Background:
I am trying to find „reasonable“ rational approximations to $ \mu^{-1} $ where
$$ \mu = 59.061175926\ldots \, \text{d} $$
is twice the long-term mean of a synodic month corresponding to the duration of a lunar phase.
First of all it shouldn’t matter too much if this is specified in days or hours.
Then the true problem is not to find rational approximations by continued fractions, but to find „reasonable“ approximations, where „reasonable“ is difficult to define and to quantify.
From the rational approximation gear ratios
$$ \mu^{-1} = \frac{\prod_{m=1}^M p_m}{\prod_{n=1}^N q_n} $$
should be derived, where each factor represents a gear, where the number of gears $ M + N $ should be small (two would be nice, three to four would still be ok …) and where the numbers $ p_m, q_n $ are not too large.
So the idea is to go for continued fractions.
Unfortunately, this approach provides different approximations, depending on the unit of measurement (days, hours …), so different approximations for $ \mu^{-1} $ and $ (60 \cdot \mu)^{-1} $.
And, even worse, the approach generates large primes for the $ p_m, q_n $, making the results worthless for practical use.
The search space explored by continued fractions is too restricted.
Question:
Is there a reasonable approach - besides brute force - to systematically explore a larger search space for rational approximations, preferring small numerators and denominators and therefore small primes?