Then-thyn split

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The then-thyn split was a phonemic split of the Old English phoneme /θ/ into two phonemes /ð/ and /θ/ occurring in Early Middle English which resulted in "then" and "thyn" ("thin") starting with different inital consonant, /ð/ and /θ/.

In Old English, the phoneme /θ/ had two allophones, one voiced and one voiceless, which were distributed regularly according to phonetic environment.

[ð] was used between two voiced sounds (either vowels or voiced consonants).
[θ] was spoken in initial and final position, and also medially if adjacent to another unvoiced consonant.

Although Old English had two graphemes to represent these sounds, <þ> (thorn) and <ð> (eth), it used them interchangeably, unlike Old Icelandic, which used <þ> for /θ/ and <ð> for /ð/.

In early Middle English times, a group of very common function words beginning with /θ/ (the, they, there, etc.) came to be pronounced with /ð/ instead of /θ/. Possibly this was a sandhi development; as these words are frequently found in unstressed positions they can sometimes appear to run on from the preceding word, which may have resulted in the dental fricative being treated as though it were word-internal.

Words which got phonemic /ð/ from the split include:

there
them
they
the
their
theirs
that
then
than
thou
thine
thy
thyself
their
theirs
though
themselves
this
these
those
thence
therefore
although
smooth
breathe
lathe
bathe
lithe
wreathe
loathe
teethe
soothe
mouth (verb)
sheathe

English has borrowed many words from Greek. Where the original Greek had the letter <θ> (theta), English generally retained the pronunciation /θ/, regardless of phonetic environment (an exception to this rule is rhythm).

English has lost its original verb inflections. When the stem of a verb ends with a dental fricative, this was usually followed by a vowel in Old English, and was therefore voiced. It is still voiced in modern English, even though the verb inflection has disappeared leaving the /ð/ at the end of the word. Examples are to bathe, to mouth, to breathe. Often a remnant of the old inflection can be seen in the spelling in the form of a silent <e>; viewed synchronically, this <e> may be regarded as a marker of the fact that the fricative is voiced.

As a result of these three developments, there are a very small number of minimal pairs in Modern English which demonstrate that /ð/ and /θ/ are distinct phonemes:

thigh : thy - in initial position we expect /θ/, but thy belongs to the group of Middle English anomalies.
ether : either - between two vowels we expect /ð/, but ether is borrowed from Greek (this is valid only for those individuals who pronounce the latter as /i:ðɚ/, not /aiðɚ/)
loath : loathe - in final position we expect /θ/, but the <th> in loathe was originally not final, as the <e> was once pronounced. Likewise the following words are minimal pairs: