Would it be possible to play tones with the IC if I use a digital pot to control the frequency?
Yes but check the resolution of the digital pot to see how closely this allows to to generate the tones on an even tempered scale.
Is it possible to generate tones fast enough?
Not sure what you mean by this.
You should also put the output through a gate so you can turn the tone on and off. Then you might want to look at envelope shaping of the tone by controlling the on profile called the attack and the off profile called the decay.
Yes, this is a good tone generator. The sine wave can have a constant amplitude, but that makes low notes quiet and high pitches notes LOUD. The XR2206 has an input pin to control the sine wave amplitude. You can have the Arduino change the loudness so high notes are as loud as low notes.
It is fast enough to be muted or give sound quickly.
I considered XR2206 as a candidate for my recent project, and discovered that it's required minimum 10V power line, that actually creates a lot of troubles to interface IC to arduino. You would need a digital pots be tolerable up to 15-20V, though optocouplers may be simplest solution.
Does it matter if I use a DAC or a digital pot?
For an amateur like me they seem to do the same thing.
It seem easier to find a >10bit DAC that can output 10v than a digital pot.
XR2206 was cheap but I am a bit concerned about the price of a high precision DAC/Digital pot that can output 10V.
Does it exist a cheap DAC/Digital pot or other solution to control the frequency?
Of course, the thing is that the Arduino itself can easily be used to generate tones. You can approximate sinewaves (to the extent that you need to, for music you do not necessarily wish to) by using a number of output bits and a resistor string, and you can use DDS algorithms (which are actually very simple) to generate precise frequencies.
It is not unreasonable to make such a system polyphonic since it merely involves repetition of the algorigthms and addition of the outputs.
Yes you will get better resolution from a DAC ( providing it is greater than 8 bits )
For an amateur like me they seem to do the same thing.
Not quite, a DAC simply outputs a voltage where as a digital pot has two ends and a wiper. For the case where the two ends are wired up to the power supply and the wiper is the output then yes they are the same, but a digital pot can be used to do things like feed audio in the top end, ground the bottom end and use the wiper as a volume controlled output. You can't do that easily with an ADC.
Paul I have already ordered 12 pieces of XR2206 from ebay so I feel obligated to use them
I think the problem will be finding a good DAC for a affordable price.