Old English hiehþu, Anglian hehþo "highest part or point, summit; the heavens, heaven," from root of heah "high" (see high) + -itha, Germanic abstract noun suffix (as in width, depth; see -th (2)). Compare Old Norse hæð, Middle Dutch hoochte, Old High German hohida, Gothic hauhiþa "height." Meaning "distance from bottom to top" is from late 13c. Meaning "excellence, high degree of a quality" is late 14c. Century Dictionary says "there is no reason for the distinction of vowel between high and height. The modern pronunciation with -t emerged 13c. but wasn't established until 19c.; Milton used highth and heighth is still colloquial in English. Compare Dutch hoogte, Danish hjöde.
word-forming element making verbs (such as darken, weaken) from adjectives or nouns, from Old English -nian, from Proto-Germanic *-inojan (also source of Old Norse -na), from PIE adjectival suffix *-no-. Most active in Middle English and early modern English, hence most verbs in -en are comparatively recent.
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Harper, D. (n.d.). Etymology of heighten. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/heighten