Cross-domain temperament merging - Revision history

ArrowHead294: Update links and references

2024-11-02T18:48:45Z

Update links and references

Cmloegcmluin: /* Map-merge */

2024-06-17T08:17:44Z

Map-merge

← Older revision		Revision as of 08:17, 17 June 2024
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	Now we take our mapping <math>M_1</math> and right-multiply it by this <math>B_{1∩2↔1}</math>, just like we would right-multiply it by a list of vectors:		Now we take our mapping <math>M_1</math> and right-multiply it by this <math>B_{1∩2↔1}</math> , just like we would right-multiply it by a list of vectors:


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	Now we find our other <math>B</math>, the one for 17-ET, i.e. <math>B_{2↔1∩2}</math>:		Now we find our other <math>B</math>, the one for 17-ET, i.e. <math>B_{2↔1∩2}</math> :

Cmloegcmluin: /* Map-merge */

2024-06-17T08:16:55Z

Map-merge

← Older revision		Revision as of 08:16, 17 June 2024
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	Canonicalize (HNF, remove all-zero columns, back to number list form, and make super). So that tells us that the intersected domain basis <math>B_1∩B_2</math> is 2.9.11.		Canonicalize (HNF, remove all-zero columns, back to number list form, and make super). So that tells us that the intersected domain basis <math>B_1∩B_2</math> is 2.9.11.

	Now we change the domain basis for each mapping to <math>B_1∩B_2</math>. Let's do 22-ET first. First we need to find our basis change matrix <math>B_{1↔1∩2}</math>:		Now we change the domain basis for each mapping to <math>B_1∩B_2</math>. Let's do 22-ET first. First we need to find our basis change matrix <math>B_{1↔1∩2}</math> :

Cmloegcmluin: /* Changing basis */

2024-06-17T01:12:30Z

Changing basis

← Older revision		Revision as of 01:12, 17 June 2024
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	Elsewhere, these types of matrices have been called [[subgroup basis matrices]], but that terminology will not be used here, for the same reasons as are described in the last section of this article (here: [[Domain basis#Terminology: domain basis vs. subgroup]]).		Elsewhere, these types of matrices have been called [[subgroup basis matrices]], but that terminology will not be used here, for the same reasons as are described in the last section of this article (here: [[Domain basis#Terminology: domain basis vs. subgroup]]).

	As discussed earlier, only certain domain basis changes are possible: (here: [[Cross-domain temperament merging#Application: determining whether it is possible to change the domain]]). To quickly recap here, it is only possible to change between domains where one is a subspace of the other. So when we say a given basis change matrix works like a two-way bridge, there's a more specific way to say what we mean: a basis change matrix allows us to change either ''from the subspace to the superspace'', or ''from the superspace to the subspace''. Which direction we go just depends on which side we enter the bridge from: the right side or the left side.		As discussed earlier, only certain domain basis changes are possible (here: [[Cross-domain temperament merging#Application: determining whether it is possible to change the domain]]). To quickly recap here, it is only possible to change between domains where one is a subspace of the other. So when we say a given basis change matrix works like a two-way bridge, there's a more specific way to say what we mean: a basis change matrix allows us to change either ''from the subspace to the superspace'', or ''from the superspace to the subspace''. Which direction we go just depends on which side we enter the bridge from: the right side or the left side.

	== Constructing a basis change matrix ==		== Constructing a basis change matrix ==

Cmloegcmluin: /* Changing basis */

2024-06-17T01:11:45Z

Changing basis

← Older revision		Revision as of 01:11, 17 June 2024
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	Given an interval, comma basis, or mapping — anything that has an associated domain basis — it is possible to change it from one domain basis to another. We can accomplish this using a '''basis change matrix''', an object that works like a two-way bridge between two domain bases.		Given an interval, comma basis, or mapping — anything that has an associated domain basis — it is possible to change it from one domain basis to another. We can accomplish this using a '''basis change matrix''', an object that works like a two-way bridge between two domain bases.

	In fact, there is no real difference between a ''basis matrix'',' such as we've been using to represent nonstandard domain bases in terms of the primes, and a basis ''change'' matrix. We can think of the basis matrix as a basis change matrix, from whatever domain ''to the simplest prime-only basis''. And so we use the same symbol for both of these objects, <math>B</math>. When the change is from a basis to the simplest prime-only basis, the subscript on <math>B</math> can simply be the basis being described; if however, the basis change matrix is from one nonprime basis to another nonprime basis, such as 2.49/3.5 to 2.7/3.5, then the subscript should contain the names of both bases, ideally with a "↔" symbol between them, and the superspace on the left and the subspace on the right, as that corresponds to their positions as labels on the matrix.		In fact, there is no real difference between a ''basis matrix'', such as we've been using to represent nonstandard domain bases in terms of the primes, and a basis ''change'' matrix. We can think of the basis matrix as a basis change matrix, from whatever domain ''to the simplest prime-only basis''. And so we use the same symbol for both of these objects, <math>B</math>. When the change is from a basis to the simplest prime-only basis, the subscript on <math>B</math> can simply be the basis being described; if however, the basis change matrix is from one nonprime basis to another nonprime basis, such as 2.49/3.5 to 2.7/3.5, then the subscript should contain the names of both bases, ideally with a "↔" symbol between them, and the superspace on the left and the subspace on the right, as that corresponds to their positions as labels on the matrix.

