Is it just because "ninth" has only one syllable? That wouldn't make sense, though, because saying "NINE-ith" wouldn't be worse than saying "NINE-e-tee". If we were used to "nineth", we would have no more trouble pronouncing it than "ninety" or "nineteen". Where did the "e" go, and why didn't it disappear from the other words with "nine" in them?
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The Old English spellings for the number nine were nigon or nigen or nigan (see Dutch negen) - actually written "niᵹ[oea]n" with the old Irish "g".
For instance in "The coronation of Edgar [the peaceful]" (a poem from the Anglo Saxon chronicles, composed at the end of the... 9th century) one can read:
Derived from nigon, you would find typically
It is probably fair to assume that the "-gon" part was unstressed and this explains why there are at least three variants nigon, nigen and nigan. The vowel of the second syllable was not pronounced distinctly and gradually faded away.
Actually the Middle English spelling shows that it disappeared pretty quickly. Here are a few spellings from Middle English.
1225 Ancr. R. 328
In nine you need the final "e" to suggest a pronunciation of /naɪn/ as opposed to /nin/ (nin).
As for ninth, however, it looks like the spellings nineth and ninth competed for a while and that ninth eventually prevailed. I don't think there's any rhyme in /-inθ/ in English that would justify the expense of an extra "e", the role of which would be to avoid the confusion between nineth /nainθ/ and ninth /ninθ/.
Here is an example of nineth in Modern English.
1688 R. Holme Armoury iii. 190/1
Also note: The same thing happened to transform eahtoþa into eighth: the unstressed "o" gradually became less heard and the spelling adapted.
Sources: OED and Oxford Dictionary of English Etymology 1992.
The only thing I can find that sort of covers this is in the wiki for ordinal indicator, it explains how to write them, but not why they are pronounced that way.
I think this will solve your question. I found it in the book "Vocabulary for the High School Student" by Harold Levine: