Altho I really like the large silent clock, but it would be nice if it had a tick tock option; perhaps impulse the second hand to move and tick once every second instead of running smoothly? The nano may have enough room to add a tick program and drive a tiny speaker. My wife said upon seeing the silent clock for the first time: "what, a clock that doesn't tick!" I'm sure it is a personal preference thing but thought I'd mention it.
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It seems possible. There should be plenty of processing power in the Arduino Nano to handle audio files. And there are plenty of available pins. The algorithm would need to be modified to handle both tasks at the same time.
One limitation is the Arduino Nano does not have a DAC, so the audio might end up sounding like a digital greeting card. A binary weighted resistor network could be used to make it similar to the first resistor based clock motor driver.
Another approach could be to let the gears do the ticking. Change the algorithm from smooth motion to start/stop. Move double speed for half a second, then stop and wait for the RTC to tick. There should be enough backlash in the gears to make a ticking sound. It could even run backwards for 0.1 second before advancing 1.1 seconds, similar to the old school clocks.