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Brugmann's law, named for Karl Brugmann, is a sound law stating that in the Indo-Iranian languages, the earlier Proto-Indo-European *o normally became *a in Proto-Indo-Iranian but *? in open syllables if it was followed by one consonant and another vowel. For example, the Proto-Indo-European noun for 'wood' was *d?ru, which in Vedic became d?ru. Everywhere else, the outcome was *a, the same as the reflexes of PIE *e and *a.
The theory accounts for a number of otherwise puzzling facts. Sanskrit has pitara?, m?tara?, bhr?tara? for "fathers, mothers, brothers" but svas?ra? for "sisters", a fact neatly explained by the traditional reconstruction of the stems as *-ter- for "father, mother, brother" but *swesor- for "sister" (cf. Latin pater, m?ter, fr?ter but soror). Similarly, the vast majority of n-stem nouns in Indic have a long stem-vowel, such as brahma? "Brahmins", ?v?na? "dogs" (from *?wones), correlating with information from other Indo-European languages that they were originally *on-stems. There are also some exceptions, including uk?an- "ox", which in the earliest Indic text, the Rigveda, shows forms as uka? "oxen". They were later replaced by "regular" formations (uka? and so on, some as early as the Rigveda itself), but the notion that the short stem vowel might have been from an *en-stem is supported by the unique morphology of the Germanic forms: Old English oxa nominative singular "ox", exen plural; the Old English plural stem, such as the nominative, continues Proto-Germanic *uhsiniz < *uhsenez, with e > i in noninitial syllables followed, in Old English, by an umlaut. This is the only Old English n-stem that certainly points to *en-vocalism, rather than *on-vocalism.
The rule seems to apply to only an *o that is the ablaut alternant of *e. Non-apophonic *o, with no alternant, developed into Indo-Iranian *?: *poti- "master, lord" > Sanskrit pati-, not ?p?ti (there is no such root as ?pet- "rule, dominate"). Alternatively, it is explained by the voiceless consonant after the vowel (see also Sanskrit prati < *proti), but to adopt a form of the sound law that affects only *o in open syllables, followed by a voiced consonant, seems to be a slim basis for a rule that is so general in Indo-Iranian. Limiting the original environment to that before voiceless consonants then requires levelling of long-vowel forms to perfects and nouns with final voiceless consonants in Pre-Indo-Iranian. That faces particular problems in explaining the archaic form ?na 'he/she has reached' < *h?eh?no?e, with its very idiosyncratic synchronic relation to Sanskrit ?a? 'reach'.
Several exceptions can be addressed by the laryngeal theory. The form that is traditionally reconstructed as *owis "sheep" (Sanskrit ?vi-), is a good candidate for again reconstructing, as *h?ewi- (with an o-colouring laryngeal), rather than an ablauting o-grade.
Perhaps the most convincing confirmation comes from the inflection of the perfect: a Sanskrit root like sad- "sit" has sasada for "I sat" and sas?da for "he, she, it sat". The conventional 19th century wisdom saw it as some kind of "therapeutic" reaction to the Indo-Iranian merger of the endings *-a "I" and *-e "he/she/it" as -a, but it was troubling that the distinction was found only in roots that ended with a single consonant. That is, dadar?a "saw" is both first- and third-person singular, but a form like ?dad?r?a would have been allowed by Sanskrit syllable structure. The mystery was solved when the ending of the perfect in the first person singular was reanalyzed, on the basis of Hittite evidence as *-h?e, beginning with an a-colouring laryngeal. In other words, while Brugmann's Law was still operative, a form of the type *se-sod-h?e in the first-person singular did not have an open root syllable.
A problem for the interpretation is that roots that quite plainly must have ended in a consonant cluster including a laryngeal, such as jan- < *?enh?- "beget" and therefore should have had a short vowel throughout (like dar?- "see" < *dor?-) nevertheless show the same patterning as sad-: jajana first-person singular, jaj?na third-person singular. Whether that is a catastrophic failure of the theory or just levelling is unsure, but after all, those who think the pattern seen in roots like sad- has a morphological, not phonological, origin, have their own headaches, such as the total failure of this "morphological" development to include roots ending in two consonants. Such an argument would anyway cut the ground out from under the neat distributions seen in the kinship terms, the special behaviour of "ox" and so on.
Perhaps the most worrisome data are adverbs such as Sanskrit prati, Greek pros (< *proti) (meaning "motion from or to a place or location at a place", depending on the case of the noun that it governs) and some other forms, all of which appear to have ablauting vowels. They also all have a voiceless stop after the vowel, which may or may not be significant.
For all of its charms, Brugmann's Law now has few supporters, and even Brugmann himself eventually gave up on it). Jerzy Kury?owicz, the author of the explanation of the sasada/sas?da matter (in his Études indoeuropéennes I), eventually abandoned his analysis for of an appeal to the theory of marked vs unmarked morphological categories.
Scholars who still accept Brugmann's Law include Martin Joachim Kümmel, who compares it to developments in Anatolian and Tocharian languages and to Saussure's losses of laryngeals near *o in the internal reconstruction of pre-PIE *o as longer than *e (Kümmel 2012:308).