My experience with the LCD-4, based in a two-weeks home audition, compared to an Abyss Phi with CC pads (not the TC) is that LCD-4 and Abyss are
complimentary headphones. To my ears, they do different things best. The only thing in common is that they share a fantastic bass, where the LCD-4 has bore bloom and the Abyss has more visceral impact and twack. With the Abyss you can tweak the bass easily by fiddling with the pads, by the way, if you want a even more sub-woofer like bass response, at the expense of some articulation.
Midrange is where the LCD-4 does its magic, and I have not heard another headphone which gives you such an emotional connection with vocals. Abyss is still great, of course, but in comparison sounds a tad leaner, colder and more distant in my system. And, yes, the Abyss Phi exhibit some sibilance IME depending on the recording, whereas the LCD-4 are velvety smooth. The Superconductor HP cable improves on the Abyss midrange, by adding some sweetness and body to it, while costing as much as an used LCD-4
Treble is much more airy and extended in the Abyss and this also helps soundstage and transparency, at the cost of some brightness. LCD-4 treble is rolled off in comparison, which is not a bad thing with some hot recordings. The problem with me is that treble presentation of the LCD-4 trades off transparency for smoothness too much, and the overall impression is of a slightly veiled soundscape.
Imaging is very good with both, but I would give the edge to the Abyss, as with the headstage, which is significantly wider on the Abyss, while keeping a good depth and - to a lesser extent - height resolution.
Abyss presentation is unique in that it resembles speakers more than any other headphone I have heard, while LCD-4 has a comparatively more headphone-ish, intimate and closed-in character.
Finally, PRaT (Pace Rhythm and Timing) was so much better with the Abyss, where transients attack and decay were faster and fast paced music and complex rhythmics passages were coming out far more exciting and readable.
I was tempted (and still am) to get an LCD-4 as a secondary HP for vocals and easy listening sessions (as opposed to critical listening), but then the TC came out and I have decided to invest on the upgrade of my Abyss instead.
I am eager to know from the few TC fresh owners where the TC improves over the Phi, especially when it comes to brightness / sibilance.