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 {{monzo|{{val|19 30 44}}}}.

We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{val|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 these ~~matrices~~, the ~~intersections~~ of two ~~of them~~ at ~~once~~, are ~~another level~~ of ~~specialness. We can do even better with them. There’s actually a mathematical function which when input any one of these matrices will output {{monzo|{{val|19 30 44}}}}, thus demonstrating~~ their ~~equivalence with respect to it~~.

We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{val|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.

~~Remember that~~ any one of these matrices ~~is a basis for~~ the null-space of {{monzo|{{val|19 30 44}}}}~~. So we are~~ in effect undoing ~~the effects of~~ the null-space ~~function~~. ~~Interestingly~~ enough, as you'll soon see, ~~it's about~~ the same ~~process~~ to ~~find~~ the null-space as it is to undo it. Working this out by hand goes like this (it is a standard linear algebra operation, so if you're comfortable with it already, you can skip this and other similar parts of these materials):

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 {{monzo|{{val|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 {{monzo|{{val|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.

Working this out by hand goes like this (it is a standard linear algebra operation, so if you're comfortable with it already, you can skip this and other similar parts of these materials):

First, transpose the matrix. That means the first column becomes the first row, the second column becomes the second row, etc.

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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 {{monzo|{{val|19 30 44}}}}.
-We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{val|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 these matrices, the intersections of two of them at once, are another level of specialness. We can do even better with them. There’s actually a mathematical function which when input any one of these matrices will output {{monzo|{{val|19 30 44}}}}, thus demonstrating their equivalence with respect to it.
+We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{val|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.
-Remember that any one of these matrices is a basis for the null-space of {{monzo|{{val|19 30 44}}}}. So we are in effect undoing the effects of the null-space function. Interestingly enough, as you'll soon see, it's about the same process to find the null-space as it is to undo it. Working this out by hand goes like this (it is a standard linear algebra operation, so if you're comfortable with it already, you can skip this and other similar parts of these materials):
+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 {{monzo|{{val|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 {{monzo|{{val|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.
+Working this out by hand goes like this (it is a standard linear algebra operation, so if you're comfortable with it already, you can skip this and other similar parts of these materials):
 First, transpose the matrix. That means the first column becomes the first row, the second column becomes the second row, etc.