I've noticed that the engines on the A320 make a very a distinct sound on takeoff. Is there something about the CFM-56 series that causes this?
This noise can be heard in this video.
The whining noise is generated by the rotating components in the engine. At takeoff power for an engine of that size, the fan is rotating at about 5200 rpm. This is about 86 times per second. There are also 22 blades on the fan at the front of the engine. This means a given point sees a fan blade pass about 1900 times per second. Each fan blade changes the pressure as it passes, resulting in a tone of about 2kHz. This is the whine that you hear. At takeoff the tips of the fan blades are going supersonic, each one creating a shockwave.
Unfortunately I am unable to watch your video at the moment so I can't say for sure that this is the noise you are hearing, but you have commented that the noise is a whine and not engine buzz. The A320 series is known for having a distinct whine, essentially caused by the pressure equalisation vents for the fuel tanks, which are open to the external airflow.
As Lufthansa describe here, the effect is the same as when you blow over the mouth of a bottle. Lufthansa are retrofitting vortex generators on their A320s in order to reduce this whine.