User:Zhenlige/The proportional-beating property of near-JI FDR chords
As we all know, delta-rational (DR) chords have rational beating frequency ratios between fundamentals. What is less well-known is that, when the chord is fully DR, and approximates a JI chord with the same delta signature, the proportional-beating property also appears upper in the harmonic series. This makes DR chords usable even with higher pitches or wider voicings.
Consider the isodifferential chord (0:1:2:...:(n-1))[+α] that approximate the JI chord (0:q:2q:...:(n-1)q)[+p] where p and q are integers. Let f be the delta frequency, and then the frequency of notes are [math]\displaystyle{ F_i=(i+\alpha)f }[/math] where [math]\displaystyle{ i=0,1,\dots,n-1 }[/math]. The difference between the (p+iq)-th harmonic of [math]\displaystyle{ F_j }[/math] and the (p+jq)-th harmonic of [math]\displaystyle{ F_i }[/math] is:
[math]\displaystyle{ \begin{split} &(p+iq)(j+\alpha)f-(p+jq)(i+\alpha)f\\ &=(jp+p\alpha+ijq+iq\alpha-ip-p\alpha-ijq-jq\alpha)f\\ &=(jp+iq\alpha-ip-jq\alpha)f\\ &=(i-j)(q\alpha-p)f\\ \end{split} }[/math]
Since other parameters are constant with a specific chord and its JI approximation, the beating ratio is proportional to the difference of fundamentals. The beating between fundamentals is the special case when p=1 and q=0.
If [math]\displaystyle{ \frac{p+iq}{p+jq}=\frac{u}{v} }[/math] where u and v are unequal integers, the difference between the u-th harmonic of [math]\displaystyle{ F_j }[/math] and the v-th harmonic of [math]\displaystyle{ F_i }[/math] is:
[math]\displaystyle{ (i-j)(q\alpha-p)f\cdot\frac{u-v}{(i-j)q}=\frac{(q\alpha-p)f(u-v)}{q} }[/math]
The beating ratio is proportional to [math]\displaystyle{ u-v }[/math]. By omiting notes, this conclusion can be generalized to all FDR chords.
Examples
For a DR ~4:5:6 or ~10:12:15 chord, ~3/2, ~5/4 and ~6/5 beats equally fast. This may explain why 15edo's ~4:5:6 sounds concordant to some people although it is not very accurate.
For an FDR ~4:5:6:7 chord, ~5/4, ~6/5, ~7/6 and ~3/2 beats equally fast, ~7/5 beats twice as fast, and ~7/4 beats three times as fast.