waves - How does increasing the tension on a string affect the density?

Consider a string under tension, for example, a string on a guitar. When a guitar string is plucked, it vibrates at a certain frequency. When the tension on the string is increased by twisting the tuning peg at the end of the neck, the frequency of the string increases. According to this video, increasing the tension on a string also increases the speed of the wave moving through it. According to this article, the speed of a wave is higher in a more dense medium.

Does this mean that increasing the tension also increases the density?

This does not seem very logical to me, since density is the "compactness" of a substance, or how close together the individual molecules are. By twisting the tuning peg on a guitar, you rotate the cylinder that the string is fixed to, wrapping it around more tightly. This pulls more of the string around that cylinder, similarly to when you reel in a fishing rod. Since more of the string's molecules are now wrapped around the peg, there should be fewer molecules over the same distance between the bridge and the nut (the two ends of the vibrating portion of the string), therefore bringing the density down.

Answer

Your intuition is right: the density of the string goes down a little bit when you increase the tension.

HOWEVER: the wave in a string is a transverse wave which depends on the tension and the mass per unit length. If you double the tension the mass per unit length goes down by a small amount (the string gets a bit "thinner" because it gets longer) . Both these things increase the velocity of the transverse wave which is given by

$$v = \sqrt{\frac{T}{\rho}}$$

Where T is the tension and $\rho$ the mass per unit length. Finally, the fundamental frequency is determined as the reciprocal of the round trip time of the wave along the string:

$$T = \frac{2 \ell}{v}$$

so that

$$f = \frac{1}{T} = \frac{v}{2\ell} = \frac{\sqrt{\frac{T}{\rho}}}{2\ell}$$

So to raise the frequency by an octave you need four times the tension (don't try this - you might break the neck) or a string that's half the diameter (one quarter of the area - so one quarter of the mass per unit length). This explains why different strings on the guitar have different gage - you would need too much tension to get the high range from a long thick string.