From the physical standpoint, real strings differ from ideal strings in having bending stiffness. When the ends of a string are clamped in a fixed position, that means the vibrational motion isn't a perfect parabola, but is distorted near the ends where the bending force is concentrated. The freely moving part of the string is shortened a little bit. It doesn't make much difference in the effective length of the string at the fundamental frequency, but as you start fitting more and more standing waves of shorter wavelength (higher partials) onto the string, the stiff fixed segments at the ends have a relatively greater effect, forcing the higher "harmonics" to go sharper relative to the fundamental as they go higher up the harmonic series. The closer a string is stretched to its breaking point, the more closely it resembles an ideal string with perfectly harmonic partials. So a heavy, tight string has its partials more in tune than a light, slacker one tuned to the same pitch.
"Fundamentals of Musical Acoustics" by Arthur H. Benade has a very good explanation of this phenomenon (and lots of others.) A very highly recommended book for musicians.
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