Although the notion of pitches being relatively high or low was well-established by the first century C.E., when Pliny used the terms summus, medius, and imus, there is no evidence that earlier Greek theorists espoused this metaphor. The terms νήτη (nētē, “down-located”) and ὑπάτη (hypatē, “up-located”) were used, but they referred to the physical placement of kithara strings, not to a spatial concept of pitch; in fact, the higher the pitch in our terms, the further down-located it was on the instrument.
The commonest adjectives for pitch in ancient Greek writings are ὀξύς (oxys, “sharp, piercing”) and βαρύς (barys, “heavy”). Ptolemy wrote that the former quality was a result of λεπτότης (leptotēs, “fineness”) and πυκνότης (pyknotēs, “close spacing [of notes]”, and that the latter was caused by μανότης (manotēs, “thickness”) and παχύτης ( pachytēs, “loose spacing”) . The idea of higher and lower sounds, and their eventual depiction as such in notation, was a later development.
This according to “The development of vertical direction in the spatial representation of sound” by Eleonora Rocconi, an essay included in Archäologie früher Klangerzeugung und Tonordnung: Musikarchäologie in der Ägäis und Anatolien (Rahden: Leidorf, 2002), pp. 389–392.
Many thanks to David Bloom for help with this post!
Related article: The perfect-pitch puzzle