Delta-rational chord: Difference between revisions

Line 18:

In practice these terms can loosely refer to approximations of mathematically exact PDR and FDR chords, for example in [[edo]] tunings.

== Facts ==

=== Facts ===

=== Exact DR tunings of MOSes are abundant ===

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

<math>\begin{align}

\mathbf{u} &= u_p \mathbf{p} + u_g \mathbf{g} \\

\mathbf{v} &= v_p \mathbf{p} + v_g \mathbf{g}

\end{align}

</math>

as elements of the rank-2 free <math>\mathbb{Z}</math>-module <math>\mathbb{Z}^2\langle \mathbf{p}, \mathbf{g}\rangle</math>.

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>

Then for any positive rational number <math>m/n</math> in the image <math>r_{\mathbf{u}, \mathbf{v}}(I),</math> there exists <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 +''m''+''n'' DR chord.

(TODO: Case analysis according to whether <math>r_{\mathbf{u}, \mathbf{v}}(x)</math> simplifies to a polynomial.)

== Finding approximate DR chords in edos ==

@@ Line 18: / Line 18: @@
 In practice these terms can loosely refer to approximations of mathematically exact PDR and FDR chords, for example in [[edo]] tunings.
-== Facts ==
+=== Facts ===
-<!--Given a primitive 2/1-equivalent MOS, there are infinitely many tunings for the generator that tune a given triad ('''0''', '''x''', '''y''') in the MOS as some delta-rational chord (given some mild conditions such as linear independence of '''x''' and '''y''').-->
+=== Exact DR tunings of MOSes are abundant ===
+Let ''a'', ''b'' be positive integers and suppose gcd(''a'', ''b'') = 1. Let ''p'' > 1. Consider a MOS ''a'''''L'''''b'''''s'''{{angbr|p}} with generator range <math>I \subseteq (1, \sqrt{p})</math>, and consider a triple ('''0''', '''u''', '''v''') (representing a triad in the MOS), '''0''' < '''u''' < '''v'''. Let '''p''' represent the period (octave) and let '''g''' represent the generator. Write
+<math>\begin{align}
+\mathbf{u} &= u_p \mathbf{p} + u_g \mathbf{g} \\
+\mathbf{v} &= v_p \mathbf{p} + v_g \mathbf{g}
+\end{align}
+</math>
+as elements of the rank-2 free <math>\mathbb{Z}</math>-module <math>\mathbb{Z}^2\langle \mathbf{p}, \mathbf{g}\rangle</math>.
+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>
+Then for any positive rational number <math>m/n</math> in the image <math>r_{\mathbf{u}, \mathbf{v}}(I),</math> there exists <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 +''m''+''n'' DR chord.
+(TODO: Case analysis according to whether <math>r_{\mathbf{u}, \mathbf{v}}(x)</math> simplifies to a polynomial.)
 == Finding approximate DR chords in edos ==