To jyoung999 .... I've wondered similarly, for decades. Then I slowly realized the duration of the correlation (the Fourier Series and Fourier Transform are correlators) determines what you see and how we think about energy.
Back at university, for one semester I had access to Fortran punchedcard compute power on a 1MIPS 360/65. The course included Fourier introduction, so I coded up the maths.
I made various mistakes in using Fourier algorithm. These mistakes were very puzzling, and as I shared the puzzling lineprinter ASCII output plots with classmates, the professor just smiled and remained silent as the students gathered around the many many sheets of fanfold paper stretched out on the chemlab worktables.
I boldly asked for more compute time, having already burned 10 minutes ($20 of fake costs) of 360/65 time, mostly in computing sin(2.459934 e.g.) and cosine(of the same). Other students gave me their accounts to squander. So I did.
Insight did not arrive. The professor still remained silent. Smiling.
My mistake was in thinking about time resolution; at that time I was focused on circuits and scope waveforms and mental-modeling the time behaviors of circuits.
I'd provide ONE cycle of input waveform ( rectangular or squarewave or sinwave). And I expected lots of interesting frequency information, so I coded the Fourier maths with 100 frequency bins of output spectra.
Had I provided 100 cycles of input waveform, then the 100 frequency bins would have given me legitimate information. But I did not have the computer time to do that. So I remained puzzled.
Soon graduating, I worked for a company with telemetry equipment as its primary cash cow; lots of scopes and some frequency spectrum analyzers and function generators.
Guess what? with a function generator (the output voltage turned way down) and a coax cable and the Frequency Spectrum Analyzer (HP141 model, with HP8553B RF section, HP8552 IF section), I learned a lot. So can you.
I learned that very wide bandwidths would give me the results, shown on the display tube phosphor, that I'd seen on the lineprinter output paper.
And Very Wide Bandwidths correspond to very short CORRELATION TIMES.
With very wide bandwidths, the "spectral lines" were wide blobs, vaguely of sin(x)/x amplitude but very wide.
As you turn down the bandwidth (the HP8553B RF section included expensive crystal filters of 30 Hertz and 10Hertz bandwidth --- hence the "B" in partnumber), which are akin to 33 millisecond and 100 millisecond CORRELATION TIMES, guess what happens?
The Spectral Lines appear, and the floor drops down to bottom of the display.
Summary --- the correlation time matters.
And the Fourier analysis behaves as if the correlation is over INFINITE TIME.
Which requires infinite Q in our circuits, meaning each of our circuits must be a bandpass filter of infinite Q *IF** the Fourier analysis is to completely explain the behavior.
Since none of our circuits have infinite Q (very narrow bandwidth), we need more than Fourier analysis in our thinking cap to understand behavior.
Summary ---- we think harmonics exist, because of our tools.
If you have a radio operating at 300 -- 320 MHz, sharing a PCB with a MCU of 10MHz clock, should you pick a RF channel at 310MHz or at 313 MHz.
Theory says the 310MHz channel will be impaired by a strong correlation with the 31rst multiple of the MCU clock.
And Theory says the 313MHz channel be free of such issues.
What is reality?
Now let us consider cross-over distortion of some amplifier, with class AB output pullup and pulldown transistors. Provide a clean 1,000Hz input sine. What happens? As the up and down transistors exchange roles at zero-crossings, the output drive strength briefly fluctuations as the combined Rout of each up/down device contributes, and
a very brief sampling-glitch occurs. An impulse function
We've just added a brief impulse at each and every zero crossing.
The human ear does not like this.
The math tools coerce that energy into "harmonics", because that is the only math model we have.
Now for the interesting thought about zero-crossings in class_AB circuits:
for complex inputs (music we care about), the zero crossings no longer are harmonically related to the tones, and we get broadband splatter of energy.
The expensive measurement tools can only RAISE THE NOISE FLOOR indicator, because the intermodulation is not harmonically related.
Just some thoughts.
By the way, friend of mine advised "Signals happen in time, not in frequency."
Eventually, couple years later, I rephrased his advice as "Frequency is just a measure of what periodicity has best correlation." Today, as I write this, I suppose that should become "....what periodicity has best auto-correlation".