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Explanation of ablaut and other vowel changes
A feature of Proto-Indo-European morpheme structure was a system of vowel alternations termed ablaut ("alternate sound") by early German scholars and still generally known by that term (except in French, where the term apophonie is preferred). Several different such patterns have been discerned, but the commonest one, by a wide margin, is e/o/∅ alternation found in a majority of roots, in many verb and noun stems, and even in some affixes (the genitive singular ending, for example, is attested as *-es, *-os, and *-s). The different states are called ablaut grades; e grade and o grade are together "full grades", and the total absence of any vowel is "zero grade".

Root *sed-
Thus the root *sed- "to sit (down)" (roots are traditionally cited in the e grade, if they have one) has three different shapes: *sed-, *sod-, and *sd-. This patterning is found throughout the PIE root inventory and is transparent:


 * *sed-: (Vedic), **sed-: in Latin sedeō "am sitting," Old English sittan "to sit" < *sit-ja- < *sed-; Greek hédrā "seat, chair" < *sed- (Greek systemically turns word-initial prevocalic s to h, i.e. rough breathing).
 * *sod-: in Latin solium "throne" (in Latin l sporadically replaces d between vowels, said by Roman grammarians to be a Sabine trait) = Old Irish suideⁿ /suðʲe/ "a sitting" (all details regular from PIE *sod-yo-m); Gothic satjan = Old English settan "to set" (causative) < *sat-ja- (umlaut again) < PIE *sod-eye-. PIE *se-sod-e "sat" (perfect) > Sanskrit sa-sād-a per Brugmann's law.
 * *sd-: in compounds, as *ni- "down" + *sd- = *nisdos "nest": English nest < Proto-Germanic *nistaz, Latin nīdus < *nizdos (all regular developments); Slavic gnězdo < *g-ně-sd-os. The 3pl (third person plural) of the perfect would have been *se-sd-ṛ whence Indo-Iranian *sazdṛ, which gives (by regular developments) Sanskrit sedur /seːdur/.

Roots *deh₃- and *steh₂-
In addition to the commonplace roots of consonant + vowel + consonant structure, there are also well-attested roots like *dheh₁- "put, place" and *deh₃- "give" (mentioned above): these end in a vowel, which is always long in the categories where roots like *sed- have full grades; and in those forms where zero grade would be expected, if before an affix beginning with a consonant, we find a short vowel, reconstructed as *ə, or schwa (more formally, schwa primum indogermanicum). An independent schwa, like the one in PIE *pǝter- "father," can be identified by the distinctive cross-language correspondences of this vowel that are different from the other five short vowels. (Before an affix beginning with a vowel, there is no trace of a vowel in the root, as shown below.)

Since short vowels disappear entirely in roots like *sed-/*sod-/*sd-, it was a reasonable inference that a long vowel under the same conditions would not quite disappear, but would leave a sort of residue. This residue is reflected as i in Indic while dropping in Iranian; it gives variously e, a, o in Greek; it mostly falls together with the reflexes of PIE *a in the other languages (always bearing in mind that short vowels in non-initial syllables undergo various developments in Italic, Celtic, and Germanic):


 * *deh₃- "give": in Latin dōnum "gift" = Old Irish dán /daːn/ and Sanskrit dâna- (â = ā with tonic accent); Greek dí-dō-mi (reduplicated present) "I give" = Sanskrit dádāmi; Slavic damъ 'I give'. But in the participles, Greek dotós "given" = Sanskrit ditá-, Latin datus all < *dh₃-tó-.
 * *steh₂-- "stand": in Greek hístēmi (reduplicated present, regular from *si-steh₂-), Sanskrit a-sthā-t aorist "stood", Latin testāmentum "testimony" < *ter-stā- < *tri-steh₂- "third party", Slavic sta-ti 'to stand'. But Sanskrit sthitá-"stood", Greek stásis "a standing", Latin supine infinitive statum "to stand".

