Speakers that are very accurate sounding but don't produce an emotional connection.

I have listened to a few speakers over the years that impressed me with their accuracy and presentation of the music, but just did not create an emotional response or connection.

Sunnyjim, I'm not sure I follow what you are saying here (maybe i do but i'm not sure that your & mine audio language is the same...) - usually the most accurate system is the most satisfying system as it portrays the program material with emotion & connects the listener to the music. But...... your statement above says just the opposite.
So, i'm pretty sure that what you think you heard as "accurate" was merely distortion masquerading as accuracy. The distortion was obviously coming from various places incl the speaker - you already know this: one can have the best low-distortion electronics & sub-standard speakers & you will get mediocre playback. The other way is also true - sub-standard electronics & the lowest distortion speakers will yield mediocre playback.

I have often wondered what that quality is in some speakers that produce an emotional connection with the listener.

Many of the readers here will not agree with what I'm going to write (one of them being melbguy1 who's had several arguments about this) but so be it. I'm not trying to convert anyone - to each his own - but I'm quite sure that this is one of the main aspects involved in preventing an emotional connection to the music. I've written this before & I'll say it again - the speaker must be a time-coherent speaker. Time-coherent speakers are the least distorting speakers & they happen to use a 1st-order x-over. Just the physics of the 1st-order x-over ensures that the phase relationship of the music between any 2 frequencies remains unchanged above, at & below the x-over point. No other order of x-over guarantees this. It is extremely important to retain the phase relationship of the music as it passes thru each stage of the music play-back chain. While electronics do distort the program material, the worst offender is the loud speaker. Like I wrote, I've been saying this for a while & atleast a couple of well-known Audiogoners have paid attention to this & they have improved their resp. systems several orders of magnitude. They claimed that their systems sounded good but after going time-coherent they are now in a totally different league. You can make your speakers time-coherent in one of 2 ways: buy a time-coherent speaker like Green Mountain Audio, Soundlab, Vandersteen, Eminent Technology, some Quad ESLs, some older Martin Logan ESLs to name a few, or, make use something like DEQX to make your existing system time-coherent. DEQX involves a lot more work as you have to do speaker measurements in free space & then apply the correction curves & there's a steep learning curve but the results are worth it, I'm told by reputable people.
Most speakers in the market are anything but time-coherent & inevitably the listener has no/little emotional connection to the music.
What i've found is that, with a time-coherent speaker, even if the electronics is mediocre the enjoyment of music is much more than if the speaker was non-time-coherent.
Time-coherent speakers have benign phase angles in the 20Hz-20KHz region compared to non-time-coherent speakers that have wild phase angles. It's these phase angles that create huge distortion onto the music signal that totally destroys listening pleasure.
See if you can find a time-coherent speaker in your vicinity - a fellow Audiogoner who would be willing to host you, a dealer, an audio show - and listen for yourself. The music rendered thru a time-coherent speaker is leagues ahead. Once you listen to a time-coherent speaker, you'll never go back again I'm pretty sure. FWIW. YMMV.  

No speaker does it all. Its a matter of deciding what approach you want to pursue, and being happy with the tradeoffs.

while this might be true, psag, the time-coherent speaker sacrifices the least & is very close to doing it all. Every genre of music plays superbly on it & the better one's electronics get, the better the speakers sound. Music enjoyment for the long-haul.... 

wolf_garcia, if you are referring to my post then I'd like to say that i wasn't trying to give any musician short shrift.... ;-)

…the emotional connection of hearing early Beatles on the dash mounted speaker of a car radio in 1965? Unforgettable.

the car audio systems of yester year might not have had the audiophile attributes of our present-day systems but they were low/very low phase shift loudspeakers & probably limited audio frequency range just because of the older technology. But because of the low phase shift in those dash-mounted speakers they sounded really good for their price. Thank you for bringing this up - this is exactly what i'm talking about w.r.t time-coherent loud speakers in a home setup.   

o, again, it was the phase coherent car speakers…not the Beatles…I give up.

wolf_garcia, maybe you didn't fully understand what I meant to say...
for you, listening to the Beatles at that time, of course it was the Beatles & the low phase distorting dash-mounted speakers allowed you to gain max listening pleasure i.e. they affected the music signal the least.

bombaywalla --

There are too many other factors for a phase-coherent speaker to potentially sound like the less desirable choice compared to a speaker not being phase-coherent. I’m guessing you refer to speakers being mechanically aligned to achieve named virtue, although delay via DSP is a viable solution as well - likely better than any electrical ditto. In any case, phase-coherency alone - to my ears, at least - is far from the deciding factor.

The thought experiment could be made whether a hypothetical speaker of my liking, one not being phase coherent, would gain significantly being converted into phase-coherency. For the sake of not altering the overall design of the speaker let's just say we'd modify it accordingly via a DSP solution, bearing in mind other sonic changes that could follow in the wake of this implementation. In that case, given these are speakers I'd fancy despite of phase imperfections, I gather the phase modification (into coherency) could lead to the desirable outcome. If anything, and if proved to be a big factor, the rationale could be to seek out the overall preferred speaker principle as a phase coherent design, but as such phase coherency would only be one of many factors to achieve the desired sonic goal.

phusis, i'm sorry to say but you are wrong!
you need to go back & read a couple of really great threads (if you haven't already) &/or re-read them -
* Is DEQX a game changer? 
* Sloped baffle
there is a lot of great info in these threads that will dispel a lot misconceptions you have...
A correction re. what you wrote - i'm talking about TIME-coherency (& not phase-coherency). These are 2 different things with time-coherency being the superset. A time-coherent speaker is always phase-coherent but a phase-coherent speaker is not time-coherent. 
Time-coherency in a speaker is a design paradigm; it's a parameter that you trade-off in the midst of speaker design as you are thinking. Once you make up your mind to manuf a time-coherent speaker it dictates how you build that speaker. It's a decision you make at the very start.
You are talking about "mechanical alignment" - that's but one very small aspect of time-coherency in a speaker. Merely having a sloped baffle means very little (it just means that the acoustical centers of the drivers are on a vertical plane) if the rest of the speaker design doesn't take it all the way towards making that speaker time-coherent.

Time-coherency in a speaker is a design paradigm; it's a parameter that you trade-off in the midst of speaker design as you are thinking.

typo on my part (sorry!) - I meant to write - it's NOT a parameter that you trade-off in the midst of speaker design as you are thinking.

Harbeth and some other well regarded UK (and elsewhere) speaker manufacturers often use "live" side panels in the speaker as a part of the overall sound. Stethoscope schmethoscope.

OK. so that does not mean that every panel of the speaker is live. the panels that are not supposed to be live should not resonant. And, apparently a simple tool can ascertain that.