As incredible as this might seem, we have actually found a game which generates major and minor triads.
The trick was to put the 7 natural notes in order of fifths in an "n = 3" setup of Hop-Over,
F C G D A E B with the central note "D" in the empty square,
with the 3 flat notes F C G colored red and the 3 sharp notes A E B colored blue.
The game would start with
the red flat F C G on the left,
the light grey neutral "D" in the empty square in the center, and
the blue sharp A E B on the left,
and could be played in 2 directions depending on the first move,
one game producing the major triads, disposed F A C E G B D,
and the other the minor triads, disposed D F A C E G B.
We seem to be dealing with exceptionally deep roots here,
to find an identical structure both in Harmony and in a little game for 5-year-olds.