This is fascinating! I always wondered why a piano could work so well despite in reality having slightly ill-tuned thirds. Thanks for sharing the tidbit.
[0] https://en.wikipedia.org/wiki/Stretched_tuning#Intervals_and...
The former create gentle swirling interference, the latter a rough stuttering. If you hit a fifth on a piano and listen carefully, you can hear the slow cycle in the sound.
The stretched octaves are due to high string tension causing some inharmonicity, ie. sharpening higher harmonics.
Not sure if I've explained it well, but vibrations of an ideal string (no bending stiffness) can be described by a second-order partial differential equation [0], whereas a real string with nonzero bending stiffness is actually more of a 'vibrating beam' problem, which is a fourth-order PDE [1].
EDIT: this is why low notes have greater inharmonicity (thicker strings).
[0] https://jmahaffy.sdsu.edu/courses/s17/math531/beamer/string....
[1] http://www.math.umbc.edu/~jbell/pde_notes/20_Beam%20Equation...
Well, I think of and hear a major third as the loveliest interval on a piano, not "rough stuttering"!
Basically, the harmonics that rise off of piano strings aren't exact multiples of the fundamental -- they're a little bit off, because piano strings don't behave entirely like ideal strings, they behave like metal cylinders. In the mid-range they're pretty pretty close to plain equal temperament, but in the high treble the strings get shorted but the string gauge stays almost the same, which means the ratio of diameter to length increases and they act less like strings and more like cylinders. The bass has similar issues with single and double wound strings. So, the fix is to just stretch the octave enough so that the harmonics of low notes line up better with the fundamentals of higher notes, and so on. (What we perceive as "out of tune-ness" is the wobbly sound of two frequencies played together that almost but don't quite line up, creating a beat frequency.)