There is a mathematical story to tell associated with this construction of overtone series that centers around a very important theorem of Fourier. Cast into the current context the theorem says that any vibration that the string is capable of can be expressed by a suitable addition of the fundamental modes each with some weighting coefficient. It is clear that each string of a piano will have its set of overtones whose fequencies are determined by simple ratios of whole numbers. The equal tempered G that is now there would use a factor of 1.498. So now the the equal tempered G and its overtones will beat with those of the C, where this would be minimized if the G were tuned using the 1.5 factor.
Beats happen when two tones are played together that are just slightly different in frequency.
The amplitude of the sine wave, hence it's loudness swells and dimishes. This swelling and diminishing of loudness is called beating. The more beating going on, the less consonant is the perception to the ear. Since the very basis of harmony is the interval of the fifth, some of the consonance of all harmony has been compromised by equal tempering. With the advent of computers and computer software powerful enough to handle the digital and analog manipulations of sound, the music of the future, providing there is one, can be free of the equal temperament that has been imposed on western music by the piano keyboard and still allow free modulation to maintain natural harmonic relations when wanted, and at the same time allow for music that can also be expressed using the rich nonharmonic linear language that has been created in many other cultures.