Based on the context, I would think the author is trying to portray a voice restricting emotion or very considered in an otherwise emotional context.
Tethered meaning 'restricted':
Tether noun teth·er | \ ˈte-t͟hər
1a: a line (as of rope or chain) by which an animal is fastened so as to restrict its range of movement
b: a line to which someone or something is attached (as for security)
2: the limit of one's strength or resources
A more commonly used synonym would be "controlled voice"
"Controlled voice" appears in the Corpus of Contemporary American English (COCA) several times, often in a situation where people are speaking in short, concise sentences. Often in a similarly emotional context where someone is trying restrain emotion.
Examples from COCA:
In a tightly controlled voice, Amy answered: "Listen to me very, very carefully."
"Let's send it off to experts. " Raina speaks in a trenchant, controlled voice, but in fact, she can hardly contain her excitement.
She talks in a low, controlled voice, her well-manicured hands held tightly together.
This would also make sense in the context of a Colonel or other military figure who would be used to speaking and acting in a controlled manner under stress or in emotional situations.