Conventional wisdom lined up roots of the *sed- and *deh₃- types as follows:

But there are other patterns of normal roots, such as those ending with one of the six resonants (*y w r l m n), a class of sounds whose peculiarity in Proto-Indo-European is that they are both syllabic (vowels, in effect) and consonants, depending on what sounds are adjacent:

Root *bʰer-/bʰor-/bʰṛ- ~ bʰr-

 * *bʰer-: in Latin ferō = Greek phérō, Avestan barā, Sanskrit bharāmi, Old Irish biur, Old Norse ber, Old English bere all "I carry"; Slavic berǫ 'I take'; Latin ferculum "bier, litter" < *bʰer-tlo- "implement for carrying".
 * *bʰor-: in Gothic and Scandinavian barn "child" (= English dial. bairn), Greek phoréō "I wear [clothes]" (frequentative formation, *"carry around"); Sanskrit bhâra- "burden" (*bʰor-o- via Brugmann's law); Slavic vyborъ 'choice'.
 * '- before consonants: Sanskrit '- "a carrying"; Gothic gabaurþs /gaˈbɔrθs/, Old English ġebyrd /jəˈbyɹd/, Old High German geburt all "birth" < *gaburdi- < *; Slavic bьrati 'to take'.
 * *bʰr- before vowels: Ved bibhrati 3pl. "they carry" < *; Greek di-phrós "chariot footboard big enough for two men" < *dwi-bʰr-o-.

Saussure's insight was to align the long-vowel roots like *deh₃--, *steh₂-- with roots like *bʰer-, rather than with roots of the *sed- sort. That is, treating schwa not as a residue of a long vowel, but like the *r of *bʰer-/*bʰor-/*bʰṛ-, an element that was present in the root in all grades, but which in full grade forms coalesced with an ordinary e/o root vowel to make a long vowel, with colouring of the e grade into the bargain; the mystery element was seen by itself only in zero grade forms:

( = syllabic form of the mystery element)

Saussure treated only two of these elements, corresponding to our *h₂ and *h₃. Later it was noticed that the explanatory power of the theory, as well as its elegance, were enhanced if a third element were added, our *h₁, which has the same lengthening and syllabifying properties as the other two but has no effect on the colour of adjacent vowels. Saussure did not suggest as to the phonetics of these elements; his term for them, coefficients sonantiques, did not fail, but merely the term in general use for glides, nasals, and liquids (i.e., the PIE resonants) as in roots like *bʰer-.

As mentioned above, in forms like *dwi-bʰr-o- (etymon of Greek diphrós, above), the new coefficients sonantiques (unlike the six resonants) have no reflexes at all in any daughter language. Thus the compound *'- "to 'fix thought', be devout, become rapt" forms a noun *'- seen in Proto-Indo-Iranian *mazdha- whence Sanskrit medhá- /mēdha/ "sacrificial rite, holiness" (regular development as in sedur < *sazdur, above), Avestan mazda- "name (originally an epithet) of the greatest deity".

Root *bʰendʰ
There is another kind of unproblematic root, in which obstruents flank a resonant. In the zero grade, unlike the case with roots of the *bher- type, the resonant is therefore always syllabic (being always between two consonants). An example would be *bʰendʰ- "tie, bind":


 * *bʰendʰ-: in Germanic forms like Old English bindan "to tie, bind", Gothic bindan; Lithuanian beñdras "chum", Greek peĩsma "rope, cable" /pêːsma/ < *phenth-sma < *bʰendʰ-smṇ.
 * *bʰondʰ-: in Sanskrit bandhá- "bond, fastening" (*bʰondʰ-o-; Grassmann's law) = Old Icelandic bant, OE bænd; Old English bænd, Gothic band "he tied" < *(bʰe)bʰondʰ-e.
 * '-: in Sanskrit baddhá- < *'- (Bartholomae's law), Old English gebunden, Gothic bundan; German Bund "league." (English bind and bound show the effects of secondary (Middle English) vowel lengthening; the original length is preserved in bundle.)

This is all straightforward and such roots fit directly into the overall patterns. Less so are certain roots that seem sometimes to go like the *bʰer- type, and sometimes to be unlike any other, with (for example) long syllabics in the zero grades while at times pointing to a two-vowel root structure. These roots are variously called heavy bases, disyllabic roots, and seṭ roots (the last being a term from Pāṇini's grammar. It will be explained below).

Root *ǵenh₁, *ǵonh₁, *ǵṇh₁-

 * *ǵenh₁-
 * PIE *ǵenh₁os- neut s-stem "race, clan" > Greek (Homeric) génos, -eos, Sanskrit jánas-, Avestan zanō, Latin genus, -eris.
 * Greek gené-tēs "begetter, father"; géne-sis < *ǵenh₁-ti- "origin"; Sanskrit jáni-man- "birth, lineage", jáni-tar- "progenitor, father", Latin genitus "begotten" < genatos.


