TITLE: Is there a sequence that contains every rational number once, but with the "simplest" fractions first?
QUESTION [8 upvotes]: The Calkin-Wilf sequence contains every positive rational number exactly once:
1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, ….
I'd consider 5/1 to be a "simpler" ratio than 8/5, but it appears later in the series.

Is there a mathematical term for the "simpleness" of a ratio?  It might be something like the numerator times the denominator, or maybe there are other ways to measure.
Is there a sequence that contains all the positive rational numbers, but with the "simpleness" of the ratios monotonically increasing?

(Small integer ratios are found in Just intonation, polyrhythm, orbital resonance, etc.)
If you use the Calkin-Wilf sequence with the num*den measure, for instance, it looks like this:

REPLY [2 votes]: You could measure simplicity by the sum of the numerator and denominator (the "length", as it would be known in some parts of Number Theory), breaking ties by, say, size of numerator. 
1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1,....
This is essentially the ordering you get out of the usual proof that the (positive) rationals are countable, except that in that proof you include 2/4 and 2/6 and 3/6 and so on. The price of leaving out those duplications is that you can't expect a simple formula for the $n$th rational.