Possessive

I came across this in a favorite song, A Good Year for the Roses:

And the sound of our one baby's crying goes unheard.

Now I usually listen to the Counting Crows' version, and that above is how the singer renders it. our one baby's crying.

But this is a popular song to cover, and other singers say what probably I'd say:
And the sound of our one baby crying goes unheard.

Do you see the difference? The first has "baby" as the possessive noun and "crying" as a gerund (an -ing word used as a noun).

The second has "baby" as a noun, and "crying" as a participle modifying that noun.

Let's try it structurally--
The sound of our one baby('s) crying (subject phrase)
goes unheard (verb phrase)

Then let's break down the subject phrase:
The sound (noun, actual subject)
of (preposition)
our one baby (noun, object of preposition)crying (adjectival participle, modifying the noun)
our one baby's crying (object of preposition-- possessive noun and gerund noun)

That is, in both cases, the "of" phrase is modifying "the sound"-- telling us what the sound is, and the verb phrase ("goes unheard") explains what the sound DOES.

So... are both "our one baby's crying goes unheard" and "our one baby crying goes unheard" correct?

I think so. It's actually a similar issue to:
I spoke to him about his/him speeding.

That is, is the noun there going to be possessive or not? Is the participle there a gerund (noun) or an adjective?

Minor point, I know-- both are perfectly correct. But I wonder why we'd choose one over the other.

(Yes, this is the place you come when you really want to contemplate the sound of one hand clapping. Or one's hand clapping. :)

Alicia