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Th is a digraph in the Latin script. It was originally introduced into Latin to transliterate Greek loan words. In modern languages that use the Latin alphabet, it represents a number of different sounds. It is the most common digraph in order of frequency in the English language.
The most logical use of ⟨th⟩ is to represent a consonant cluster of the phonemes /t/ and /h/, as in English knighthood. This is not strictly a digraph, since technically a digraph is a pair of letters representing a single phoneme.
Aspirated stop /tʰ/
⟨th⟩ is used in academic transcription systems to represent letters in oriental alphabets that have the value /tʰ/. According to the Royal Thai General System of Transcription, for example, ⟨th⟩ represents a series of Thai letters with the value /tʰ/.
Alveolar stop /t/
Because neither /tʰ/ nor /θ/ were native phonemes in Latin, the Greek sound represented by ⟨th⟩ came to be pronounced /t/. The spelling retained the digraph for etymological reasons. This practice was then borrowed into German, French and other languages, where ⟨th⟩ still appears in originally Greek words, but is pronounced /t/. See German orthography. Interlingua also employs this pronunciation.
In early modern times, French, German and English all expanded this by analogy to words for which there is no etymological reason, but for the most part the modern spelling systems have eliminated this. Examples of unetymological ⟨th⟩ in English is the name of the River Thames from Middle English Temese and the name Anthony (the ⟨th⟩ is sometimes pronounced /θ/ under the influence of the spelling) from Latin Antonius.
Dental stop /t̪/
In the transcription of Australian Aboriginal languages ⟨th⟩ represents a dental stop, /t̪/.
Voiceless fricative /θ/
During late antiquity, the Greek phoneme represented by the letter ⟨θ⟩ mutated from an aspirated stop /tʰ/ to a fricative /θ/. This mutation affected the pronunciation of ⟨th⟩, which began to be used to represent the phoneme /θ/ in some of the languages that had it. The Old English Latin alphabet adapted the runic letters ⟨þ⟩ (thorn) and ⟨ð⟩ (eth) to represent this sound. The digraph ⟨th⟩ gradually superseded these letters in Middle English.
Languages besides English that use ⟨th⟩ for /θ/ include Old High German, Albanian and Welsh. Albanian and Welsh treat it as a distinct letter and alphabetize it between ⟨t⟩ and ⟨u⟩. Old High German used it before the final phase of the High German consonant shift, in which /θ/ and /ð/ came to be pronounced /d/.
Voiced fricative /ð/
English uses ⟨th⟩ to represent the voiced dental fricative /ð/. This unusual extension of the digraph to represent a voiced sound is caused by the fact that, in Old English, the sounds /θ/ and /ð/ stood in allophonic relationship to each other and so did not need to be rigorously distinguished in spelling. The letters ⟨þ⟩ and ⟨ð⟩ were used indiscriminately for both sounds, and when these were replaced by ⟨th⟩ in the 15th century, it was likewise used for both sounds. (For the same reason, ⟨s⟩ is used in English for both /s/ and /z/.)
In Irish and Scottish Gaelic, ⟨th⟩ represents the lenition of /t/. In most cases word-initially, it is pronounced /h/. For example: Irish and Scottish Gaelic toil [tɛlʲ] 'will' → do thoil [də hɛlʲ] 'your will'.
This use of digraphs with ⟨h⟩ to indicate lenition is distinct from the other uses which derive from Latin. While it is true that the presence of digraphs with ⟨h⟩ in Latin inspired the Celtic usage, their allocation to phonemes is based entirely on the internal logic of the Celtic languages.
The Irish and Scottish Gaelic lenited /t/ is silent in final position, as in Scottish Gaelic sgith /skiː/ "tired". And, rarely, it is silent in initial position, as in Scottish Gaelic thu /uː/ "you".
In English the ⟨th⟩ in asthma is often silent.