Why are "bath" and "bathe" pronounced differently?

I'm specifically talking about British English. In British English, "bath" (noun) has a long vowel ([ɑː]) while the verb "bathe" has a diphthong ([eɪ]) and sounds more like the American version of "bath".

Etymonline on "bath" says:

"bath (n.) Old English bæð "an immersing of the body in water, mud, etc.," also "a quantity of water, etc., for bathing," from Proto-Germanic *badan (source also of Old Frisian beth, Old Saxon bath, Old Norse bað, Middle Dutch bat, German Bad), from PIE root *bhē- "to warm" + *-thuz, Germanic suffix indicating "act, process, condition" (as in birth, death). The etymological sense is of heating, not immersing."

And on "bathe":

"bathe (v.) Old English baþian "to wash, lave, place in a bath, take a bath" (transitive and intransitive), from root of bath (q.v.), with different vowel sound due to i-mutation. Related: Bathed; bathing. Similar nouns in Old Norse baða, Old High German badon, German baden."

However it doesn't say anything about why "bath" has a long a vowel /bɑːθ/ and "bathe" has a vowel like American "bath" /beɪð/. Googling didn't help much. I also read different articles (like grammarly) but to no avail.

There is also a question in this platform but that does not explain this difference: "to bath" vs "to bathe"

Does anyone know the reason they are pronounced differently?

EDIT after the two answers:

I did not know the symbols that are used to represent the vowel sounds so I confused them.

I should not have compared the vowels in American "bath" and British "bathe". Also pointed out by the commenters.

Solution 1:

I assume you're talking about Southern British English. Because 'bath' in Northern British English is pronounced /bæθ/, not /bɑ:θ/.
In Southern British English, 'bath' is pronounced /bɑ:θ/ and 'bathe' is pronounced /beɪð/.

I don't know what happened to these words in Old and Middle English but I'm going to apply some general pronuncation rules (Modern English).

The pronuncation of 'bath' was originally /bæθ/ in the South but due to a split known as trap bath split, the short vowel [æ] shifted to the long vowel [ɑː] before some consonants (/θ/, /sk/, /st/, /ft/ etc).

Examples:

• Ask: /æsk/ in the North and /ɑ:sk/ in the South.
• Draft: /dræft/ in the North and /drɑ:ft/ in the South.
• Fast: /fæst/ in the North and /fɑ:st/ in the South.
• Bath: /bæθ/ in the North and /bɑːθ/ in the South.

According to Pronunciation Studio:

The split is found in many very common words like: GLASS, CAST, ENHANCE and PATH. It only happens in words that are spelt with an ‘a’ which appears before one of the following consonant sounds or clusters:

+/f/ LAUGH, STAFF
+/ft/ RAFT, AFTER
+/mpl/ EXAMPLE, SAMPLE
+/nd/ COMMAND, DEMAND
+/ns/ ANSWER, FRANCE
+/n(t)ʃ/ BRANCH, RANCH
+/nt/ AUNT, ADVANTAGE
+/s/ GRASS, PASS
+/sk/ TASK, MASK
+/sp/ GRASP, CLASP
+/st/ FAST, NASTY
+/θ/ BATH, AFTERMATH

The split appeared in southern English pronunciation in the mid-17th century and it’s not clear why it affected some words and not others, though there is a clear tendency for it to appear in shorter and more commonly used words.

When we add the silent e to a word, it usually changes the vowel in a word to a diphthong or a long vowel.

Examples:

• Mat /mæt/ -> mate /meɪt/
• Rat /ræt/ -> rate /reɪt/
• Bit /bɪt/ -> bite /baɪt/ etc.

The /æ/ vowel seems to have a special relationship with /eɪ/.

When a word has the /æ/ vowel and we add the silent e to it, the /æ/ vowel often changes to /eɪ/.

This relationship can also be seen in words like profane and profanity.

The silent e often indicates a long vowel or a diphthong.

So the pronunciation of 'bath' was /bæθ/ and the silent e at the end of 'bathe' indicates that it has either a long vowel or a diphthong.

The Wikipedia article on silent e has also mentioned it:

Digraphs are sometimes treated as single letters for purposes of this rule:

bath, bathe (/bæθ/, /beɪð/)
breath, breathe (/bɹɛθ/, /bɹið/)
cloth, clothe (/klɔθ/, /kloʊð/)

The trap bath split did not affect 'bathe' so its pronuncation remained unaffected (i.e. /beɪð/).

