Equivalence continuum: Difference between revisions
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Say that the vector '''v''' (which depends on ''T'') defining this unique line has components (''v''<sub>1</sub>, ''v''<sub>2</sub>, ''v''<sub>3</sub>), so that the plane associated with the rank-2 temperament has equation ''v''<sub>1</sub>''x'' + ''v''<sub>2</sub>''y'' + ''v''<sub>3</sub>''z'' = 0. [We may further assume that ''v''<sub>1</sub>, ''v''<sub>2</sub>, ''v''<sub>3</sub> are integers with gcd 1, since the condition of being perpendicular to two integer vectors is defined by a system of linear equations with integer coefficients, thus has a unique rational solution up to scaling.] One coordinate ''v''<sub>i</sub> is always guaranteed to be nonzero, for any temperament. Assuming ''v''<sub>1</sub> ≠ 0, we can scale '''v''' by 1/''v''<sub>1</sub>, then the resulting vector '''v'''/''v''<sub>1</sub> = (1, ''v''<sub>2</sub>/''v''<sub>1</sub>, v<sub>3</sub>/''v''<sub>1</sub>) = (1, ''s'', ''t'') points in the same direction as '''v''' and describes two rational (or infinite) parameters ''s'' and ''t'' which defines any temperament with ''v''<sub>1</sub> ≠ 0 on 31edo's 7-limit rank-2 continuum uniquely. Two-dimensional coordinates can similarly be assigned for the set of all temperaments such that ''v''<sub>2</sub> ≠ 0 and the set of all temperaments such that ''v''<sub>3</sub> ≠ 0.<!-- Note that this continuum is actually part of a mathematical manifold with a more complicated topology and needs to be described using more than one local chart (coordinate system) constructed like this; unlike for the ''k'' − ''r'' = 1 case, a single circle won't define every point on this 2-dimensional continuum, just like a single circle won't define every point on a 2-dimensional sphere.--> | Say that the vector '''v''' (which depends on ''T'') defining this unique line has components (''v''<sub>1</sub>, ''v''<sub>2</sub>, ''v''<sub>3</sub>), so that the plane associated with the rank-2 temperament has equation ''v''<sub>1</sub>''x'' + ''v''<sub>2</sub>''y'' + ''v''<sub>3</sub>''z'' = 0. [We may further assume that ''v''<sub>1</sub>, ''v''<sub>2</sub>, ''v''<sub>3</sub> are integers with gcd 1, since the condition of being perpendicular to two integer vectors is defined by a system of linear equations with integer coefficients, thus has a unique rational solution up to scaling.] One coordinate ''v''<sub>i</sub> is always guaranteed to be nonzero, for any temperament. Assuming ''v''<sub>1</sub> ≠ 0, we can scale '''v''' by 1/''v''<sub>1</sub>, then the resulting vector '''v'''/''v''<sub>1</sub> = (1, ''v''<sub>2</sub>/''v''<sub>1</sub>, v<sub>3</sub>/''v''<sub>1</sub>) = (1, ''s'', ''t'') points in the same direction as '''v''' and describes two rational (or infinite) parameters ''s'' and ''t'' which defines any temperament with ''v''<sub>1</sub> ≠ 0 on 31edo's 7-limit rank-2 continuum uniquely. Two-dimensional coordinates can similarly be assigned for the set of all temperaments such that ''v''<sub>2</sub> ≠ 0 and the set of all temperaments such that ''v''<sub>3</sub> ≠ 0.<!-- Note that this continuum is actually part of a mathematical manifold with a more complicated topology and needs to be described using more than one local chart (coordinate system) constructed like this; unlike for the ''k'' − ''r'' = 1 case, a single circle won't define every point on this 2-dimensional continuum, just like a single circle won't define every point on a 2-dimensional sphere.--> | ||
==== Example (7-limit rank-2 temperaments in 31edo) ==== | ==== Example (7-limit rank-2 temperaments in 31edo) ==== | ||
Let's look at where some well-known 7-limit rank-2 temperaments supported by [[31edo]] live in the 2-dimensional equivalence continuum C(2, 7-limit 31edo). Choose the basis '''u'''<sub>''x''</sub>, '''u'''<sub>''y''</sub>, '''u'''<sub>''z''</sub> = 81/80, 126/125, 1029/1024 to define (''x'', ''y'', ''z'') coordinates on the kernel of 7-limit [[31edo]]. Then: | Let's look at where some well-known 7-limit rank-2 temperaments supported by [[31edo]] live in the 2-dimensional equivalence continuum C(2, 7-limit 31edo). Choose the basis '''u'''<sub>''x''</sub>, '''u'''<sub>''y''</sub>, '''u'''<sub>''z''</sub> = 81/80, 126/125, 1029/1024 to define (''x'', ''y'', ''z'') coordinates on the kernel of 7-limit [[31edo]]. Then: | ||