Where does the phrase "run the gamut" come from?

It means "Extend over an entire range"

Gamut is a contraction of gamma and ut, the lowest and highest notes respectively

CK,

To run the gamut was a form of punishment, where the person being punished was made to run through two rows of people with clubs in there hands. Your were sentenced to run the gamut for your crime, you can guess how severe this could turn out to be.

TDCWH

to run the gamut = to cover a whole range

gamut = a complete range or extent: a face that expressed a gamut of emotions, from rage to peaceful contentment.; the entire series of recognized notes.

when you cover the whole range, you go from the first to the last, just as you complete the gamut.

not sure...will check for answer