Folk history alert, none of the following is true, except maybe all of it. ;)

One of the most popular edible mushrooms in North America is the chanterelle (cantharellus cibarius.) The diminutive is “chanty,” no “e.”

Held one way the mushroom looks like a funnel or flute...

“late 18th century: from French, from modern Latin cantharellus, diminutive of cantharus, from Greek kantharos, denoting a kind of drinking container.” [OED online]

“Inverted” it's a bell...

“f. The treble, in singing; also, a treble string, or bell; also, a small bell for a chyme.”
[A Dictionarie of the French and English Tongues, (London: 1650)]

No idea about the French minstrel or chantyman's fiddle strings though.

Apropos nothing at all, another name for shantytown is... mushroom town.