Is there any rule in Old English / Modern English a/o (ham/home, ban/bone, stan/stone), a/oa (fam/foam, hlaf/loaf, gat/goat) transformation?
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1The Middle English vowel in all of these was /ɔː/. The modern spelling differences seem to be arbitrary. – Cairnarvon Jun 18 '20 at 0:59
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1I don't think there's a special term for this change (it involves backing, rounding and lowering). Minkova 2014 informally (?) refers to it the Southern Vowel Shift. for more info see Stenbrenden 2016 doi.org/10.1017/CBO9781107295469.002 – Alex B. Jun 25 '20 at 20:29
This difference is arbitrary: all of goot, gote, goote, goate, goat are attested by the end of Middle English in the 16th century, and the standardisation that happened afterward does not reflect any diachronic phonological difference.
The oCV orthography (as in gote, stone) is the older one, as it reflects the consequence of early Middle English /o/ undergoing open syllable lengthening to /ɔː/.
The "competing" value for the orthographic o, the closed /oː/, generally came from long vowels in early Midde English, so had settled into oo as its orthography.
The oa orthography is a 15th century feature, evidently by analogy with ea for /ɛː/. Compare the following, where the first two in each triplet are open-mid, and the third is close-mid.
BOAT - BOTE - BOOT
MEAT - METE - MEET
These equivalent orthographies have had different outcomes in Modern English, a merger having happened between /ɛː/ and /eː/ (the MEAT-MEET merger) in the Great Vowel Shift.