In the West Germanic variety that gave rise to Old English, a-mutation did not affect the second element of the diphthong */eu/ (for which the earliest Old English texts have eu): treulesnis "faithlessness", steup- "step-" (Epinal Glossary 726, 1070); but in other branches of West Germanic */eu/ eventually became */eo/ unless followed by */w/, e.g. Old Saxonbreost "breast" vs. treuwa "fidelity." In most variants of Old Norse, */eu/ > /jɒu/ > /ju:/ or /jo:/, without regard to a-mutation, e.g. Old Icelandic djúpr.
Effects of a single nasal consonant
Old English derives from a type of Germanic in which single */m/ had the same effect on preceding */u/ and */e/ as a nasal stop followed by another consonant. The effect occurs in other West Germanic languages, though more erratically, and sometimes in Old Norse.
Old Norse nema, Old High German neman : Old Frisiannima, nema, Old Saxon niman, neman : Old English niman "to take"
Old High German gi-noman, Old Frisian nomen : Old Norse numinn, Old English numen, Old Saxon numan "taken" (past participle)
Old High German gomo "man", Old Frisian gomo : Old Norse gumi, Old English guma, Old Saxon gumo
a-mutation was also sometimes blocked before single */n/, again with much variation among languages.
Old Saxon honig, -eg, Old High German hona(n)g : Old English hunig (for older -æg), Old Frisian hunig, Old Norse hunang
Alternative ideas
A number of scholars have questioned the traditional model of Proto-Germanic a-mutation in whole or in part. In particular, the rare a-mutation of */i/ to */e/ "as a Proto-Germanic phenomenon has always been contested." Lloyd, for example, proposed an alternative explanation for all apparent instances of a-mutation of */i/; he suggested that "the partial overlapping in Germanic of the two phonemes /i/ (represented in all environments by [i]) and /e/ (with the allophones [e] and [i]) led to the occasional development of an e-allophone of i by systemic analogy". Cercignani, on the other hand, argued that "no 'umlaut' phenomena can be assumed for Proto-Germanic", preferring to ascribe these changes to "the prehistory of the individual languages."
Notes
References
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Collitz, Hermann (1918), "Early Germanic vocalism", Modern Language Notes 33:6, pp. 321–333.
Gordon, E. V. (1957). An Introduction to Old Norse. Second Edition revised by A. R. Taylor. Oxford University Press.ISBN0-19-811184-3.
Grønvik, Ottar (1983). Die dialektgeographische Stellung des Krimgotischen und die krimgotische cantilena. Oslo, Universitetsforlaget.
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Lloyd L. (1966). "Is there an a-umlaut of i in Germanic?", Language 42:4, pp. 738–745.
Sturtevant (1956). "The a-umlaut of the radical vowel i in Old Norse monosyllabic stems", Modern Language Notes 71:3, pp. 194–200.
Wright (1917). Grammar of the Gothic Language. Oxford University Press.