Why does pitch change with change of intensity?

 At your risk.

For example playing a 440Hz louder gives the perception of a lower pitch.

Why?

Not clear to me.

A piano can convert frequency to distance. For example by pressing the right pedal we leave all strings undamped. Then then if we whistle for example, the higher the frequency the more to the right strings are vibrated by resonance.

The basilar membrane converts frequency to length. 

Is it a nature's way of protecting the ear? Pitch change by stimulus intensity preventing the same hair cells being fired  along the length of the basilar membrane?

Is the change of position at the basilar membrane due to some non linearity in amplitude response. creating a DC offset?

Similar to demodulation of AM signals by the rectification effect of a diode?

(Side chain signal of a compressor created by diode rectification? Does the movement of the 3 tiny bones in the middle ear play a nonlinearity part too?)

Similar to DC bias change of single ended amplifiers with cathode current bias, RC (cathode resistor).

Similar to expansion in solids by temperature increase, there is nonlinearity in the vibration of molecules, the displacement to the right is higher that the displacement to the left.