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A silly what-if question that sounds a bit mad: I am curious as to why the letter "H" in English and some other European languages is used as a modifier to make diglyphs represent a single phoneme (ch, sh, th, dh, kh, gh etc.).

I cannot figure out what is the technical term for a letter used as modifier in diglyphs, which is half the problem.

The use of H is not universal in Latin script languages: some opt for diacritic marks and others for other letters. In Polish, the letter "Z" is used as a modifier like an H. In Italian CI is used as /ʧ/. While in Czech and Slovakian a caron diacritic is used instead of a H. While a dot above was used in Irish, but now an H due to foreign-made typewriters.

In runic alphabets, dots or marks within the letters were used. I understand that Germanic language orthographies were not translitterated from runes, but there were some borrowings, say wynn and thor, which survived in English despite French influence. So I would have predicted diacritics would have ended up being used...

A mirrored C means "con" in Latin, while I am pretty sure I read that some script used rotated printing types for some phonemes. Which also seems more logical.

Also, the use of H seems arbitrary. H is rare, but so are Q, X, Y, Z. And before printing presses, there were all sorts of symbols we don't use (9=orum, long esse etc.). Any of them could have been used.

I hope I made my logic of why I don't understand why 'H'...

To my understanding, it comes from TH and PH.

In Ancient Greek, there were "aspirated" consonants written Θ and Φ, which literally sounded like "t followed by h" and "p followed by h". So when words with these consonants were borrowed into Latin, which didn't have these sounds natively and thus lacked special letters for them, they were respelled as TH and PH.

Then sound changes happened, and the sounds written Θ and Φ in Greek changed to be pronounced like modern English "th" and "f". But they were still (usually) written as TH and PH in Latin, because that was the established convention. That's how we end up with words like "phone", with the "ph" representing /f/: it comes from a Latinized Greek word that originally had Φ.

Later, when Germanic languages with that same "th" sound (written /θ/ in the International Phonetic Alphabet) started being written in the Latin alphabet, scribes needed to find a way to represent this non-Latin sound. Some borrowed a runic letter, "thorn" (Þ), which is still used in Icelandic. But others looked at Greek, and the conventions for spelling Greek words with that same /θ/ sound, and wrote it TH.

Now the idea of putting an H after a letter to change its sound had been introduced, and it caught on. In English, which at one point had an alternation between /k/ and /t͡ʃ/ (both written C), they started putting an H after the second one to make CH.

But this is all, as you can see, basically an accident. Polish uses Z instead because that was Latin's most common affricate-marking letter. Italian uses I because that's the same way they represent /j/, and their [t͡ʃ] is historically and perhaps underlyingly /kj/. Hebrew uses a slash-like stroke after the letter just because it's unambiguous. And so on and so forth.

The "main" letter of a digraph is called the base, and I have never heard "the other letter" being given a name, though modifier seems most appropriate. As to "why H?", I think one should start with "when H?". The Latin alphabet derives from the Greek alphabet via a chain of historical modifications. Crucially, the Latin alphabet modifies the Etruscan alphabet: Etruscan is said to have aspirated versus unaspirated stops, and their alphabet was a version of the western Greek alphabet. So in Etruscan, 𐌙 represented [kʰ], 𐌈 represented [tʰ] and 𐌘 represented [pʰ]. When the Etruscan alphabet was adapted to Latin, these letters were dropped because Latin did not have aspiration as a feature of consonants (no point in retaining letters that you have no use for).

Use for some means of indicating aspiration arose when the Romans found need to learn Greek. There is a natural phonetic relationship between the consonant [h] and aspiration, thus φ was rendered as ph, and so on. The thing to bear in mind is that at the relevant stage, Greek φ θ χ were aspirated stops. However: their pronunciation in Greek changed so that they became affricates and then fricatives, hence "ph" came to be associated with the pronunciation "f". This gets the ball rolling. "Rh" similarly arises from the fact that initial r in Greek was voiceless and written in Greek with the aspiration diacritic.

Subsequently, the letters of the Latin alphabet were fixed (sort of), which meant that any novel sound had to be represented as a multigraph. "H" has an advantage because it was already used as a modifier in Latin. The spelling of Italian multigraphs is influenced by its historical phonology: [tʃ] is represented as "ci" because [tʃ] arose from palatalization of k (spelled "c"). The specific history of "h" as a modifier is very complex, and depends on the particular writing system.