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

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A good way to explain why we don’t need three of these commas is that if you had three of them, you could use any two of them to create the third, and then subtract the result from the third, turning that comma into a zero vector, or a vector with only zeroes, which is pretty useless, so we could just discard it.

And a potentially helpful way to think about why any other interval arrived at through linear combinations of the commas in a basis would also be a valid column in the basis is this: any of these interval vectors, by definition, is mapped to zero steps by the mapping. So any combination of them will also map to zero steps, and thus be a comma that is tempered out by the temperament.

When written with the {{val|}} notation, we’re expressing maps in “covector” form, or in other words, as the opposite of vectors. But we can also think of maps in terms of matrices. If vectors are like matrix columns, maps are like matrix rows. So while we have to write {{monzo|-4 4 -1}} vertically when in matrix form, {{val|19 30 44}} stays horizontal.

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 A good way to explain why we don’t need three of these commas is that if you had three of them, you could use any two of them to create the third, and then subtract the result from the third, turning that comma into a zero vector, or a vector with only zeroes, which is pretty useless, so we could just discard it.
+And a potentially helpful way to think about why any other interval arrived at through linear combinations of the commas in a basis would also be a valid column in the basis is this: any of these interval vectors, by definition, is mapped to zero steps by the mapping. So any combination of them will also map to zero steps, and thus be a comma that is tempered out by the temperament.
 When written with the {{val|}} notation, we’re expressing maps in “covector” form, or in other words, as the opposite of vectors. But we can also think of maps in terms of matrices. If vectors are like matrix columns, maps are like matrix rows. So while we have to write {{monzo|-4 4 -1}} vertically when in matrix form, {{val|19 30 44}} stays horizontal.