Synonyms for on high
Old English heh (Anglian), heah (West Saxon) "of great height, lofty, tall, exalted, high-class," from Proto-Germanic *haukhaz (cf. Old Saxon hoh, Old Norse har, Danish høi, Swedish hög, Old Frisian hach, Dutch hoog, Old High German hoh, German hoch, Gothic hauhs "high;" also German Hügel "hill," Old Norse haugr "mound"), perhaps related to Lithuanian kaukara "hill." Spelling with -gh represents a final guttural sound in the original word, lost since 14c.
Of sound pitch, late 14c. Of roads, "most frequented or important," c.1200. Meaning "euphoric or exhilarated from alcohol" is first attested 1620s, of drugs, 1932. Sense of "proud, haughty, arrogant, supercilious" (c.1200) is reflected in high hand (late 14c.) and high horse. High seas first attested late 14c., with sense (also found in the Latin cognate) of "deep" as well as "tall" (cf. Old English heahflod "deep water," also Old Persian baršan "height, depth"). Of an evil or a punishment, "grave, serious, severe" (e.g. high treason), c.1200 (Old English had heahsynn "deadly sin, crime").
High pressure (adj.) is from 1824, of engines, 1891, of weather systems, 1933, of sales pitches. A child's high chair is from 1848. High school "school for advanced studies" attested from late 15c. in Scotland; by 1824 in U.S. High time "fully time, the fullness of time," is from late 14c. High noon is from early 14c.; the sense is "full, total, complete." High and mighty is c.1200 (heh i mahhte). High finance (1905) is that concerned with large sums. High and dry of beached things (especially ships) is from 1783. High-water mark is what is left by a flood or highest tide (1550s); figurative use by 1814.