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What might be the pronunciations of PIE "plain velar" series *k *g *gʰ, the "palatovelar" series *ḱ *ǵ *ǵʰ, and the "labiovelar" series *kʷ *gʷ *gʷʰ ?

Was the *gʰ same as the modern day Hindi ?

And were the palatovelar" series *ḱ *ǵ similar to Russian soft (palatalized) consonants? For instance кь and гь?

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    We don’t know. There are dozens of different theories, all with their strengths and weaknesses. Personally, I ascribe to the theory that they were simply /k ɡ ɡʰ kʲ ɡʲ ɡʲʰ kʷ ɡʷ ɡʷʰ/, as indicated by the transliteration, and that the aspirated stops were probably breathy voice like in most modern Indian languages. But there are many other variants including uvular, pharyngealised, pre-glottalised, ejective, assibilated and various other articulations. – Janus Bahs Jacquet Jun 15 at 8:21
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It's difficult (and possibly perilous) to be too assertive about the actual nature of these PIE phonemes.
To begin with, the three-series system is more or less decalqued from Old Indian, which is an issue as such a system is usually thought to be typologically impossible, when the extra fourth series *kh is removed.
The voiceless *k/*ḱ were probably just that: i.e voiceless. They are frequent and unmarked.
There is a recurrent pattern in loanwords that voiced phonemes are not taken over as "voiced", but "voiced aspirated", which suggests the so-called "voiced aspirated" *gʰ/*ǵʰ were just plain voiced, and that the so-called "voiced" were something else than voiced, pre-glottalized is a serious option.
examples:
Semitic *barz(il) "iron" > Latin ferrum (not **berrum), English brass (not **prass)
Afrasian *bide "monkey" > Greek pith-Ekos (not **bid-Ekos)
Sumerian gud "luth" > Greek kith-ara (not **gid-ara) etc.
Now, if you accept that Nostratic makes sense, then the velars are probably in fact uvular and the palatovelars are velars, reflecting the contrast between *q vs *k, which is fairly frequent world-wide, from Berber to Eskimo.
Now it's clear that the traditional symbols and names are deeply entrenched and people will not change them easily...

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    I find the Sumerian one less convincing than the others, because Sumerian g is usually reflected as /k/ in loans (e.g. Sumerian é.gal > Akkadian ekallum). – Draconis Jun 15 at 16:19
  • What about the pronunciation of PII consonants ? Are we a little bit more certain about PII compared to PIE ? – Nikos Jun 18 at 10:42
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It's very much uncertain, and the subject of some debate

The labiovelars are least controversial, likely being labialised velars e.g. *kʷ was /kʷ/

It's the palatovelars and plain velars where things get tricky. The palatovelars seem to have been either palatalised, or further front than the plain velars, as evidenced by their tendency to affricate first. This was the reason for their names, which would suggest that *k & *ḱ were /k/ & /kʲ~c/ respectively (this roughly corresponds to Russian кь)

The problem with this is that the plain velars are vastly rarer than the palatovelars, the opposite of what we'd expect if this identification were the case. To avoid this, many people reconstruct *k & *ḱ as /q/ & /k/ respectively instead. This would make the observed distribution typologically typical, but runs into conflicts with some (less common) reconstructions of the laryngeals. This view also sometimes suggests that *kʷ may have had variation between velar and uvular i.e. *kʷ was /kʷ~qʷ/. If *kʷ were truly velar like the so-called palatovelars rather than uvular though, this could explain why it palatalises in Albanian & Hellenic in positions where *k is preserved

My two cents here would be that the *ḱ *k *kʷ were /k/ /q/ /kʷ/

The voicing distinction is even more controversial. There are two main schools of thought, one of which is more diverse than the other

The traditional model holds that *t *d *dʰ were pronounced /t/ /d/ /dʱ/. This essentially identical to their reflexes' pronunciation in most modern Indo-Aryan languages, but is very unusual typologically (note that those Indo-Aryan languages have an additional /tʰ/ series)

The other main school of thought is the glottalic theory. The original form of which held that *t *d *dʰ were pronounced /t ~ tʰ/ /tʼ/ /d ~ dʱ/, a situation preserved more-or-less intact in Classical Armenian. This is slightly less typologically unusual, but has been argued to make some of the sound changes producing the attested daughters less typologically plausible. Since then there have been various revisions, which have little in common, other than that the mediae (the "plain voiced" stops of the traditional model) have some form of glottalisation

For my two cents, I'd probably follow Haspelmath and say that phonemes aren't as good a tool for typology as phones, and stick to the /t/ /d/ /dʱ/ pronunciation, but with the assumption that the Indo-Aryan voiceless aspirates are in fact a retention of what was originally allophonic, and not an innovation. Most other branches lost laryngeals earlier than Indo-Iranic, and so lost one of the main deciders of the allophonic variation, and simply lost the aspiration distinction, resulting in the varied disintegration of the voicing system. Indo-Aryan on the other hand preserved the early allophony as a new phonemic distinction once the Indo-Iranian laryngeals were lost

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