Delta-rational chord: Difference between revisions

Line 48:

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

Then, provided that the positive rational number <math>m/n</math> lies in the image <math>r_{\mathbf{u}, \mathbf{v}}(I)</math> and yields a nondegenerate equation, we can solve for the frequency ratio <math>g \in I</math> that satisfies <math>r_{\mathbf{u}, \mathbf{v}}(g) = m/n,</math> making the specified chord {{nowrap|('''0''', '''u''', '''v''') a + ''n'' + ''m''}} DR chord.

The existence of an exact tuning for a delta signature specification is only guaranteed to hold when we only care about a ratio between ''two'' terms in the delta signature being exact. If we want to optimize an arbitrary specified delta signature (with some deltas possibly held free), we can use a least-squares-error solution instead to minimize the error.

@@ Line 48: / Line 48: @@
 <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, provided that the positive rational number <math>m/n</math> lies in the image <math>r_{\mathbf{u}, \mathbf{v}}(I),</math> we can solve for the frequency ratio <math>g \in I</math> that satisfies <math>r_{\mathbf{u}, \mathbf{v}}(g) = m/n,</math> making the specified chord {{nowrap|('''0''', '''u''', '''v''') a + ''n'' + ''m''}} DR chord.
+Then, provided that the positive rational number <math>m/n</math> lies in the image <math>r_{\mathbf{u}, \mathbf{v}}(I)</math> and yields a nondegenerate equation, we can solve for the frequency ratio <math>g \in I</math> that satisfies <math>r_{\mathbf{u}, \mathbf{v}}(g) = m/n,</math> making the specified chord {{nowrap|('''0''', '''u''', '''v''') a + ''n'' + ''m''}} DR chord.
 The existence of an exact tuning for a delta signature specification is only guaranteed to hold when we only care about a ratio between ''two'' terms in the delta signature being exact. If we want to optimize an arbitrary specified delta signature (with some deltas possibly held free), we can use a least-squares-error solution instead to minimize the error.