The other answers are excellent, however I want to address this question:
How does a multiplexer play into this?
Just as an example of how multiplexing helps, I opened up a scrolling LED sign which has 504 individual LEDs which "appear" to be all on at once (if wanted).
Internally were 10 x 74HC595 serial shift registers. 9 of them were hooked up to the individual columns (giving 72 columns), and the last one was connected to the 7 rows.
9 * 8 = 72 columns
72 * 7 = 504 LEDs
Thus we could turn on 504 LEDs. However to have individual LEDs appear to come on the code has to turn on one row (out of the 7) and then output the pattern for that row to the 72 columns. Then it switches to another row, does the next pattern, and so on.
I have a video of this effect on Vimeo.
If you do it fast enough, all the LEDs appear to be one at once.
Now you can do this for switches too, just connect all the individual positions together, and then apply "power" to each switch in sequence, and read the results. This time you want an input shift register, eg. the 74HC165 for the reading, and an output register for powering each switch.
The problem is with switches though, that you would get "phantom" reads because switches aren't diodes (or, worse, the inactive switches would short the power to ground). So you would need a diode in series with each switch to prevent that. Ah well, diodes are cheap.
This is exactly the same concept as the way keypads work. Say, a 16-key keypad will have 8 wires: 4 rows and 4 columns.
Having said all that, you will still need quite a few chips unless you set up quite a complex "simulated" keypad array. Maybe it wouldn't be too bad.
You have 17 switches so that would be 2 x 74HC595 for activating each switch, plus one spare pin on the Arduino. And the switches seem to have around 12 positions, so that would be 2 x 74HC165 to read which position it was at.
So, for each switch, you make that one HIGH and the others LOW. Current will flow through the diodes and the rotary switch, presenting a HIGH on one of the 12 "reading" pins. The diodes stop the current from shorting the HIGH to the LOW through other switches.
You take a reading, and move onto the next switch. This can be done quickly, in the same way the LED sign scrolls so fast you can't spot any flickering.
This schematic illustrates the idea:
In this example we apply a HIGH to the common part (rotor) of the left switch. It goes through the diode, and the selected position, to make input "read 1" high. The diode on the other switch prevents the HIGH from being shorted to the LOW, even though both switches are in the same position.
You would need a pull-down resistor for each of the 12 "reading" lines, so that if they weren't being forced HIGH, they would read LOW. And one diode per rotary switch. So, not too many extra parts. And the wiring is simple, just connect each corresponding switch pin together (except for the common one, of course).
I think this will work, unless anyone can see a major flaw. :)