User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
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Because vals can be multiplied by an arbitrary scalar they represent lines that pass through the origin. Because one endpoint of the line is fixed the resulting space of interest is 2-dimensional in the case of three component vals. This is known as the val space and is isomorphic to the [[Projective tuning space|projective tuning space]]. | Because vals can be multiplied by an arbitrary scalar they represent lines that pass through the origin. Because one endpoint of the line is fixed the resulting space of interest is 2-dimensional in the case of three component vals. This is known as the val space and is isomorphic to the [[Projective tuning space|projective tuning space]]. | ||
3D lines that pass through the origin find representation in the [ | 3D lines that pass through the origin find representation in the [[Wikipedia:Geometric Algebra|Geometric Algebra]] <math>\mathcal G(3,0)</math>. Our vals are now vectors, or points if you think projectively, in this space. | ||
:<math> < 5, 8, 12 ] \mapsto 5e_1 + 8e_2 + 12e_3 =: \overleftarrow{5} </math> | :<math> < 5, 8, 12 ] \mapsto 5e_1 + 8e_2 + 12e_3 =: \overleftarrow{5} </math> | ||
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To find the '''period''' (in n dimensions). Calculate <math>d = gcd(\mathbf{W}_{12}, \mathbf{W}_{13}, \ldots, \mathbf{W}_{1n})</math>, where <math>\mathbf{W}_{1n}</math> are all of the bivector components associated with <math>e_1</math> (i.e. with prime 2) of the 2-wedgie for the temperament. Your period will be 1\d in backslash notation i.e. one step of d equal divisions of the octave. | To find the '''period''' (in n dimensions). Calculate <math>d = gcd(\mathbf{W}_{12}, \mathbf{W}_{13}, \ldots, \mathbf{W}_{1n})</math>, where <math>\mathbf{W}_{1n}</math> are all of the bivector components associated with <math>e_1</math> (i.e. with prime 2) of the 2-wedgie for the temperament. Your period will be 1\d in backslash notation i.e. one step of d equal divisions of the octave. | ||
To find the '''generator'''. Solve the equation <math>c_2\mathbf{W}_{12} + c_3\mathbf{W}_{13} + \ldots + c_n\mathbf{W}_{1n} = d</math> for the coefficients <math>c_2,\ldots,c_n</math> (using some algorithm such as the [ | To find the '''generator'''. Solve the equation <math>c_2\mathbf{W}_{12} + c_3\mathbf{W}_{13} + \ldots + c_n\mathbf{W}_{1n} = d</math> for the coefficients <math>c_2,\ldots,c_n</math> (using some algorithm such as the [[Wikipedia:extended Euclidean algorithm|extended Euclidean algorithm]]). Then one valid generator for the temperament is <math>\overrightarrow{g}=c_2e_2 + c_3e_3 + \ldots + c_ne_n</math>. Thus (the tempered version of) the generator represents <math>q_2^{c_2} \cdots q_n^{c_n}</math> where 2.''q''<sub>2</sub>.(…).''q''<sub>n</sub> is the [[JI subgroup]]. | ||
This generator might not be the most meaningful musically, but it's enough to build [[MOS]] scales for rank-2 temperaments which along with the POTE tuning is the main thing you need to compose music in the chosen temperament. | This generator might not be the most meaningful musically, but it's enough to build [[MOS]] scales for rank-2 temperaments which along with the POTE tuning is the main thing you need to compose music in the chosen temperament. | ||