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As far as I'm aware, languages which do not have contrastive tongue root advancement/retraction can still differ in terms of tongue root position. Are there any acoustic measurements that can be used to determine this position? Also, is there any known difference between English and French?

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There are a number of studies which attempt to discern the phonetic properties of so-called ATR vs. non-ATR vowels, starting with Lindau (1975). One problem which has been evident from the start is that languages differ in how the transcriptional difference [i] vs [ɪ], even just looking at African languages with phonologically-established ATR harmony systems. The typical formant-based approach is to plot the vowels in a standard vowel space (F2-F1 against inverse F1). When the supposedly "advanced" vowels systematically appear to the left of the corresponding non-advanced vowels, people tend to conclude that there is a tongue root distinction at work. However, this only works for vowels of "comparable heights", for example, the [+ATR] vowels of Akan are generally "to the left" of the [-ATR] vowels when considered on a pairwise basis, but [e] and [ɪ] are horizontally the same. But in Igbo, regardless of height, the [+ATR] vowels are all "to the left" of the [-ATR] vowels when you separate the vowels into front vs. back sets. In Ateso, [ɪ] is much further left than [e] ([i] is a bit further to the left of [ɪ]); but [o,u] are horizontally the same, as are [ɔ,ʊ] (which are somewhat to the right of [o,u]). In other words, there is little systematic formant pattern even for languages with full-blown ATR distinctions and active ATR harmony.

There have been some attempts to relate "ATR" to spectral slope and formant bandwidth, see this article by Hess. This is motivated by a widespread auditory observation – limited as far as I know to African languages – that [-ATR] vowels have a subjectively "brighter" quality and [+ATR] vowels have a "muffled" quality, which usually corresponds to a flatter source spectrum in the [+ATR] case (more high-frequency energy) and a rapid downward slope of the source in [+ATR] vowels. (A topic studied in the 80's in various studies at UCLA). And yet... sometimes you can call on this muffled / bright distinction to ferret out an ATR contrast (which can be really hard to perceive), and sometimes (usually) you can't.

In lieu of a minimal contrast in ATR, there is no practical means of using an acoustic measurement to recover tongue root position. The problem as I understand it is that there is no good way to determine "this effect is because of the tongue root" from the acoustics. Lindau's dissertation is very useful betcause it also provides x-ray tracings, where it is crystal clear that all of the [+ATR] vowels withing a front/back set have a tongue root position that is advanced relative to any [-ATR] vowel. But, front-back tongue position does also affect tongue-root position (in back vowels, target vowel height influences jaw height quite a lot, which in turn carries the tongue root along). Theoretically, if you can control all of the physiological parameters for vowel production and can model a vocal tract shape, you could compute the resonance properties of such a vocal tract, and maybe discern an acoustic consequence of "tongue root advancing". This is well beyond what ordinary mortals can do at present. Just at the level of production, I don't think we have a rich enough theory of production, especially in terms of the lower end of the vocal tract (I just learned of this recent publication which expands out knowledge of the lower end, in a relevant way).

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