This is exactly it. If you have two identical pianos, both precisely equally tempered, but one tuned a half step lower, and then you played notes of the same pitch on both pianos (the notes are a half step apart, the pitch is not), they will sound different. Disregarding all the differences surrounding the natural wood and other materials, and the fact that it's impossible to use the same velocity, and the pianos aren't in the same exact position relative to the room, and everything like that -- you will be left with just the differences that have to do with tension.
On an FFT graph, you'll have the same fundamental and a very similar next few partials, but eventually you'll see (and hear) the results of the different tensions. Those differences represent what people like and don't like about different tunings -- nothing to do with the slightly different pitch between 440/442/438/432/etc!
On an FFT graph, you'll have the same fundamental and a very similar next few partials, but eventually you'll see (and hear) the results of the different tensions. Those differences represent what people like and don't like about different tunings -- nothing to do with the slightly different pitch between 440/442/438/432/etc!