**Core Concept**
The cricothyroid muscle plays a crucial role in adjusting the pitch of the voice by altering the tension of the vocal cords. This muscle is innervated by a specific nerve that originates from the superior laryngeal nerve.

**Why the Correct Answer is Right**
The cricothyroid muscle is supplied by the external branch of the superior laryngeal nerve. This nerve branch arises from the vagus nerve (cranial nerve X) and innervates the cricothyroid muscle, allowing for precise control over vocal cord tension. The external branch of the superior laryngeal nerve is responsible for transmitting motor signals to the cricothyroid muscle, enabling changes in pitch.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because the recurrent laryngeal nerve, which is also a branch of the vagus nerve, primarily innervates the intrinsic muscles of the larynx except for the cricothyroid muscle. It does not supply the cricothyroid muscle.

**Option B:** This option is incorrect because the internal branch of the superior laryngeal nerve is responsible for sensory innervation of the larynx, particularly the mucous membranes above the vocal folds. It does not supply the cricothyroid muscle.

**Option C:** This option is incorrect because the hypoglossal nerve (cranial nerve XII) primarily innervates the extrinsic muscles of the tongue, not the cricothyroid muscle.

**Clinical Pearl / High-Yield Fact**
It is essential to remember that the external branch of the superior laryngeal nerve is responsible for the motor innervation of the cricothyroid muscle. This is crucial for surgeons and anesthesiologists to avoid damaging this nerve during laryngeal procedures.

**Correct Answer:** C. External branch of the superior laryngeal nerve