This, I think, is the sone. To be exact, it's not a folk song, but a school song for children.

TSUKI ("The Moon")

(MIDI (with illustration) is HERE.)

Deta deta tsuki ga
Marui marui manmarui
Bon no yoona tsuki ga.

Kakureta kumo ni
Kuroi kuroi makkuroi
Sumi no yoona kumo ni.

Mata deta tsuki ga
Marui marui manmarui
Bon no yoona tsuki ga.

(Rough translation: "1. The moon has risen [or has appeared], A perfect round moon, It is like a bon ('round tray'). 2. The moon is behind the clouds, Pitch-black clouds, They are like charcoal [or India ink]. 3. The moon appears again, A perfect round moon, It is ike a bon ('round tray').")

Found other sites:
(1) Click here. With romanized lyrics & MIDI.
(2) Click here. With translation and singing of the first verse.

~Masato