Delta-rational chord: Difference between revisions

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=== Facts ===

==== "Exact DR tunings" of MOS scales are abundant ====

Let ''a'', ''b'' be positive integers and suppose gcd(''a'', ''b'') = 1. Consider a MOS ''a'''''L'''''b'''''s'''{{angbr|~~2/1~~}} with generator range <math>I \subseteq (1, \sqrt{2})</math>, and consider a triple ('''0''', '''u''', '''v''') (representing a triad in the MOS), '''0''' < '''u''' < '''v'''. Let '''p''', '''g''' be a basis formally representing the MOS scale's period and generator. Write

Let ''a'', ''b'' be positive integers and suppose gcd(''a'', ''b'') = 1. Let ''E'' > 1 be the ratio for the equave. Consider a MOS ''a'''''L'''''b'''''s'''{{angbr|E}} with generator range <math>I \subseteq (1, \sqrt{E})</math>, and consider a triple ('''0''', '''u''', '''v''') (representing a triad in the MOS), '''0''' < '''u''' < '''v'''. Let '''p''', '''g''' be a basis formally representing the MOS scale's period and generator. Write

<math>\begin{align}

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Define the rational function <math>r_{\mathbf{u}, \mathbf{v}} : I \to (0,\infty)</math> by

<math>\displaystyle{r_{\mathbf{u}, \mathbf{v}}(x) = \frac{2^{v_p}x^{v_g}- 2^{u_p}x^{u_g}}{2^{u_p}x^{u_g} - 1} }.</math>

<math>\displaystyle{r_{\mathbf{u}, \mathbf{v}}(x) = \frac{E^{v_p}x^{v_g}- E^{u_p}x^{u_g}}{E^{u_p}x^{u_g} - 1} }.</math>

Then for any positive rational number <math>m/n</math> in the image <math>r_{\mathbf{u}, \mathbf{v}}(I),</math> we can solve for <math>g \in I</math> that satisfies <math>r_{\mathbf{u}, \mathbf{v}}(g) = m/n,</math> making the specified chord ('''0''', '''u''', '''v''') a +''n''+''m'' DR chord.

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 === Facts ===
 ==== "Exact DR tunings" of MOS scales are abundant ====
-Let ''a'', ''b'' be positive integers and suppose gcd(''a'', ''b'') = 1. Consider a MOS ''a'''''L'''''b'''''s'''{{angbr|2/1}} with generator range <math>I \subseteq (1, \sqrt{2})</math>, and consider a triple ('''0''', '''u''', '''v''') (representing a triad in the MOS), '''0''' < '''u''' < '''v'''. Let '''p''', '''g''' be a basis formally representing the MOS scale's period and generator. Write
+Let ''a'', ''b'' be positive integers and suppose gcd(''a'', ''b'') = 1. Let ''E'' > 1 be the ratio for the equave. Consider a MOS ''a'''''L'''''b'''''s'''{{angbr|E}} with generator range <math>I \subseteq (1, \sqrt{E})</math>, and consider a triple ('''0''', '''u''', '''v''') (representing a triad in the MOS), '''0''' < '''u''' < '''v'''. Let '''p''', '''g''' be a basis formally representing the MOS scale's period and generator. Write
 <math>\begin{align}
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 Define the rational function <math>r_{\mathbf{u}, \mathbf{v}} : I \to (0,\infty)</math> by
-<math>\displaystyle{r_{\mathbf{u}, \mathbf{v}}(x) = \frac{2^{v_p}x^{v_g}- 2^{u_p}x^{u_g}}{2^{u_p}x^{u_g} - 1} }.</math>
+<math>\displaystyle{r_{\mathbf{u}, \mathbf{v}}(x) = \frac{E^{v_p}x^{v_g}- E^{u_p}x^{u_g}}{E^{u_p}x^{u_g} - 1} }.</math>
 Then for any positive rational number <math>m/n</math> in the image <math>r_{\mathbf{u}, \mathbf{v}}(I),</math> we can solve for <math>g \in I</math> that satisfies <math>r_{\mathbf{u}, \mathbf{v}}(g) = m/n,</math> making the specified chord ('''0''', '''u''', '''v''') a +''n''+''m'' DR chord.