The intuitive explanation is a symmetry argument. A wave that is equal to itself when shifted half a cycle has only even harmonics, while a wave that is opposite to itself when shifted half a cycle has only odd harmonics. If the duty cycle of a square wave is not exactly 50%, it will have both even and odd harmonics.
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Sum of 1st, 3rd, 5th, and 7th harmonics approximates square wave. Sum of 1st, 3rd, 5th, 7th and 9th harmonics approximates square wave. The end result of adding the first five odd harmonic waveforms together (all at the proper amplitudes, of course) is a close approximation of a square wave.
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