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

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And there’s our {{map|{{vector|4 -4 1}}}}. Feel free to try reversing the operation by working out the mapping-row basis from this if you like. And/or you could try working out that {{map|{{vector|4 -4 1}}}} is a basis for the null-space of any other combination of ETs we found that could specify meantone, such as 7&12, or 12&19.

And there’s our {{map|{{vector|4 -4 1}}}}<ref>Interestingly, the other two columns in the bottom half of this matrix are valuable too. They tell you prime count vectors that would work for your generators. In this case, the two vectors are {{vector|1 0 0}} and {{vector|-8 5 0}}, so that tells you that the octave and the [[diesis]] could generate meantone.</ref>. Feel free to try reversing the operation by working out the mapping-row basis from this if you like. And/or you could try working out that {{map|{{vector|4 -4 1}}}} is a basis for the null-space of any other combination of ETs we found that could specify meantone, such as 7&12, or 12&19.

It’s worth noting that, just as 2 commas were exactly enough to define a rank-1 temperament, though there were an infinitude of equivalent pairs of commas we could choose to fill that role, there’s a similar thing happening here, where 2 maps are exactly enough to define a rank-2 temperament, but an infinitude of equivalent pairs of them. We can even see that we can convert between these maps using Gauss-Jordan addition and subtraction, just like we could manipulate commas to get from one to the other. For example, the map for 12-ET {{map|12 19 28}} is exactly what you get from summing the terms of 5-ET {{map|5 8 12}} and 7-ET {{map|7 11 16}}: {{map|5+7 8+11 12+16}} = {{map|12 19 28}}. Cool!

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-And there’s our {{map|{{vector|4 -4 1}}}}. Feel free to try reversing the operation by working out the mapping-row basis from this if you like. And/or you could try working out that {{map|{{vector|4 -4 1}}}} is a basis for the null-space of any other combination of ETs we found that could specify meantone, such as 7&12, or 12&19.
+And there’s our {{map|{{vector|4 -4 1}}}}<ref>Interestingly, the other two columns in the bottom half of this matrix are valuable too. They tell you prime count vectors that would work for your generators. In this case, the two vectors are {{vector|1 0 0}} and {{vector|-8 5 0}}, so that tells you that the octave and the [[diesis]] could generate meantone.</ref>. Feel free to try reversing the operation by working out the mapping-row basis from this if you like. And/or you could try working out that {{map|{{vector|4 -4 1}}}} is a basis for the null-space of any other combination of ETs we found that could specify meantone, such as 7&12, or 12&19.
 It’s worth noting that, just as 2 commas were exactly enough to define a rank-1 temperament, though there were an infinitude of equivalent pairs of commas we could choose to fill that role, there’s a similar thing happening here, where 2 maps are exactly enough to define a rank-2 temperament, but an infinitude of equivalent pairs of them. We can even see that we can convert between these maps using Gauss-Jordan addition and subtraction, just like we could manipulate commas to get from one to the other. For example, the map for 12-ET {{map|12 19 28}} is exactly what you get from summing the terms of 5-ET {{map|5 8 12}} and 7-ET {{map|7 11 16}}: {{map|5+7 8+11 12+16}} = {{map|12 19 28}}. Cool!