	Elsewhere, these types of matrices have been called [[subgroup basis matrices]], but that terminology will not be used here, for the same reasons as are described in the last section of this article (here: [[Domain basis#Terminology: domain basis vs. subgroup]]).		Elsewhere, these types of matrices have been called [[subgroup basis matrices]], but that terminology will not be used here, for the same reasons as are described in the last section of this article (here: [[Domain basis#Terminology: domain basis vs. subgroup]]).

Cmloegcmluin: /* Application: determining whether it is possible to change the domain */

2024-06-17T00:32:44Z

Application: determining whether it is possible to change the domain

← Older revision		Revision as of 00:32, 17 June 2024
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	So from these two opposing bulleted facts above, we can conclude that for any pair of domains where neither one is a subspace of the other, there would be no way for us to express ''either intervals or maps'' from one in the other. And that's why we could say that they're incomparable domains.		So from these two opposing bulleted facts above, we can conclude that for any pair of domains where neither one is a subspace of the other, there would be no way for us to express ''either intervals or maps'' from one in the other. And that's why we could say that they're incomparable domains.

	Notably, there is still a way to get intervals to a subspace, and maps to a superspace, but it's indirect. To do so, take the [[dual]] of your object, then change basis, then take the [[dual]] again. So for a mapping you would use the nullspace function to convert it to its corresponding comma basis, change domain basis, and then use the nullspace function to convert it back to its corresponding mapping on the other side.		Notably, there is still a way to get intervals to a subspace, and maps to a superspace, but it's indirect. To do so, [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_exploring_temperaments#Nullspace\|take the dual]] of your object, then change basis, then take the dual again. So for a mapping you would use the nullspace function to convert it to its corresponding comma basis, change domain basis, and then use the nullspace function to convert it back to its corresponding mapping on the other side.

	== General method to determine whether a domain is a subspace of another ==		== General method to determine whether a domain is a subspace of another ==

Cmloegcmluin: /* Application: determining whether it is possible to change the domain */

2024-06-17T00:30:12Z

Application: determining whether it is possible to change the domain

← Older revision		Revision as of 00:30, 17 June 2024
Line 63:		Line 63:
	So from these two opposing bulleted facts above, we can conclude that for any pair of domains where neither one is a subspace of the other, there would be no way for us to express ''either intervals or maps'' from one in the other. And that's why we could say that they're incomparable domains.		So from these two opposing bulleted facts above, we can conclude that for any pair of domains where neither one is a subspace of the other, there would be no way for us to express ''either intervals or maps'' from one in the other. And that's why we could say that they're incomparable domains.

	Notably, there is still a way to get intervals to a subspace, and maps to a superspace, but it's indirect. To do so, take the [[dual]] of your object, then change basis, then take the [[dual]] again. So for a mapping you would use the nullspace function to convert it to its corresponding comma basis, change domain basis, and then use the ~~antinullspace~~ function to convert it back to its corresponding mapping on the other side.		Notably, there is still a way to get intervals to a subspace, and maps to a superspace, but it's indirect. To do so, take the [[dual]] of your object, then change basis, then take the [[dual]] again. So for a mapping you would use the nullspace function to convert it to its corresponding comma basis, change domain basis, and then use the nullspace function to convert it back to its corresponding mapping on the other side.

	== General method to determine whether a domain is a subspace of another ==		== General method to determine whether a domain is a subspace of another ==