It's worth noting that the vowel in 'bathe' is not the vowel in American English 'bath' as pointed out by Peter Shor in his comment.

The [ð] in 'bathe' is because of intervocalic fricative voicing.

Solution 2:

bathe: the a was lengthened in Middle English, which results in the Modern English "face" vowel [eɪ]

“Bathe” is pronounced with [eɪ] (which is not the sound in the American version of "bath") because [eɪ] is what a Middle English long [aː] sound turned into. The vowel in the verb bathe was lengthened during the Middle English period because the verb originally had a vowel after the ”th” consonant sound: when this vowel sound was lost (or before it was lost), it caused lengthening of the [a] in the preceding syllable. The same kind of vowel alternation shows up in the noun/verb pairs grass/graze and glass/glaze (also in some other word pairs such as brass/brazen and staff/staves).

bath: short vowel in Middle English, lengthened during the Modern English period, in only some dialects

The vowel in “bath” was lengthened later on in some (but not all) dialects by a process that also lengthened a before the other voiceless fricatives /s/ and /f/ in certain contexts. But because the lengthening of a before voiceless fricatives happened later, it resulted in a different quality of the vowel. In southern British English, lengthened a in this context has the quality of a back vowel, [ɑː].

In some American English accents, "ath" words like "bath" instead show a lengthened or "tense" vowel with a front quality, which is realized phonetically in a variety of ways (e.g. [æə̯] or [eə̯]). But however it's pronounced, I have not heard of American English speakers merging the vowel in bath with the face vowel (which is [eɪ] or [e]).

A more detailed history

You only asked about the difference between the vowels, but here is an overview of the entire history of other differences between the words.

Here's a chart:

Abbreviations: PG = Proto-Germanic, OE = Old English, ME = Middle English, ModE = Modern English, SBE = Southern British English

Between PG and OE, the sound changes of "Anglo-Frisian brightening" and "A-restoration" applied, creating the difference in Old English between æ in bæþ and a in baþian. Although marked in writing, the distinction between short æ and short a in Old English was barely contrastive and was lost in Middle English. (The Old English long vowels ǣ and ā, on the other hand, had a stronger contrast that did endure in later forms of the language.) The quoted Etymonline entry is wrong: i-mutation did not apply to either of these words. Even though baþian has an i, the i in the Class II weak verb suffix -ian did not cause i-mutation; this may be because it came from the Proto-Germanic suffix -ōną which had no *i. Fricative sounds like þ had "allophonic voicing" in Old English: at the end of a word, þ was pronounced [θ], while in the middle of a word between vowels, þ was pronounced [ð].

Between OE and Early Middle English, the vowel merger mentioned above turned the vowel in both words into short [a]. Vowel reduction caused the infinitive ending -ian to eventually become something like [ən]; word-final [n] could also be lost in this context, leaving only a schwa [ə]. Word-final schwa in Middle English was written with the letter "e", so Middle English forms like [baðə] (and others like it) are what's behind the "silent e" spelling pattern of "bathe". (I wrote a more detailed answer about "silent e" spellings here.)

During the Middle English period, final schwa sounds came to be lost, but not before causing a preceding short [a], [e] or [o] sound to lengthen to [aː], [ɛː] or [ɔː] respectively. (Lengthening of [i] and [u] happened only sometimes: the modern English "silent e" spelling patterns for the letters "i" and "u" have more complicated origins.) So by the Late Middle English period, bathe might be pronounced [baːð]. By this point, the distinction between [θ] and [ð] would be considered phonemic rather than allophonic as both sounds could occur in the same environment (at the end of a word). The loss of word-final schwa created a number of other pairs of words with alternations between voiced and voiceless word-final fricatives.

Between Middle English and Early Modern English, the Great Vowel Shift turned [aː] into [eɪ]. There were similarly large changes in the pronunciation of many other Middle English long vowel sounds. Middle English short [a] usually came to have a short front pronunciation [æ] in Modern English, but in some contexts it was backed and lengthened to [ɑː]. The history of that lengthening is really a separate question as it isn't relevant to why bath and bathe are pronounced differently.