Schwa | schwa indogermanicum
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The comparative method establishes six short vowels for Proto-Indo-European. The phonetics of the typical reflexes make five vowels easy to arrange in a common system ("the Latin five"): i e a o u.
However, a sixth correspondence set is not so simple, ə in Indo-European languages (if it survives at all; in medial syllables, it is lost in Baltic and Slavic and reflected as u, in Germanic, if it is not lost; in Indic, the reflex is i, and in Iranian, the vowel is lost):
(1) Gothic fadar "father", Latin pater, Greek patḗr, Old Irish athair /ˈaθirʲ/, but Vedic pitár-, Avestan pta, ta nominative singular (the form pita scans as a monosyllable and is presumably an orthographic artifact). (2) Gothic dauhtar (Old High German tohter and similar old Germanic forms), Old Church Slavic dŭšti, Lithuanian duktė, Vedic duhitár-, Avestan duγðar but Greek thugátēr.
The obvious slots were all taken by five short vowel reconstructions with strong phonetic claims, and the etymon for the sixth vowel was put into the most available space, phonetically speaking: not high, not low, not front, not back, not rounded: *ə "schwa".
That was not such a bad guess: in Indic, there are "prop-vowels" for otherwise impossible final consonant sequences, and they too become Vedic i: Vedic hā́rdi nominative singular "heart". The original Indo-European paradigm was based on a neuter root-noun *ḱerd-/*ǵherd- whose endingless nominative singular, pre-Indo-European **ḱerd, **ǵherd had become Proto-Indo-European *ḱēr, *ǵhēr by simplification of the final cluster with compensatory lengthening of the vowel: Greek kêr, Hittite HEART-er; in Indic, the root-final *d was restored in the nominative singular, based on all the other cases but at a cost: a word-final cluster /rd/ is phonologically impossible in Indic, a problem resolved by a prop vowel. Any vowel would have done the job, but a neutral vowel is a usual choice: Proto-Indo-Iranian *źhārd-ə from which, by regular sound laws, hā́rdi. Another example is Vedic ákṣi nominative singular neuter "eye" from *akṣ (oblique stem akṣṇ-), root *okʷ (*H₃ekʷ).
This schwa primum indogermanicum was, however, always slightly odd. Seemingly independent occurrences, as in the "father" words, were rare. More commonly, *ə alternated with long vowels, in a clearly patterned system, parallel to the alternation between a short vowel and zero: the root *sed "sit" has forms as such in Sanskrit (sadati "is sitting"), but the reduplicated present, sīdati "sits down" reflects *si-sd- with zero grade of the root: the vowel has dropped. Compare the Indic root sthā "stand", with such forms as ásthāt aorist "he stood", but the participle, where the root vowel should drop, is sthi-tá- "stood" with -i- from schwa.
Eventually, schwa indogermanicum was radically reinterpreted as the reflexe of the syllabic "laryngeals" (consonants), and what is now known as the laryngeal theory was developing into its current form. It then was often referred to as the "theory of consonantal schwa".
There is also a schwa secundum (usually, the indogermanicum is unsaid), which is some kind of reduced state of an originally short vowel. The reconstruction or reconstructions (two different schwas are commonly deployed) of 6 is only a stopgap. Its supposed reflexes are various and unpredictable, and the occurrence of the vowels has no morphological anchor, unlike the whole rest of the ablaut (vowel alternation) system. In terms of linguistic reconstruction, therefore, it has no explanatory value, being a case of putting the rabbit into the hat for the purpose of taking it back out again. In more technical terms, a schwa secundum in a reconstruction is actually a case of removing an attested mystery into the protolanguage and replacing one mystery by another. Most cases of schwa secundum are not really problems at all, being ordinary cases of levelling, or the phenomena have other and better explanations. For example, the occurrence of -u- in Greek for expected -o-, as in núx "night" and phúllon "leaf" (cf. Latin nox, folium) seems to be regular when the expected o is between a labial and a resonant consonant (núx reflects *nokʷt-s).
The Indo-European kinship terms built to a suffix that looks like *-ter-, "father, mother, brother, daughter," and "husband's brother's wife" (Sanskrit yātar-), are actually formed by a suffix *-əter-, i.e. -h₂ter-. That is, *pəter- is morphologically *p-h̥₂ter-, and the subscript ring means "syllabic", *māter- "mother" is actually *ma-h₂ter- etc.