This music book I'm reading taught me about frequency ratios as the basis for scales, but it doesn't explain it well enough for me, so I'm trying to learn more about it on my own. I want to know how they're applied to strings. I thought that since 3:2 (P5) is 1.5:1, that it would appear exactly halfway between the end of the string and the halfway point (octave), but that isn't the case, which I realized is due to the fact that pitches aren't evenly spaced throughout the string, a difference of pitch corresponds to a shorter length of string the farther "up" the string you get. And what actually does appear to be located halfway between the end and the octave is actually 4:3 (P4). And I'm really struggling to find a way to convert ratios, starting with 1:1 at the end of the string and ending at 2:1 halfway, to string length, starting at 0 and ending at the octave, which on my instrument seems to be around 15 1/2 inches. Why does 3:2 seem to appear at 2/3s of the string (up to the octave), while 4:3 shows up at 1/2? What's the conversion process I'm supposed to use here?
I'm really not very knowledgeable about math and I have trouble understanding things a lot of the time, just as a warning.
Edit: I should mention that the instrument I'm using for reference is a shamisen, which has no frets.