If you want to see our Web in Spanish click here 1. A fatter string has more mass more mass means the sound wave travels slower, so it produces a deeper sound at the same tension more tension produces a higher note.
Stiffness is a factor it takes more energy to move the string so the sound decays faster.
The closer you get to the tensile strength of a material, the more efficient the energy is used to vibrate up to a certain point. The scale length is out for a certain pitch and thickness of string you can vary the material sizes but not a lot. A fatter string gives more tension. A thinner string gives less at the same pitch. If you just doubled the size of the string by using something twice as thick you would also raise the tensile strength, this can work in some cases but not in most you could do it with a long scale, but not with a short scale.