Out of curiosity - what makes one sub "musical" and one "cinematic?" If they reproduce sound accurately at similar SPLs, what possible difference is there? I happen to have one of your subs, and it sounds fine on both, as did the previous sub - which just wasn't as good.
I guess that's a fair and somewhat complicated question. As
@Audionaut helpfully linked to above, I've answered briefly earlier, but let's elaborate a bit.
I understand that the difference between "musical" as opposed to something suitable for home theater (not sure what to call that) can be a bit unclear. A subwoofer obviously doesn’t know whether it is playing a kick drum or an explosion. Its job is simply to reproduce low frequencies accurately. If the main goal is high SPL at very low frequencies, it's natural to look at large, high excursion drivers, and also large, ported enclosures. The result will be something the size of a small refrigerator, aka not well suited for a small, European living room.
Another approach (closer to what I do) is a smaller, sealed enclosure, potentially still with a relatively large driver, or two drivers, but more moderate excursion. The result will typically be more linear response, but less capacity in the lowest frequencies (below say 30-35hz where high SPL start to require very high excursion. With the 10D (and maybe the Sentinel?) I also use a dual opposed configuration, which means vibration and noise from the cabinet is essentially eliminated, further reducing mechanical noise and reducing the risk of being able to locate the subwoofer even with high crossovers.
So any loudspeaker or subwoofer is a set of compromises. What the designer/manufacturer chooses to focus on will affect what it is more or less suitable for. Where the term
“musical” usually comes from I suspect is when distortion tends to remain low in the upper bass (roughly 40–100 Hz), and even beyond that if you cross high. Typical "home theater", very high excursion subwoofer drivers aren't well suited above 100-150hz. The ones we currently use are essentially linear to 1000hz.
The new driver we are working on will have a similar focus on being well behaved well beyond the intended frequency band. It will be different in several ways to a typical home theater subwoofer driver, and also different than the PA-type subwoofer drivers that have been mentioned earlier in the thread. The reason I'm looking into custom is because I simply have not been able to find an OEM driver that is close enough to what I am looking for.
Subwoofers designed primarily for maximum deep-bass output may prioritise absolute displacement and amplifier power so they can produce very high SPL at the lowest frequencies. They are not necessarily super linear even at moderate excursion (the stated xmax is often well into distortion land), but that's not necessarily a practical problem, especially for movies. So the differences people hear usually come from how the system is designed: driver compromises, maximum displacement, linearity, cabinet size, driver configuration, DSP strategy, and so on.
To get slightly more technical;
A lot of the character of a subwoofer comes from how linear the motor and suspension remain as the cone moves. If those stay symmetrical and change slowly with excursion, we get less distortion and better sound quality. Further, voice coil inductance and how it changes with excursion can affect the upper bass region. Designs that keep inductance low and stable is preferred. System Q, and DSP strategy also affect behaviour. A sealed system with moderate Q and DSP extension tends to have a very predictable roll-off, good transient behaviour and low group delay.
With regards to the 10D, choices has been made to ensure it works very well with music, and it has a lot of displacement for its size, so one of the key benefits is how much power it has for its size. Interestingly, a review of the 10D on a home theater oriented site was actually released just yesterday. It's obvious through his review that a single 10D was not quite enough to satisfy him with regards to deep bass capacity, but as he points out;
"it (the 10D) packs the equivalent of a 15 inch driver in a cabinet that looks like it should only hold one 10 inch." 
www.beyma.com
