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How does one write exceptions into formal sound change notation?

For example, Pre-Old Japanese seems to have undergone vowel raising of /e/ and /o/ to /i/ and /u/ respectively everywhere except at the end of words (where they instead became /ʲe/ and /ʷo/ respectively).

I can write the general rules as:

e > i
o > u

But if I wanted to specify that this doesn't apply for the environment _#, how would I do this?

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The notion of "sound change notation" is a bit of a problem. It is a (not unreasonable) attempt to use a device of a different domain of linguistics, but you have identified a case where the concepts of historical linguistics do not translate into the phonological analogs that you are asking about.

There is some relationship between a given synchronic phonological rule of a language, where a rule e.g. k → č /__ i in the phonology of some language is related to some series of historical changes that we might notate as k > č / __ i. In that case, you're just replacing → with > in the informal notation of a phonological rule. There is a notation for "except when" in phonology, so you could adopt that notation in your rule. It is known as the "complement notation", and is typically indicated by prefixing the symbol "~" to the relevant expression, thus e,o > i,u /__~#.

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