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The human throat cannot produce a completely pure tone. The bones and tissues of the skull do not resonate perfectly with all harmonics, resonating more strongly with the lower harmonics. This magnification of lower harmonics distorts the perceived tone of sound coming from inside the head.

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Nothing about that is going to make the fundamental pitch of a sound you hear seem like a different frequency than it is. It will affect your perception of the complex overtones as you say, but matching the fundamental frequency you're singing with the resonant frequency of the glass does not involve complex overtones.

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