What counts are "distinquishable" sounds. If you use a sound that cannot be distinquished from a similar sound represented by a different symbol, it is not a new sound. Therefore the answer is "finite".
So the number of possible sounds depends on both the speaker and the listener. Listen to a spoken foreign language, and it is quite difficult to distinquish a wide variety of sound. But if you grew up conversant in the language, it would be rich in sound.
Now to bring this thread back towards astronomy and god, what happens as you grow up speaking and hearing a particular language, your subconscious, I would say your superconscious, learns to process and derive meaning from the sounds below (above?) the level of your consciousness.
Scientific measurements indicate that it takes 1/2 second to realize consciously a sensation on your skin if you do not see it happening. Also, when you decide to perform some act, scientists measure neuron activity in your brain one full second before the act is initiated physically. Now if we allow for 1/2 second for transmission from the brain to the body, as the prior test suggests, then your consciousness operates on a time-scale of about 1/2 second.
Some people's consciousness operates faster than this, and some slower. Being in the slower category, I can readily recognize people who talk much faster than I. So fast that I cannot keep up trying to process their speech.
The point is that none of us operate at like one sound every half second. That's more like how long we pause before thinking of what to say. So our speech derives from something beyond our consciousness. At least in my own experience, i just sort of guide the speech with will or intention, and it sort of comes out on its own.
It may be that we have something like a tape of everything we need to say in life. And when a particular need arises we just do a tape dump of the required words.
But the bottomline here is that I think there is an aspect, a quantum computer aspect of your superconsciousness that operates much much faster than the 1/2 second time-scale of consciousness. You are just not conscious of it.