Douglas Blumeyer's RTT How-To: Difference between revisions
Cmloegcmluin (talk | contribs) →linear temperaments: add additional diagram |
Cmloegcmluin (talk | contribs) →multicovectors: add Wolfram link |
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In natural language, that’s each element of the first row times the determinant of the square matrix from the other two columns and the other two rows, summed but with an alternating pattern of negation beginning with positive. If you ever need to do determinants of matrices bigger than 3×3, see [https://www.mathsisfun.com/algebra/matrix-determinant.html this webpage]. Or, you can just use an online calculator. | In natural language, that’s each element of the first row times the determinant of the square matrix from the other two columns and the other two rows, summed but with an alternating pattern of negation beginning with positive. If you ever need to do determinants of matrices bigger than 3×3, see [https://www.mathsisfun.com/algebra/matrix-determinant.html this webpage]. Or, you can just use an online calculator. | ||
{| class="wikitable" | |||
|+WolframAlpha code ([https://www.wolframalpha.com/input/?i=Determinant%5B%7B%7B1%2C0%2C1%7D%2C%7B0%2C1%2C1%7D%2C%7B0%2C0%2C-2%7D%7D%5D try it]) | |||
!input | |||
!output | |||
|- | |||
|<code>Determinant[{{1,0,1},{0,1,1},{0,0,-2}}]</code> | |||
|<nowiki>-2</nowiki> | |||
|} | |||
And so our results are <span><math>-2</math></span>, <span><math>3</math></span>, <span><math>1</math></span>, <span><math>-11</math></span>. At first glance we don’t have a GCD, but there is a slight trick here. We prefer for the first term to be positive. It doesn’t make a difference, but is a way to normalize things (we could have found the result where the first term came out positive by simply changing the order of the rows of our mapping, which doesn’t affect how the mapping works at all). So, we can simply think of this as extracting a GCD of -1, and our list ends up as <span><math>2</math></span>, <span><math>-3</math></span>, <span><math>-1</math></span>, <span><math>11</math></span>. Finally, set these inside triply-nested brackets, because it’s a tricovector for a rank-3 temperament, and we get {{multival|rank=3|2 -3 -1 11}}. | And so our results are <span><math>-2</math></span>, <span><math>3</math></span>, <span><math>1</math></span>, <span><math>-11</math></span>. At first glance we don’t have a GCD, but there is a slight trick here. We prefer for the first term to be positive. It doesn’t make a difference, but is a way to normalize things (we could have found the result where the first term came out positive by simply changing the order of the rows of our mapping, which doesn’t affect how the mapping works at all). So, we can simply think of this as extracting a GCD of -1, and our list ends up as <span><math>2</math></span>, <span><math>-3</math></span>, <span><math>-1</math></span>, <span><math>11</math></span>. Finally, set these inside triply-nested brackets, because it’s a tricovector for a rank-3 temperament, and we get {{multival|rank=3|2 -3 -1 11}}. | ||
As for getting from the multicovector back to the mapping, you can solve a system of equations for that. Though it’s not easy and there may not be a unique solution. And you probably will never have the multicovector without the mapping anyway. | As for getting from the multicovector back to the mapping, you can solve a system of equations for that. Though it’s not easy and there may not be a unique solution. And you probably will never have the multicovector without the mapping anyway. | ||
=== multivectors === | === multivectors === | ||