Delta-rational chord: Difference between revisions

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 <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.

== DR and RTT ==

One may be able to tune a rank-2 regular temperament in such a way that a chord of interest exactly "inherits" its delta signature from a simple JI preimage thereof. This is done by setting up an algebraic equation relating the intervals in the chord to a generator and then solving for the generator that produces proportionally-beating triads. If we want to optimize a 4:5:6 triad in Meantone, for instance, we want a 1:1 ratio of frequency deltas between the major third and minor third. The minor third can be written as <math>g-g^4/4</math>, and the major third can be written as <math>g^4/4-1</math>. Therefore we must find the roots of the polynomial <math>g^4-2g-2</math> (the difference between the two, simplified to make all coefficients integers). This results in a generator of 1.49453, or about 695.6 cents. However, the equation to solve depends on what chord you want to tune as equal-beating. For example, assuming pure octaves, Meantone admits an equation for tuning the 3:4:5 as equal-beating: <math>g^4+2g-8=0.</math> The latter equatoon has solution g = 1.496 = 697.3c.