Explanation of 3 Tone Control Circuit
I'm not going to try and give you a full treatise but give you a high-level overview.



simulate this circuit &ndash; Schematic created using CircuitLab
Figure 1. A simplified variable gain inverting amplifier.
The gain of Figure 1a is \$ - \frac {R_f}{R_i} \$. Figure 1b allows adjustment of the gain by adjusting the pot. The gain increases as the wiper moves towards the IN. In practice a fixed value resistor would be added at each end of the pot to set minimum and maximum gains and present &quot;silly&quot; gain values being set by the pot at either end.
It should be fairly obvious to see that your circuit consists of three such circuits around one op-amp. What may not be so obvious is that the inverting op-amp's inverting terminal is a virtual earth (ground) for the audio signal due to the feedback and the configuration makes it a summing amplifier. The main trick of the circuit now is to feed in the three bands of interest to be summed.
We'll use a slightly different schematic for the discussion for reasons that will be made clear.

Figure 2. A better circuit? Source: Interface Bus.

Looking at the treble circuit we see that the signal to the summing point is
fed through a 5 nF capacitor. The capacitor will block low frequencies and pass high frequencies so this part of the circuit will only affect the treble.
Looking at the bass we see that there is a high frequency bypass on the pot so there is no high-frequency potential difference across it and so it will only affect bass frequencies.
The mid-range is a combination of the bass and treble and forms a band-pass filter for the mid-range.

Now spot the difference: your circuit has no bypass capacitor on the bass control. I suspect that the result would be that the bass control would work as a volume control only and that to boost the bass one would have to turn the &quot;bass&quot; control up and turn the other two down. This may be rather unsatisfactory as there may be a &quot;hump&quot; in between the middle and treble ranges.

I really want to understand how this circuit works intuitively/mathematically ...

I've never done the maths for this so you'll have to do some digging. If you analyse the circuit one band at a time (and removing the other two) you may be able to figure out the equations yourself.