This is Peter Pringle, an expert in ancient music. Some years ago, he says, “I built a small lyre based on a 1200 BCE artifact” found in the ruins of Megiddo. Now he uses it to accompany himself as he sings Psalm 23 in Hebrew.
No music texts from ancient Israel survive, so he had to compose a melody himself. He doesn’t claim authenticity… but this might be pretty close to what it sounded like when David first sang it.