If a humanoid had a second pair of vocal cords or similarly functioning structures under the primary vocal cords, what would be the effect on the voice? Would this allow the humanoid to sing with two voices with the proper nerve placement? Would there be no effect at all? I am aware that a syrinx would provide the effect of two voices, but I am wondering about the effect of two pairs of vocal cords.
@JBH's answer is great; it deals with questions of biology, and it sets out the different possibilities for parallel/series vocal cords. However, it's confused around what series vocal cords entail. Here I'll take a purely acoustic perspective to flesh that out.
With vocal cords in parallel (ie. one for each lung, with the airstream combined further up), your humanoids could speak/sing with two pitches at once. Since they still share the same articulators (everything in the oral cavity, etc.) they couldn't speak or sing different words, produce different vowel sounds, or anything like that. But they could sing, say, a major third interval. So it depends on your definition of voice, but basically yes.
With vocal cords in series (one after the other in the trachea), they could still produce more than one simple pitch, but not in such a straightforward way. If you've ever sung into a wind instrument, you'll have noticed that the note you sing and the note you're fingering on the instrument combine in a weird way. The pitches of the two vocal cords would combine in the same way as that. In fact, these humanoids might end up sounding like Daleks!
So how exactly do the pitches interact?
Long answer: summing vs. modulating
All the answers so far have expressed some intuition that the vocal cords in series do something different. @Willk says it'd be like overtone singing. @RonJohn says they'd interfere and block each other. @JBH goes as far as to say that they could achieve anything from wah to reverb. Actually @RonJohn comes closest in saying they interfere. More precisely, they modulate each other, though this is more productive than he suggests.
When two people sing an interval( or if two sets of vocal cords with different pulmonary sources vocalise into the same downstream air passage) the pressure waves are summed: that is, at every point in time the value of the output is the sum of the values of the inputs. That's because the airflow through each is independent of the other, so each just contributes some air pressure to the common passage at their output.
When we talk about a signal having two frequencies in it, or when we talk about mixing two sounds, that's always a sum.
So what happens in the case of vocal cords in series? Here, when one vocal cord is constricted, the other can't produce any sound, so the amplitude of one modulates the other. When both of the oscillations are at audible frequencies, this is called ring modulation. Rather than being summed, the signals are multiplied. Note the differences between the above and below graphs. In the above, it's as if the faster wave rides along the path of the slower wave. In the graph below, the faster wave is enveloped by the slower wave. (Caveat: unlike the electrical signals graphed here, acoustic signals are unipolar and so the multiplication would look a bit different.)
So why does it sound different?
The result of ring modulating two frequencies is the so-called 'sum and difference' of those frequencies. That is, the frequencies you get out aren't the ones you get in, and crucially, any harmonic relations between the input frequencies don't necessarily hold in the output frequencies. Harmonic relations (one frequency being an integer multiple of another) are very important to our perception of sounds (it's why we like harmonic intervals, like thirds and fifths), so this inharmonicity sounds weird, and can even make it hard to resolve a pitch to the sound. All the more so for the complex signals produced by vocal cords.
So this is a bit whackier than the suggestion of overtone singing, which involves your oral cavity forming a resonator that picks out and amplifies one harmonic frequency from your voice so much that it's perceived as a separate tone. But it's not quite as whacky as being able to achieve reverb effects.
On a final note
Ring modulation is unintuitive to us, and thus hard to control. However, control will come naturally to these creatures. Most examples of ring modulation used in sound effects (such as Dalek voices) involves a constant tone modulating a complex signal, or (in the case of the flute singing example) two melodies a constant interval apart. If the bivocal humanoids had mastery over dynamically varying both vocal cords at once, I'd imagine they could create all sorts of interesting polyphonies.
What would be the evolutionary advantage to this?
A wider range of sound frequencies may (maybe) allow for a more complicated communication ... but we already have languages that range from very simple to unbelievably complex. So, while I provide some insight with my answer, please bear in mind that this evolutionary development doesn't actually make sense.
I can envision a second set of chords systemically in three ways
Series: The two vocal chords are in series, meaning one immediately follows the other in the tracheal path. This solution means the second set of chords modulates the first set of chords. I imagine the result to be little different from passing an audio signal through a sound post-processor. You can easily obtain a drone effect (one or more harmonic tones in the background of the primary chord) or produce secondary effects such as shifting tone, producing reverbration, or maybe even the vocal equivalent of a cosmic wah. You could make yourself sound like your own talk box.
Parallel: The two vocal chords are in parallel, meaning the tracheal path splits to deliver 50% of the air to each chord, which ostensibly could be actuated independently. I imagine this could give rise to two independent frequencies, but only insofar as those frequencies are not shaped by the hollow of the mouth and throat, which are common to both vocal chords. This could give rise to a very rich voice, capable of providing its own contrasting harmony. Note that I do not believe this would achieve the idea of singing with two voices. It would only add depth to the single voice and complexity to communication with distinct secondary frequencies following a primary frequency. For example, English-speaking humans ask a question and usually raise the frequency of the sound as the question comes to an end. A parallel-chord being could add a second frequency to the first, providing more information about the nature of the asked question.
Integrated: An integrated dual vocal chord is a single voice box in the trachea that contains two separate sets of chords — or, perhaps more obviously, it has twice the number of "chords." I'll be honest with you, I'm not sure how this would sound, other than to provide a greater range of frequencies (adding chords to the same basic space requires the new chords to be smaller, suggesting a higher frequency range).
More air needed to vocalize, suggesting that we would all speak more quietly. (That would be nice....)
More space is needed in the throat, suggesting necks would be bulkier.
You just doubled the ability for something to go wrong (e.g., laryngitis), and I'm not sure if biologically having two sets of chords means the two sets would ever be 100% independent of one another. In other words, if you lost your voice in one box, you'd sound like a frog regardless.
There would be a difference, but I don't believe it would be a great difference. Talented singers would produce richer sound and our communication would be more nuanced, but in general, it wouldn't be a dramatic difference.
On the other hand, the price we'd pay to get that minor difference is very high with a much more complicated throat structure and all the possibilities for problems that would go with it.
There are ways to use other throat / mouth structures as additional sound generation organs analogous to vocal cords. You can get an idea of what your 2-vocal cord creatures might sound like by listening to cats and overtone singers.
Overtone singing. https://en.wikipedia.org/wiki/Overtone_singing Tuvan throat singing is the most famous of these styles. The link has sound clips example. The singers align structures in the throat (or mouth) such that they vibrate with harmonic frequencies of the main note produced by the vocal cords.
Most of the throat singers you find on the web sound like metal dudes. But the sounds this lady makes do not sound like human sounds. https://youtu.be/2kzPek5XbUg?t=266
Cats do something like overtone singing when they purr.
The mechanism of how cats purr is elusive. This is partly because cats do not have an unique anatomical feature that is clearly responsible for this vocalization. One hypothesis, supported by electromyographic studies, is that cats produce the purring noise by using the vocal folds and/or the muscles of the larynx to alternately dilate and constrict the glottis rapidly, causing air vibrations during inhalation and exhalation. Combined with the steady inhalation and exhalation as the cat breathes, a purring noise is produced with strong harmonics.
Your being with 2 sets of vocal cords would need to run them both off the same airstream. One resonating structure (like one guitar string or one set of vocal cords) can produce one tone at a time. Your creatures could use them together or one at a time. As with overtone singers the second set could vibrate according to a harmonic of the first. Or, like cats, the second set could be used to produce vibrations with a different tonal range than the first.