Okay, so first, a few clarifications. Moundbuilding in the eastern US (there are some mounds out west, but they’re not included in the cultures usually associated with moundbuilders) is a practice that spans millennia, and was the product of likely hundreds of different cultures and peoples. The earliest mounds in the US are at Watson Brake in Louisiana, dating to around 3500 BC, and moundbuilding in indigenous communities didn’t end in some regions until around the 17th century. Arguably it’s still going on, since the First Americans Museum (run by indigenous people and created in tandem with the tribes) built a mound of its own back in 2021. But let’s start from the beginning.
The earliest earthen monuments in the United States predate Stonehenge, the Egyptian pyramids, and the first Mesoamerican civilizations. Around 3500 BC, earthen mound centers began appearing around the lower Mississippi River, most famously at Watson Brake. A lot of these discoveries were made very recently, so there’s a lot we don’t know about the first moundbuilders. Our knowledge starts to improve around the time of Poverty Point.
Around two thousand years after Watson Brake (~1700 BC), a new set of earthen mounds emerged to the northeast, at a site on Bayou Maçon today known by the unfortunate name of Poverty Point. This was an astounding center of trade and ritual. Its centerpiece, Mound A, stands at 72 feet (22 m) tall with a massive base and is still one of the largest mounds in North America. Poverty Point was the centerpiece of a vast continental trade network, with objects sourced from as far as Iowa and the Appalachian foothills.
The earliest mounds in North America are heavily concentrated in the lower Mississippi. Eventually, however, the practice appears to have encompassed the entirety of eastern North America. The next great heartland of moundbuilding appeared not in Louisiana, but far to the northeast in the river valleys of modern Ohio, Kentucky, and West Virginia. Here, the Adena culture thrived from around 500 BC, building massive and conical burial mounds. (Conical mounds are a rarity elsewhere). It’s very likely the Adena were also responsible for Ohio’s famous Serpent Mound, an effigy mound built to resemble the shape of a serpent. (Effigy mounds are primarily concentrated in the northern half of the US, later especially around Wisconsin and Iowa).
The Adena were followed by the spectacular Hopewell culture. The Hopewell built extraordinary earthen complexes following astronomical alignments, such as the Newark Earthworks, one of the largest earthen structures anywhere on the planet and which functioned, so far as we can tell, essentially as an ancient observatory. They also continued the Adena practice of using mounds for burials, and their connections were extraordinary, spanning across most of the eastern United States and with artifacts from as far away as California being found in the region. The time also appears to have been a period of relative peace - something that would change dramatically after the fall of the Hopewell.
Okay, so let’s briefly stop and look at your question. First, on names. All of these are modern, archaeological names. There were no people that called themselves Adena, Hopewell, or Poverty Point; they are modern concepts to describe cultures that existed without writings. So we have no way of knowing what language(s) they spoke, whether they were a unified culture or several different cultures, or even necessarily who their most direct descendants might be. The Hopewell culture seems to have ended around 500 AD; it would be another millennium before written language entered North America. So we can’t really speak of a lingua franca. We know they communicated, given obvious trade ties, but how that occurred is entirely unclear.
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