ㅏㅑㅓㅕㅗㅛㅜㅠㅡㅣ
The first four vowels in this illustration, with the vertical lines, were incompatible with the second four vowels, the ones with the horizontal lines. The last two vowels were neutral, as was another one, not shown here, which was written as a single dot and which has since fallen out of use. [some dude]
Nononono. There are two things wrong here. The first one is that the [in]compatibility stated above, in regard of syllable composition, is plain wrong. ㅗ and ㅏ could and still can combine — that’s how we write /wa/. ㅜ and ㅓ, ditto for /we/. Moreover, ㅛ and ㅑ, and ㅠ and ㅕ combined in Middle Korean, at least in the “漢字正音” [correct pronunciation of sinograms] usage to produce diphtongs that are not possible anymore today — and that could probably be reconstructed as */ywa/ and */ywe/.
The second one is that vowel harmony in Middle Korean, like in Mongolian or Turkish — and the main link, on that matter and others, with Korean is Mongolian, not Turkish — is based on two exclusive sets of vowels, {ㅏ, ㅗ, ㆍ} [a, o, ʌ] vs {ㅓ, ㅜ, ㅡ} [e, u, ɯ]. Neither Neither ㆍ nor ㅡ were neutral. ㅣ /i/ was indeed neutral and combined in different ways with both sets of vowels: ㅏ+ㅣ -> ㅐ, ㅜ+ㅣ -> ㅟ, ㅣ+ㅗ -> ㅛ, etc… I don’t have my Middle Korean dictionaries with me in HK, but I can remember at least one example: /kasʌy/ thorn vs /kusɯl/. The harmony rule extended to the grammatical particles that are suffixed to nouns, adjectives and verbs: /mʌl + ʌn/ [the] horse [as subject] vs /mɯl + ɯn/ [the] water [as subject].