= Vocal tract
{title2=$f_n\simeq(2n-1)c/(4L)$}
The <vocal tract> is the air passage above the sound-producing region of the <larynx>, including the pharyngeal, oral and, when acoustically coupled, nasal cavities. Movements of the <tongue>, <lips>, <jaw> and soft <palate> change its shape. These changes help turn a laryngeal <sound> source or a turbulent air stream into distinguishable <speech> <sounds>.<Voice, Speech, and Language: What are they? by NIDCD>[Voice and speech definitions: vocal-fold source and articulatory shaping]{ref}
A simple acoustic model treats an unbranched tract of length $L$ as a uniform tube closed at one end and open at the other. A <pressure> maximum at the closed end and <pressure> minimum at the open end allow odd quarter <wavelengths>, giving
$$
f_n=\frac{(2n-1)c}{4L},\qquad n=1,2,\ldots.
$$
With $L=0.17$ m and <sound speed> $c=340$ m/s, these <resonances> are $500$, $1500$ and $2500$ Hz. A real <vocal tract> is neither uniform nor permanently closed at the glottis; its varying cross-section and branching shift the <resonances>. These spectral <resonances>, associated with <formants>, differ from the repetition <frequency> of vocal-fold vibration. This is why changing articulation can change a <vowel> without requiring the same change in voice pitch.
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