Douglas Blumeyer's RTT How-To: Difference between revisions

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There’s nothing special about the pairing of meantone and magic. We could have chosen meantone|hanson, or magic|negri, etc. A matrix formed out of the intersection of any two of these commas will capture the same exact null-space of {{vector|{{map|19 30 44}}}}.

We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{map|19 30 44}}; we learned this bit in the very first section: all we have to do is make sure that the comma maps to zero steps in this ET. But that's not a special relationship between 19-ET and any of these commas ''individually''; each of these commas are tempered out by many different ETs, not just 19-ET. The special relationship 19-ET has is to a null-space which can be expressed in basis form as the intersection of ''two'' commas (at least in the 5-limit; more on this later). In this way, the comma ~~basis matrices~~ which represent the intersections of two commas are greater than the sum of their individual parts.

We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{map|19 30 44}}; we learned this bit in the very first section: all we have to do is make sure that the comma maps to zero steps in this ET. But that's not a special relationship between 19-ET and any of these commas ''individually''; each of these commas are tempered out by many different ETs, not just 19-ET. The special relationship 19-ET has is to a null-space which can be expressed in basis form as the intersection of ''two'' commas (at least in the 5-limit; more on this later). In this way, the comma bases which represent the intersections of two commas are greater than the sum of their individual parts.

We can confirm the relationship between an ET and its null-space by converting back and forth between them. There exists a mathematical function which — when input any one of these comma ~~basis matrices~~ — will output {{vector|{{map|19 30 44}}}}, thus demonstrating the various bases' equivalence with respect to it. If the operation called "taking the null-space" is what gets you from {{vector|{{map|19 30 44}}}} to one basis for the null-space, then ''this'' mathematical function is in effect ''undoing'' the null-space operation.

We can confirm the relationship between an ET and its null-space by converting back and forth between them. There exists a mathematical function which — when input any one of these comma bases — will output {{vector|{{map|19 30 44}}}}, thus demonstrating the various bases' equivalence with respect to it. If the operation called "taking the null-space" is what gets you from {{vector|{{map|19 30 44}}}} to one basis for the null-space, then ''this'' mathematical function is in effect ''undoing'' the null-space operation.

And interestingly enough, as you'll soon see, the process is almost the same to take the null-space as it is to undo it.

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To understand why, we have to cover a few key points:

# Just as a vector is the dual of a covector, we also have a '''multivector''' which is the dual of a multicovector. Analogously, we call the thing the multivector represents a '''multicomma'''.

# We can calculate a multicomma from a comma basis ~~matrix~~ much in the same way we can calculate a multimap from a mapping matrix

# We can calculate a multicomma from a comma basis much in the same way we can calculate a multimap from a mapping matrix

# We can convert between multimaps and multicommas using an operation called “taking the '''complement'''”<ref>Elsewhere on the wiki you may find the complement operation called "taking [[the dual]]", or even the dual of a multimap being called simply "the dual". In these materials, I am using the dual to refer to the general case, while the specific case of the dual of a multimap is a multicomma and the operation to get from one of these to its dual is called taking the complement (whereas to get to the dual of a mapping, which is a comma basis, the operation is called taking the null-space).</ref><ref>You may also sometimes see "Hodge dual" used where you'd expect to see the complement operation. The Hodge star operation, or Hodge dual operation, is not another name for the complement operation. It is a linear algebra operation which works as a limited substitute for the exterior algebra operation. The limitation is that it only works when the rank is 2. This is because when rank is 2, bicovectors can be represented as skew-symmetric matrices (see: https://en.wikipedia.org/wiki/Bivector#Matrices), which gives you access to some extra linear algebra utilities such as Hodge star.</ref>, which basically involves reversing the order of terms and changing the signs of some of them.

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 There’s nothing special about the pairing of meantone and magic. We could have chosen meantone|hanson, or magic|negri, etc. A matrix formed out of the intersection of any two of these commas will capture the same exact null-space of {{vector|{{map|19 30 44}}}}.
-We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{map|19 30 44}}; we learned this bit in the very first section: all we have to do is make sure that the comma maps to zero steps in this ET. But that's not a special relationship between 19-ET and any of these commas ''individually''; each of these commas are tempered out by many different ETs, not just 19-ET. The special relationship 19-ET has is to a null-space which can be expressed in basis form as the intersection of ''two'' commas (at least in the 5-limit; more on this later). In this way, the comma basis matrices which represent the intersections of two commas are greater than the sum of their individual parts.
+We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{map|19 30 44}}; we learned this bit in the very first section: all we have to do is make sure that the comma maps to zero steps in this ET. But that's not a special relationship between 19-ET and any of these commas ''individually''; each of these commas are tempered out by many different ETs, not just 19-ET. The special relationship 19-ET has is to a null-space which can be expressed in basis form as the intersection of ''two'' commas (at least in the 5-limit; more on this later). In this way, the comma bases which represent the intersections of two commas are greater than the sum of their individual parts.
-We can confirm the relationship between an ET and its null-space by converting back and forth between them. There exists a mathematical function which — when input any one of these comma basis matrices — will output {{vector|{{map|19 30 44}}}}, thus demonstrating the various bases' equivalence with respect to it. If the operation called "taking the null-space" is what gets you from {{vector|{{map|19 30 44}}}} to one basis for the null-space, then ''this'' mathematical function is in effect ''undoing'' the null-space operation.
+We can confirm the relationship between an ET and its null-space by converting back and forth between them. There exists a mathematical function which — when input any one of these comma bases — will output {{vector|{{map|19 30 44}}}}, thus demonstrating the various bases' equivalence with respect to it. If the operation called "taking the null-space" is what gets you from {{vector|{{map|19 30 44}}}} to one basis for the null-space, then ''this'' mathematical function is in effect ''undoing'' the null-space operation.
 And interestingly enough, as you'll soon see, the process is almost the same to take the null-space as it is to undo it.
@@ Line 1,247: / Line 1,247: @@
 To understand why, we have to cover a few key points:
 # Just as a vector is the dual of a covector, we also have a '''multivector''' which is the dual of a multicovector. Analogously, we call the thing the multivector represents a '''multicomma'''.
-# We can calculate a multicomma from a comma basis matrix much in the same way we can calculate a multimap from a mapping matrix
+# We can calculate a multicomma from a comma basis much in the same way we can calculate a multimap from a mapping matrix
 # We can convert between multimaps and multicommas using an operation called “taking the '''complement'''”<ref>Elsewhere on the wiki you may find the complement operation called "taking [[the dual]]", or even the dual of a multimap being called simply "the dual". In these materials, I am using the dual to refer to the general case, while the specific case of the dual of a multimap is a multicomma and the operation to get from one of these to its dual is called taking the complement (whereas to get to the dual of a mapping, which is a comma basis, the operation is called taking the null-space).</ref><ref>You may also sometimes see "Hodge dual" used where you'd expect to see the complement operation. The Hodge star operation, or Hodge dual operation, is not another name for the complement operation. It is a linear algebra operation which works as a limited substitute for the exterior algebra operation. The limitation is that it only works when the rank is 2. This is because when rank is 2, bicovectors can be represented as skew-symmetric matrices (see: https://en.wikipedia.org/wiki/Bivector#Matrices), which gives you access to some extra linear algebra utilities such as Hodge star.</ref>, which basically involves reversing the order of terms and changing the signs of some of them.