I'm no mathematician, and so I humbly ask this to those who are better versed in the art than I.
I'm hoping for an "objective" measure of a ratio's complexity, so I can compare two or more ratios in a meaningful way. For example, 1/1 is the simplest I can think of. 1/2 is the next simplest. 3/2 is perhaps of equivalent simplicity to 1/2. But, 4/3 is more complex as is 7/4 while 40/27 is much more complex.
But how much more? And which of these two is more complex: 35/17 or 40/21? And by how much?
Intuitively, I was thinking of multiplying the numerator by the denominator, and perhaps getting the log of that. But if I use that approach, then a ratio and its reciprocal would be the same. For example, with the ratios 5/6 and 6/5 it seems to me that 5/6 is more complex, but I can't quite explain why, maybe because it's closer to 1.
Any opinions? Has this been answered before?
Thanks.
EDIT– apologies for the confusion in the comments (thanks for commenting); I didn't communicate clearly enough in my example.
So a clearer example. My intuitive ranking from simple to more complex (limited to numbers between 1/2 and 2, which is what I'm interested in):
1/1, 2/1, 1/2, 3/2, 2/3, 5/3, 4/3, 3/5, 3/4, 5/4, 4/5, 6/5, 7/4, 7/5, 8/5, 5/7, 4/7, 9/5, 5/6, 7/6, 8/7, 11/6….
…as I continue the series my intuition becomes less certain: this series (as intuited) may be 'wrong", but the first few terms are definitely "right".
Best Answer
The length of a continued fraction makes a fraction more (subjectively) complex.
All rational numbers can be written as a finite continued fraction, as there is an algorithm that computes continued fractions, but only stops if the number is rational.
There is evidence to suggest that certain cultures find rational numbers with simple continued fraction representations more tuning. In Western music, the preferred interval for the minor third is $6:5$, or $[1; 5]$ in continued fraction notation. However, this interval can also be tuned as $32:27$, or $[1; 5, 2, 2]$. The first interval clearly has a more compact representation, so to Western ears, it sounds "nicer". (Wikipedia)
When comparing two continued fractions of the same length, the fraction with a larger final term might be considered "uglier" as it is closer to a rational approximation with fewer terms.
On the contrary, as Gerry Myerson suggests, fractions with large denominators might have short continued fraction representations and vice versa. This suggests there needs to be a trade-off between the length of the continued fraction and the size of the denominator.