The most pleasing and simplest combination is 1:2, meaning the one string should be half or double of the other. We make 'doubling' or 'halving' our basic interval, and we can expand in both directions if we want, by repeated doubling or halving. (Two notes that are half or double of each other form an 'octave'.) But an octave is a pretty big jump. We'd like to use many more notes in between. So within the octave, we'll choose the four simplest ratios between ½ and 1. We'll choose ⅗, ⅔, ¾, and ⅘. This gives us a ladder or scale of five jumps, and all of the notes should sound pretty good together. But the jumps are not all the same size. In particular, the first and last jumps are especially large. (Just imagine humming the Do-Re-Mi scale without Re or Ti. It just doesn't sound right.) So we'll cut the two large jumps in half by inserting two more fractions: we'll pick 8/9 and 8/15 as two more notes that don't sound too horrible with the rest of them, and then we'll have a ladder or a scale of eight notes that we can use together as a more-or-less harmonious set to make music. (After the 1:2 octave ratio, the next best ratios are 2:3 and 3:4. These are the fifth and fourth notes of our final eight-note octave ladder, and in modern terms, any pairs of notes in the ratio 2:3 or 3:4 form a 'perfect fifth' or 'perfect fourth'.)
Unfortunately, the jumps in this elementary scale are still a bit irregular, which makes transposing and coordinating multiple instruments a bit difficult. Is there a way to construct a ladder or scale of notes, in which all of the jumps are evenly spaced, but the scale still contains all of the 'best' simple integer ratios? The modern solution is the 'even-tempered scale', which you can see displayed visually in the ladder of frets on a guitar neck...but that's another story.