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

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=== null-space ===

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 meet of any two of these commas will capture the same exact null-space of {{vector|{{map|19 30 44}}}}.

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 meet of any two of these particular 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 mapping-row {{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 meet of ''two'' commas (at least in the 5-limit; more on this later). In this way, the comma bases which represent the meet of two commas are greater than the sum of their individual parts.

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 === null-space ===
-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 meet of any two of these commas will capture the same exact null-space of {{vector|{{map|19 30 44}}}}.
+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 meet of any two of these particular 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 mapping-row {{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 meet of ''two'' commas (at least in the 5-limit; more on this later). In this way, the comma bases which represent the meet of two commas are greater than the sum of their individual parts.