To get the true sound of the instrument ( guitar, piano, trumpet, etc.) Combine all those harmonics which will have different sound levels due to the instrument wood, metal, finish, scale, size, shape and how you pluck the string and the combined response is what you hear as the instruments "acoustic" sound. How pickups, effects, amplifier preamp and output stage amplifer add their unique influence on the various harmonics is the difference in sound from a classical acoustic tone to heavy metal.
For those with an electronics background the signal you see on an oscilloscope is the time signature of the sound waveform. To appreciate the tonal signature you need a spectrum analyzer to identify visually the various frequency components (harmonics) and their relative sound level.
For those with a math background connecting the time waveform to the frequency spectrum is the Fourier Transform. That math function had a lot of uses in engineering and science but the obvious application is how it relates sound time signature to frequency spectrum.
You don't need to know much math to appreciate the significance of the Fourier Transform. The digital versions referred to as the "Fast Fourier Transform" is what is resident on many digital mixers, audio playback software on computers, etc where you can watch a display of the time and/or frequency sprectum while enjoying music while doing your taxes and occasionally looking at pretty waveforms associated with the audio app!
Certainly the connection of these things to music was one of biggest reasons I became an engineer. The digital coding, processors and software we all have at our finger tips today is unimaginable from a person growing up in the 60's.
Thank goodness for the USSR's Sputnik! It got the USA focused on the space race and the scientists of that time knew solid state technology and the associated massive weight reduction and functionality as compared to vacuum tubes was a fundamental building block to any serious space program. That national investment made everything possible that we take for granted today.
Our government leaders of today need to appreciate the vision of those leaders 50 + years ago and overcome political diffences to make decisions that have a similar positive effect on the generation 50 years from now. Not only did that investment make USA the leader it improved the technology of the world!
M