the real piano string
When thinking about the piano strings, or other strings for that matter, it’s basically assumed that there’s fundamental frequency and that the partials occur at harmonics of the fundamental frequency. From here on, classical music theory people go on about the consonance of the intervals being related to the internal partials of the music instruments, and how therefore western classical music theory is universal, based on scientific reasoning, and should therefore be imposed on everyone else (and it is, thorough the school system).
Off course this is not so. The piano string for example is slightly inharmonic and furthermore other (also inharmonic) partials called “phantom partials” , which are cause by the longitudinal waves that cross the string, also contribute to the sound.
Very nice examples of the influence of the longitudinal modes on the piano string sound can be found here.