My ultimate goal is to be able to predict from external factors whether and what kind of vowel quality correlates an /a–a:/ contrast might have in a language, and specifically to determine which on is more likely to be front or back.
It's common knowledge that vowel length contrasts are typically augmented, most commonly with tenseness (μμ > [+tense]). This in turn has consequences on the vowel's realisation and possibly further phonetic differentiation.
Basic physiology suggests that the long /a:/ will be more open, and thus more back than the short one, and this seems to be supported e.g. by Lombard: front when short, back when long; also Parisian French at the turn of the last century, and modern Quebecois (with rounding and even diphthongisation). But there are opposite examples like Hungarian, with its /ɒ~ɔ/ vs /a:/. In both these cases no tensess seems to be involved (at least synchronically: I'm clueless about the latter's historical phonology). Proto-Slavic had the same: whether or not there was systemic vowel tenseness I cannot say, but two reduced vowels there were.
In languages with vowel reduction, with or without a length contrast, the shorter an /a/ gets the less back it becomes, and a drift towards /æ~ɛ/ is possible (e.g. historic reduction in Latin). Conversely, in a final unchecked syllable Oscan exhibits backing, with /a > ɔ/ (presumably: it's transcribed Ú, like /o/, despite the letter O being unused in the writing system) but not elsewhere. It seems reduction might have something to do with the presence of a phonetically non-back /a/: Latvian has extremely short, sometimes barely audible short unstressed vowels, yet they're not phonologically reduced as in Russian, nor lax, but only subject to physical articulatory constraints. It's tempting to link this to its highly consistent maximally open /ɑ/.
It starts to get interesting when tenseness (or is it tongue root?) comes into play. Dutch has a front tense /a~æ:/ vs. back lax /ɑ/, but apparently some dialects reverse this (Wiki). German has short /a/ vs. long /ɑ:/, neither of which is ostensibly lax, but seems to have a back short lax [ɑ] in open unstressed syllables (Wiki). In Irish, the long /a:/ but not the short one exhibits regional variation in being canonically either back or front. In Danish, /a:/ but not /a/ is front [+tense] [æ̝ ~ ɛ̞] (comment thread). In Middle English, the tensing and raising of both the short and the long low vowel to a low-mid one suggests that both were non-back (Kiparsky below).
So far, it looks like the tensing (or +ATR) of the low vowel /a/ may be associated with either fronting or backing, but the latter seemingly more rarely, perhaps only if it's [+back]. Kiparsky in Joseph & Janda (2003), citing himself (1974), gives a diagram where æ and ɔ2 (lower than the [-tense] ɔ1) are the [+tense] pairs to a and ɒ. So if a system doesn't allow the [+tense] ɐ, one of these two are likely to be chosen as the tense allophone - and I'm interested in the factors that determine which one it is.
So, how does the length/tensenss contrast in the vowel /a/ correlate with its phonetic realisation? Have some universal phonological correspondences and factors been found that determine which one is likely to be phonetically back or front? Is it possible to look at the phonology of a language and predict the phonetic realisation of its low vowel? Or to look at its phonetic realisation and learn something about its phonology? Can one determine either the phonology or the phonetics by looking at two states of a phonological system before and after its restructuring (eg. lack of differentiation of the long and short /a/ in Romance?).
Albeit my grasp on this is mainly intuitive, I welcome technical descriptions of any depth, autosegmental, particle phonology, the more the merrier.