Not much is remembered of the shrI-kula traditions before the universal sweep by the pa~nchadashI and structurally related mantras, like manu vidyA or skanda vidyA. These may be termed the classical shrIvidyA mantras and examples include:
The manmatha vidyA or kAdi vidyA:
ka e I la hrIM | ha sa ka ha la hrIM | sa ka la hrIM
This celebrated mantra is seen even in tiru-mantiraM of sundaranAtha (tirumUlar) the celebrated siddha who migrated from Kashmir to the drAviDa desha.
The lopAmudra vidyA or the hAdi vidyA:
ha sa ka la hrIM | ha sa ka ha la hrIM | sa ka la hrIM
The skanda vidyA (with 18 syllables):
ha sa ka la hrIM | ha sa ka sa ka la hrIM | sa ha ka ha ka hrIM |
The manu vidyA (with 18 syllables):
ka ha e I la hrIM | ha ka e I la hrIM | sa ka e I la hrIM
Now even within the classic mantras mainly the kAdimata and to a smaller extant the hAdimata dominate.
However, there are two independent mantras of shrIkula that appear to have existed as parallel shrIvidyA traditions before the pa~nchadashI-like formulae became universal. Only fragments of these texts now survive giving the minimal sketch of mantraic deployment. One of them appears to parallel the archaic kubjikA tradition, where the mantras are combined with five vaktra mantras for shiva. This is also called the nityaklinnA vidyA. The second appears to be a stand alone form of tripurasundarI. While in both these versions the 8 bhairavas are present, not all of the 16 nityA-s are invoked (at least in the surviving fragments). vArAhI (danDanathA) is however worshipped as in the pa~chadashi based forms.
It appears that sundaranAtha (tirumUlar) appears to have been aware of these parallel forms as he mentions a shakti-pIThaM tripurA chakra, which from the sketchy details seems to be chakra that goes along with the above version.