Cmloegcmluin: \colorbox -> \style

2023-01-16T19:06:39Z

\colorbox -> \style

← Older revision		Revision as of 19:06, 16 January 2023
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	<math>		<math>
	\left[ \begin{array} {rrrrr}		\left[ \begin{array} {rrrrr}
	1 & 0 & 0 & \~~colorbox~~{~~pink~~}0 & \~~colorbox~~{~~pink~~}0 \\		1 & 0 & 0 & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{0} \\
	0 & 1 & 0 & \~~colorbox~~{~~pink~~}0 & \~~colorbox~~{~~pink~~}0 \\		0 & 1 & 0 & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{0} \\
	0 & 0 & 1 & \~~colorbox~~{~~pink~~}0 & \~~colorbox~~{~~pink~~}0 \\		0 & 0 & 1 & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{0} \\
	\hline		\hline
	0 & 0 & 0 & \~~colorbox~~{~~yellow~~}1 & \~~colorbox~~{~~yellow~~}0 \\		0 & 0 & 0 & \style{background-color:#FFF200;padding:5px}{1} & \style{background-color:#FFF200;padding:5px}{0} \\
	0 & 1 & 0 & \~~colorbox~~{~~yellow~~}0 & \~~colorbox~~{~~yellow~~}2 \\		0 & 1 & 0 & \style{background-color:#FFF200;padding:5px}{0} & \style{background-color:#FFF200;padding:5px}{2} \\
	0 & -1 & 0 & \~~colorbox~~{~~yellow~~}0 & \~~colorbox~~{~~yellow~~}{-2} \\		0 & -1 & 0 & \style{background-color:#FFF200;padding:5px}{0} & \style{background-color:#FFF200;padding:5px}{{-2}} \\
	\end{array} \right]		\end{array} \right]
	</math>		</math>
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	<math>		<math>
	\left[ \begin{array} {rrrrrrrr}		\left[ \begin{array} {rrrrrrrr}
	1 & 0 & 0 & 0 & 0 & \~~colorbox~~{~~pink~~}0 & \~~colorbox~~{~~pink~~}0 & \~~colorbox~~{~~pink~~}0 \\		1 & 0 & 0 & 0 & 0 & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{0} \\
	0 & 1 & 0 & 0 & 0 & \~~colorbox~~{~~pink~~}0 & \~~colorbox~~{~~pink~~}0 & \~~colorbox~~{~~pink~~}0 \\		0 & 1 & 0 & 0 & 0 & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{0} \\
	0 & 0 & 1 & 0 & 0 & \~~colorbox~~{~~pink~~}0 & \~~colorbox~~{~~pink~~}0 & \~~colorbox~~{~~pink~~}0 \\		0 & 0 & 1 & 0 & 0 & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{0} \\
	0 & 0 & 0 & 1 & 0 & \~~colorbox~~{~~pink~~}0 & \~~colorbox~~{~~pink~~}0 & \~~colorbox~~{~~pink~~}0 \\		0 & 0 & 0 & 1 & 0 & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{0} \\
	0 & 0 & 0 & 0 & 1 & \~~colorbox~~{~~pink~~}0 & \~~colorbox~~{~~pink~~}0 & \~~colorbox~~{~~pink~~}0 \\		0 & 0 & 0 & 0 & 1 & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{0} \\
	0 & 0 & 0 & 0 & 0 & \~~colorbox~~{~~yellow~~}1 & \~~colorbox~~{~~yellow~~}0 & \~~colorbox~~{~~yellow~~}0 \\		0 & 0 & 0 & 0 & 0 & \style{background-color:#FFF200;padding:5px}{1} & \style{background-color:#FFF200;padding:5px}{0} & \style{background-color:#FFF200;padding:5px}{0} \\
	0 & 1 & 0 & 0 & 0 & \~~colorbox~~{~~yellow~~}0 & \~~colorbox~~{~~yellow~~}2 & \~~colorbox~~{~~yellow~~}0 \\		0 & 1 & 0 & 0 & 0 & \style{background-color:#FFF200;padding:5px}{0} & \style{background-color:#FFF200;padding:5px}{2} & \style{background-color:#FFF200;padding:5px}{0} \\
	0 & 0 & 1 & 0 & 0 & \~~colorbox~~{~~yellow~~}0 & \~~colorbox~~{~~yellow~~}0 & \~~colorbox~~{~~yellow~~}0 \\		0 & 0 & 1 & 0 & 0 & \style{background-color:#FFF200;padding:5px}{0} & \style{background-color:#FFF200;padding:5px}{0} & \style{background-color:#FFF200;padding:5px}{0} \\
	0 & 0 & 0 & 0 & 0 & \~~colorbox~~{~~yellow~~}0 & \~~colorbox~~{~~yellow~~}0 & \~~colorbox~~{~~yellow~~}0 \\		0 & 0 & 0 & 0 & 0 & \style{background-color:#FFF200;padding:5px}{0} & \style{background-color:#FFF200;padding:5px}{0} & \style{background-color:#FFF200;padding:5px}{0} \\
	0 & 0 & 0 & 0 & 0 & \~~colorbox~~{~~yellow~~}0 & \~~colorbox~~{~~yellow~~}0 & \~~colorbox~~{~~yellow~~}1 \\		0 & 0 & 0 & 0 & 0 & \style{background-color:#FFF200;padding:5px}{0} & \style{background-color:#FFF200;padding:5px}{0} & \style{background-color:#FFF200;padding:5px}{1} \\
	\end{array} \right]		\end{array} \right]
	</math>		</math>

