That's how harmonics work. Harmonics are basically amplification of oscillation where two or more oscillation periods coincide. They can't occur at fractional ratios, because the oscillations have to be in-phase, IE, whole-number multiples of each other.
Think about making waves in a bathtub: when you splash at a regular frequency, the waves get bigger because each new impulse coincides with the crest or trough of the existing waves.
I just realized harmonics are really difficult to explain.

Our hypothetical 12-bladed fan produces a harmonic oscillation for every balanced pair of blades. Essentially, axes of symmetry on a rotating mass create oscillations at regular intervals. Non-prime numbers of blades will result in more axes of symmetry.
So the 12-blade fan oscillates at w(rate of rotation)/1, w/2, w/3, w/4, w/6, and w/12.

A 13-bladed fan will only oscillate at w/1 and w/13, and so will seem much quieter. The two oscillation frequencies do not have any common factors, so no harmonic amplification, so less vibration, so less noise.

Clear as mud?

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Except, regardless of how many blades it has, if they are all equal in length and spaced out evenly they will amplify like crazy. That's because each one would have the same wavelength relative to the last and following blade, they would all be generating identical pulses.

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Good point. I was thinking purely in terms of mechanical oscillation, not the pulse each blade makes as it cuts through the air.

You can make physics say anything is possible if you leave out the right bits from your theoretical system.

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RIP:
2011 E350 BlueTEC, Schwartz 81k (totaled by uninsured drunk driver)
1964 Unimog 404.1 (wrongfully towed by the city)
1994 E320, Brilliantsilber 208k (transmission shuffled off after sudden catastrophic fluid loss)
1982 300D, Silberblau under a blue repaint 256k (sold for parts)
1995 E320 Polarweiss, 131k (sold for parts)
1995 E300D Polarweiss, 287k (totaled by drunk driver)

yes but at only one frequency, if it is a 12 blade it will amplify like crazy at 5 frequencies. think of it this way, a 12 blade fan has 6 2 blade fans, 3 4 blade fans, 4 3 blade fans and 2 6 blade fane buried inside of it, each of these produces its own amplified noise frequencies.

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