What's causing the audible 7/8th rhythm in Qi chargers?
To keep the received voltage where the receiver wants it, the receiver sends data packets back to the transmitter. When the voltage is close, the packets are sent about every 250ms. When the voltage is not close, the packets are sent about every 32ms. If the transmitter does not detect these packets, it will shut off the power and assume the device is gone.

Several factors may be involved in the audio effects you observed. One is the transmitter changing its modulation in response to these requests it is receiving from the receiver. Another is the data itself, which is transmitted by modulating the load the receiver applies.

The transmitter uses a PID algorithm to adjust its output power to provide the receiver the voltage it requires. The load drawn by the receiver varies with this voltage, creating a system that is a bit difficult to stabilize. Apparently, this algorithm is dithering in a loop that includes a portion that is somewhat far off from the desired voltage.

Getting PID algorithms to smooth out is notoriously difficult. It's somewhat analogous to a bad driver who notices he's too far from the car in front of him and so starts pressing on the accelerator but then doesn't notice he's too close until he needs to tap on the brake, repeating the cycle.

Qi's need to support a variety of chargers and devices probably made it difficult to get it to perfectly zero in on a stable setting.

The "chip-chip-chip" you are hearing is likely these coarse adjustments. The faster "tatata" sound is likely when it gets far enough off that more frequent corrections are required. As expected, the "chip"s are roughly 250ms apart. The "tatata" is likely three fast adjustments, which we would expect to be 3x32ms or .092 seconds. This fits your measurements nicely.