Update; Approx GCD's as Phase Modulated JI Intervals.

I think I've found the missing requirement. I've stated the motivation many times so let me just go through the mathematical steps:

1. The extrema of the wave f(t) = sin(2pi at) + sin(2pi bt) occurs at the points where this wave meets with its envelope +/-2cos(pi(a - b)t). Taking f(t) = 2sin(pi(a + b)t)cos(pi(a - b)t) = +/-2cos(pi(a - b)t) we get sin(pi(a + b)t) = +/-1 and solving for time gives
t = (2k + 1)/2(a + b), k = 0,1,2,3...

2. The larger maxima in f(t) occurs at the times t = (2k + 1)/2(a + b) that are closest to the largest maxima of the envelope. Since the first of these occurs at t = 0 then the closest will be at k = 0 and t1 = 1/2(a + b). This is true independent of our choice of a and b.

3. Here's the step that was missing (and I *think* it solves the problem). We obtain the value for k = K simply by rounding off (a + b)/(a - b) to the nearest integer. This makes sense if we take it in the form 1/2(a + b)Hz x 2(a - b)seconds = cycles, for we are taking the closest whole-numbered cycle. The end of the 'approx GCD' cycle will be at t2 = (2K + 1)/2(a + b), the difference is the period t2 - t1 = T = K/(a + b) and the frequency is 1/T = (a + b)/K.

4. The rest I've said before. We obtain the unique integer pair (p, q) by dividing our original frequencies (a, b) with this approx GCD and taking it in quotient plus remainder form.

aK/(a + b) = p + r/(a + b),
bK/(a + b) = p - r/(a + b),

where r = aq - pb. Adding these we see that K = (p + q) and our 'GCD' becomes 1/T = (a + b)/K = (a + b)/(p + q).

5. We retrieve a and b again by multiplying the two equations in 4. by this frequency

a = p[(a + b)/(p + q)] + r/(p + q),
b = q[(a + b)/(p + q)] - r/(p + q).

6. We obtain

f(t) = sin(2pi(p[(a + b)/(p + q)]t + rt/(p + q))) +
sin(2pi(q[(a + b)/(p + q)]t - rt/(p + q))).

Since (p, q) are whole then this is our simpler 'JI' wave wit an equal and opposite phase mod in each component.

7. Or this could be written as an AM'd average wave

f(t) = 2sin(pi(a + b)cos(pi[(p - q)(a + b)/(p + q)]t + 2rt/(p + q))).

It is seen that this can be seen as an AM with a PM. Note that if (p, q) are epimoric, then this AM is just the approx GCD frequency + phase.

Please let me know if you see any obvious problems. Thanks

Rick