I get that we can gain some instrumental insight from instrument artifacts, but was there a way to write music that we can understand in the same manner as today’s scale? Thanks!
Can you define what period and/or cultures you’re talking about, specifically?
Unfortunately, we don't quite know. Staff notation dates back only to the 11th century when Guido of Arezzo developed a notation that included the rhythm to allow for faster training with Gregorian chants.
Notation before this would only include the melody (the individual notes being played) and whether they went up or down in pitch, a basic skeleton to use as reference for someone who had already learned the piece by rote. In many cases we have notation but no knowledge of exactly what tones the notes are referring to! Reconstructing ancient music without the rhythm or knowledge of exact notes is thus a lot of imprecise guesswork.