cm}

	\left[ \begin{~~array} {rrr~~}		\left[ \begin{matrix}
	-4 \\		-4 \\
	4 \\		4 \\
	-1 \\		-1 \\
	\end{~~array~~} \right]		\end{matrix} \right]

	\text{comma-merge}		\text{comma-merge}

	\left[ \begin{~~array} {rrr~~}		\left[ \begin{matrix}
	7 \\		7 \\
	0 \\		0 \\
	-3 \\		-3 \\
	\end{~~array~~} \right]		\end{matrix} \right]

	→		→

	\left[ \begin{~~array} {rrr~~}		\left[ \begin{matrix}
	-4 & 7 \\		-4 & 7 \\
	4 & 0 \\		4 & 0 \\
	-1 & -3 \\		-1 & -3 \\
	\end{~~array~~} \right]		\end{matrix} \right]

	\text{which canonicalizes to}		\text{which canonicalizes to}

	\left[ \begin{~~array} {rrr~~}		\left[ \begin{matrix}
	-19 & -15 \\		-19 & -15 \\
	12 & 8 \\		12 & 8 \\
	0 & 1 \\		0 & 1 \\
	\end{~~array~~} \right]		\end{matrix} \right]

	</math>		</math>


	== Application ==		== Application ==

Fredg999: Beginner template

2023-07-08T19:02:34Z

Beginner template

← Older revision		Revision as of 19:02, 8 July 2023
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	~~This page gives a general introduction to this concept; for a more mathematical take on this, see [[~~Meet and join~~]].~~		{{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: '''map-merge''', which works by merging the temperaments' [[mapping]]s, and '''comma-merge''', which works by merging the temperaments' [[comma basis\|comma bases]].

Cmloegcmluin: \colorbox → \style

2023-01-16T19:07:32Z

\colorbox → \style

← Older revision		Revision as of 19:07, 16 January 2023
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	\left[ \begin{array} {r\|r\|r\|r}		\left[ \begin{array} {r\|r\|r\|r}

	\~~colorbox~~{~~pink~~}0 & -49 & -45 & -36 \\		\style{background-color:#F2B2B4;padding:5px}{0} & -49 & -45 & -36 \\
	\~~colorbox~~{~~pink~~}0 & 31 & 27 & 21 \\		\style{background-color:#F2B2B4;padding:5px}{0} & 31 & 27 & 21 \\
	\~~colorbox~~{~~pink~~}0 & 0 & 1 & 0 \\		\style{background-color:#F2B2B4;padding:5px}{0} & 0 & 1 & 0 \\
	\~~colorbox~~{~~pink~~}0 & 0 & 0 & 1 \\		\style{background-color:#F2B2B4;padding:5px}{0} & 0 & 0 & 1 \\

	\end{array} \right]		\end{array} \right]