to give off a foul stench, and so to become offensive and hated
TranslatabilityApproximateCertaintySettled
At its root this word is about smell. It describes fish rotting in the Nile, frogs piled up after a plague, food crawling with worms, and wounds festering on the body. In every such case it means to give off a foul, unmistakable stench.
From that physical picture the word takes an easy and frequent leap into the world of relationships. To become a stench to someone is to become hateful and repugnant to them, the way people back away from a bad smell. Jacob fears his sons have made him offensive to the surrounding peoples; David is reported to have made himself odious to his own men; envoys who insult a king make themselves a stench to him. The smell is gone, but the reaction, disgust and rejection, is the same.
So the two uses are not really separate words but one image working on two levels. A thing that reeks and a person who has become hateful are described with the same verb, because in both cases the response is to turn away in revulsion.
keeps alive the concrete image of a stench, so becoming hated is pictured as turning into something people physically recoil from1Sa 27:12
Both
both describe a person who has become intensely repugnant to others2Sa 10:6
Only odious
the English "odious" is abstract and formal, naming the hatred without any hint of the underlying smell
“abhorred”
Comparison data is not available for this English gloss yet.
How ba'ash differs from similar Hebrew words
צַחֲנָהThe noun for a stench itself, the reek that rises from rotting flesh; this word is the verb of giving off that smell and, unlike the noun, frequently turns figurative for becoming hated.
תָּעַבTo abhor or detest something, seen from the side of the person who is disgusted; this word describes the opposite pole, becoming the repulsive thing that others abhor.
ba'ash in the other biblical languages
ὄζωGreek“to stink”The Greek verb for giving off a smell, used of Lazarus's body in the tomb; it stays literal and never makes the leap to becoming socially hateful.
βδελύσσομαιGreek“to abhor”The Greek verb for loathing or detesting, again from the disgusted person's side; this word describes becoming the object of that loathing.
Describes a literal, physical bad smell: rotting fish in the Nile, spoiled manna breeding worms, festering wounds, and dead flies decaying in perfume.Exo 7:18Exo 16:20Psa 38:5Ecc 10:1
Extends that stench to relationships, where a person or group makes itself odious and hateful in another's sight.Gen 34:301Sa 13:42Sa 10:6
Both
Both belong to the world of a revolting stench and the revulsion it stirs: the Hebrew word names the offensive smell that makes someone hated, while the Greek word names the disgust that recoils from what is foul.Exo 5:21Rom 2:22
Names the inner act of loathing something else: the subject feels disgust and turns away from the object, as Paul says of the person who detests idols.Rom 2:22
In its passive form it brands a person or thing as morally polluted and repugnant, one of the vile who are shut outside.Rev 21:8
What scholars say about ba'ash
The verb is treated as a single root meaning "to have a bad smell, stink" (BDB; HALOT), the intransitive Qal used literally of rotting fish, frogs, manna, and festering flesh (Exod 7:18, 21; 8:14; 16:20; Ps 38:5;Isa 50:2). The causative Hiphil covers both the literal ("cause to stink," Eccl 10:1) and, far more often, the figurative "make odious" (Gen 34:30;Exod 5:21;1 Sam 27:12), while the Niphal and Hithpael are reflexive-passive, "become / make oneself odious" (1 Sam 13:4;2 Sam 10:6;2 Sam 16:21;1 Chr 19:6). Lexicographers treat the figurative sense as a natural metaphorical extension of the olfactory one rather than a separate homonym (TWOT 195; DCH). The syntax of Prov 13:5, where the verb stands without an object, is read either reflexively (the wicked makes himself offensive) or transitively (he causes shame and disgrace), a minor point that does not affect the lexical sense.
Sources
BDB, s.v. בָּאַשׁlexicon
HALOT, s.v. בָּאַשׁlexicon
DCH, s.v. בָּאַשׁlexicon
R. L. Harris, G. L. Archer, B. K. Waltke, eds., Theological Wordbook of the Old Testament (TWOT), §195reference