I'd like to give a rigorous analysis of the "dissonance" of musical intervals -- say, a perfect fourth versus a perfect fifth. I think one way to get "rigor" here is to consider the frequencies and amplitudes of the overtone series for both pitches, and then appeal to the physiology and psychology of acoustical processing.
One relatively parsimonious way of approaching this problem is to think about playing these intervals on a monochord. Contemporary "demonstration monochords" have two strings, in fact, not one. I can pluck one string without a bridge and use that as the fundamental, and then use the monochord's bridge to tune the second string to a perfect fourth or perfect fifth.
Throughout this thought experiment, assume I pluck all strings at the middle of their lengths. (The plucking location affects which harmonics are sounded, but I'm not sure how sensitive the resulting dissonances are to changes in plucking location.)
String 1
Say the fundamental frequency for string 1 is $f$. Because it is plucked in the middle, the even harmonics aren't sounded. The frequencies are thus odd multiples of the fundamental, with amplitudes decaying inverse to the square:
$$\begin{eqnarray} \text{frequency} &:& f, 3f, 5f, 7f \ldots \\ \text{amplitude} &:& 1, 1/9, 1/25, 1/49 \ldots \end{eqnarray} $$
Perfect Fourth above
Now say I use the bridge to tune string 2 to a perfect Pythagorean fourth (frequency ratio 4 to 3) above the fundamental of string 1. I will assume the amplitude of the fundamental in this series is the same as the amplitude of the fundamental for string 1's series, but I'm not sure whether this is a safe assumption. Again, because the string is plucked in the middle, even harmonics are missing:
$$\begin{eqnarray} \text{frequency} &:& \frac{4}{3}f, 4f, \frac{20}{3}f, \frac{28}{3}f\ldots \\ \text{amplitude} &:& 1, 1/9, 1/25, 1/49 \ldots \end{eqnarray} $$
Perfect Fifth above
This time, tune string 2 to a perfect Pythagorean fifth (frequency ratio 3 to 2) above the fundamental of string 1. Again I assume a common amplitude for the fundamental, and again, because the string is plucked in the middle, even harmonics are missing:
$$\begin{eqnarray} \text{frequency} &:& \frac{3}{2}f, \frac{9}{2}f, \frac{15}{2}f, \frac{21}{2}f \ldots \\ \text{amplitude} &:& 1, 1/9, 1/25, 1/49 \ldots \end{eqnarray} $$
Question: How can I use this simple analysis (if it is more or less correct) to explain the "dissonance" of the perfect fourth versus the perfect fifth?
In particular, how can I finish the explanation to arrive at the standard conclusion that the perfect fifth is more "consonant" than the perfect fourth?
Perhaps one doesn't need overtone series for that conclusion at all -- maybe the biology says it's enough to look at the dissonance of 4/3f versus f compared to 3/2f versus f, the former giving a smaller "amplitude fluctuation" -- but I thought I might as well look at the overtone series, too.
I know I now need to import some facts from biology, so the question isn't strictly physics.