How does this inductive musical-keyboard function?
Firstly, get a cheap analogue 20MHz scope at least otherwise you'll have great difficulty debugging any project involving these boards. You're seeing 2.5V as the average of a 5V square wave at high frequency. (Either that or the oscillator is very broken).







simulate this circuit &ndash; Schematic created using CircuitLab

Start by examining U3/A at the center of this. Consider a DC analysis (pretend C7 does not conduct). The inverter's output is connected to its input through R19. This cannot be stable and it will tend to oscillate.

L1 is a representative inductor with its multiplex transistor turned on (not shown in diagram).

An edge from U3/A will be inverted again and fed into the LC oscillator tank L1/C6 (there's another capacitor and a tuning pot I've not bothered with). Oscillations coming out of the tank are fed back into the inverter pair through the comparatively small R18, so the LC tank pushes the inverter oscillator around rather than vice versa. Exact frequency analysis of this is going to be a pain but I suspect the delay through the inverters is roughly the period of 3MHz. 

So the output at F0 is a square(ish) wave whose frequency depends on the inductance of L1, which in turn depends on the position of the key. Quite a neat bit of analogue electronics - high noise immunity and good sensitivity.

How do you read this thing? Get a moderately fast microcontroller with a timer/counter unit. Use two counters: one to count input edges, one to count elapsed time. Convert to a frequency. Measure experimentally what range of values you see across the depression of a key. Not sure whether this will be linear. Try not to use long wires and avoid solderless breadboard.

(You could also do frequency-to-voltage in the analogue domain, but that's a pain)