Cmloegcmluin: temper out → vanish

2022-12-27T18:16:58Z

temper out → vanish

← Older revision		Revision as of 18:16, 27 December 2022
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	<blockquote style="background-color: #fbfbfb; border: 1px solid #eeeeee; padding: 10px 20px;">		<blockquote style="background-color: #fbfbfb; border: 1px solid #eeeeee; padding: 10px 20px;">
	Map-merging produces a temperament that ~~tempers out~~ ''only'' the commas that are ~~tempered out~~ by ''all'' of the input temperaments.		Map-merging produces a temperament that makes to [[vanish]] ''only'' the commas that are made to vanish by ''all'' of the input temperaments.
	<br>		<br>
	Conversely, comma-merging produces a temperament that ~~tempers out~~ ''every'' comma ~~tempered out~~ by ''any'' of the input temperaments.		Conversely, comma-merging produces a temperament that makes to vanish ''every'' comma that is made to vanish by ''any'' of the input temperaments.
	</blockquote>		</blockquote>

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	''For a comma-merge'', because the output temperament deals with every comma, then its domain basis must be capable of supporting this: specifically, it must include every [[basis element]] from any of the input temperaments' domain bases. Think of it this way: for any given temperament, its domain basis elements are the building blocks for its commas, and so in order to express every comma in the merged temperament, we will need every input temperament's building blocks gathered in one place. In other words, we must find the merge of all the input domain bases.		''For a comma-merge'', because the output temperament deals with every comma, then its domain basis must be capable of supporting this: specifically, it must include every [[basis element]] from any of the input temperaments' domain bases. Think of it this way: for any given temperament, its domain basis elements are the building blocks for its commas, and so in order to express every comma in the merged temperament, we will need every input temperament's building blocks gathered in one place. In other words, we must find the merge of all the input domain bases.

	''For a map-merge'', then, because the output temperament will deal only with tempered commas shared by every input temperament, then its domain basis only needs to include the basis elements that are present in all of the input domain bases. Here's why: a comma built using a basis element that isn't shared by all input temperaments couldn't even be ''built'' in all input temperaments, let alone ~~tempered out~~ by all of them. And so, to build the set of commas that are ~~tempered out~~ by all temperaments, we only need the building blocks that can be found in all of them. In mathematical terms, we must find the intersection of the input domain bases.		''For a map-merge'', then, because the output temperament will deal only with tempered commas shared by every input temperament, then its domain basis only needs to include the basis elements that are present in all of the input domain bases. Here's why: a comma built using a basis element that isn't shared by all input temperaments couldn't even be ''built'' in all input temperaments, let alone made to vanish by all of them. And so, to build the set of commas that are made to vanish by all temperaments, we only need the building blocks that can be found in all of them. In mathematical terms, we must find the intersection of the input domain bases.

	<gallery heights=450px widths=450px>		<gallery heights=450px widths=450px>

Cmloegcmluin: update link

2022-12-15T00:15:26Z

update link

← Older revision		Revision as of 00:15, 15 December 2022
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	# Canonicalize.		# Canonicalize.

	The reason this works is that wherever the corresponding top-half columns are all zeros, this was achieved through linear combinations of vectors from both domain bases, which means the information below them represents vectors that are in both of them. In other words, if <math>(x, x) + (y, 0) = (0, z)</math> and <math>x</math> is in <math>B_1</math> and <math>y</math> is in <math>B_2</math>, then we must have <math>x + y = 0</math> and <math>z = x</math><ref>credit this explanation to Tom Price on Discord</ref>. We're sort of abusing HNF as a way to solve a system, kind of like [[Douglas Blumeyer's RTT ~~How-To~~#Nullspace\|when we calculate the nullspace]]<ref>Credit this explanation to Sintel on Discord</ref>.		The reason this works is that wherever the corresponding top-half columns are all zeros, this was achieved through linear combinations of vectors from both domain bases, which means the information below them represents vectors that are in both of them. In other words, if <math>(x, x) + (y, 0) = (0, z)</math> and <math>x</math> is in <math>B_1</math> and <math>y</math> is in <math>B_2</math>, then we must have <math>x + y = 0</math> and <math>z = x</math><ref>credit this explanation to Tom Price on Discord</ref>. We're sort of abusing HNF as a way to solve a system, kind of like [[Dave Keenan & Douglas Blumeyer's guide to RTT: exploring temperaments#Nullspace\|when we calculate the nullspace]]<ref>Credit this explanation to Sintel on Discord</ref>.

	=== But first: a gentle introduction ===		=== But first: a gentle introduction ===