 * *ǵonh₁-
 * Sanskrit janayati "beget" = Old English cennan /kennan/ < *ǵonh₁-eye- (causative); Sanskrit jána- "race" (o-grade o-stem) = Greek gónos, -ou "offspring".
 * Sanskrit jajāna 3sg. "was born" < *ǵe-ǵonh₁-e.


 * *ǵṇh₁-
 * Gothic kuni "clan, family" = OE cynn /kynn/, English kin; Rigvedic jajanúr 3pl.perfect < *- (a relic; the regular Sanskrit form in paradigms like this is jajñur, a remodelling).
 * Sanskrit jātá- "born" = Latin nātus (Old Latin gnātus, and cf. forms like cognātus "related by birth", Greek kasí-gnētos "brother"); Greek gnḗsios "belonging to the race". (The ē in these Greek forms can be shown to be original, not Attic-Ionic developments from Proto-Greek *ā.)

Discussion
The startling reflexes of these roots in zero-grade before a consonant (in this case, Sanskrit ā, Greek nē, Latin nā, Lithuanian ìn) is explained by the lengthening of the (originally perfectly ordinary) syllabic resonant before the lost laryngeal, while the same laryngeal protects the syllabic status of the preceding resonant even before an affix beginning with a vowel: the archaic Vedic form jajanur cited above is structurally quite the same (*') as a form like *' "they saw" < *.

Incidentally, redesigning the root as *ǵenH- has another consequence. Several of the Sanskrit forms cited above come from what look like o-grade root vowels in open syllables, but fail to lengthen to -ā- per Brugmann's law. All becomes clear when it is understood that in such forms as *ǵonH- before a vowel, the *o is not in fact in an open syllable. And in turn, that means that a form like jajāna "was born," which apparently does show the action of Brugmann's law, is a false witness: in the Sanskrit perfect tense, the whole class of seṭ roots, en masse, acquired the shape of the aniṭ 3sing. forms. (See Brugmann's law for further discussion.)

Other roots
There are also roots ending in a stop followed by a laryngeal, as *'- "spread, flatten," from which Sanskrit '- "broad" masc. (= Avestan pərəθu-), pṛthivī- fem., Greek platús (zero grade); Skt. prathimán- "wideness" (full grade), Greek platamṓn "flat stone." The laryngeal explains (a) the change of *t to *th in Proto-Indo-Iranian, (b) the correspondence between Greek -a-, Sanskrit -i- and no vowel in Avestan (Avestan pərəθwī "broad" fem. in two syllables vs Sanskrit pṛthivī- in three).


 * Caution has to be used in interpreting data from Indic in particular. Sanskrit remained in use as a poetic, scientific, and classical language for many centuries, and the multitude of inherited patterns of alternation of obscure motivation (such as the division into seṭ and aniṭ roots) provided models for coining new forms on the wrong patterns. There are many forms like tṛṣita- "thirsty" and tániman- "slenderness", that is, seṭ formations to unequivocally aniṭ roots; and conversely aniṭ forms like píparti "fills", pṛta- "filled", to securely seṭ roots (cf. the "real" past participle, pūrṇá-). Sanskrit preserves the effects of laryngeal phonology with wonderful clarity but looks upon the historical linguist with a threatening eye: for even in Vedic Sanskrit, the evidence has to be weighed carefully with due concern for the antiquity of the forms and the overall texture of the data. (It is no help that Proto-Indo-European itself had roots which varied in their makeup, as *ǵhew- and *ǵhewd-, both "pour"; and some of these root extensions are, unluckily, laryngeals.)

Stray laryngeals can be found in isolated or seemingly isolated forms; here the three-way Greek reflexes of syllabic *h₁, *h₂, *h₃ are particularly helpful, as seen below. (Comments on the forms follow.)


 *  in Greek ánemos "wind" (cf. Latin animus "breath, spirit; mind," Vedic aniti "breathes") < *anə- "breathe; blow" (now *h₂enh₁-). Perhaps also Greek híeros "mighty, super-human; divine; holy," cf. Sanskrit iṣirá- "vigorous, energetic."
 *  in Greek patḗr "father" = Sanskrit pitár-, Old English fæder, Gothic fadar, Latin pater. Also * "big" neut. > Greek méga, Sanskrit máha.
 *  in Greek árotron "plow" = Welsh aradr, Old Norse arðr, Lithuanian árklas.