There seems to be some misunderstanding here about the purpose of the "full Boehm" clarinet. It has two distinct aspects that it covers:

1) The low Eb key allows for the performance of A clarinet parts on a Bb clarinet. That's the only reason it was extended, similar to the extension of the "regular" bass clarinet from low E to low Eb. (Older basses from the late 1800's only ranged down to E, but the obvious lack of A bass clarinets (few have been made over the centuries) and the obvious logistic difficulties in having both an A and Bb bass forced the issue there.
Most orchestral clarinet players have both an A and Bb soprano, so the need was not as pressing for the extension on the soprano. However, for those who wanted it, it was there.
I do not like the Bb in the staff that the low Eb key plus register key allows. Once in a blue moon, I will play a passing note in a fast passage that way, but for the most part it's pinch for the fast passages and A and side trill key for the slow ones. Your mileage may differ.

2) The other elements of the "full Boehm" design are various keys added to make certain articulations more facile. They include:

LEFT HAND Eb LEVER:
This one duplicates the full key set up for both little fingers. It adds zero "complexity" to the instrument, and is recognized as a useful addition by makers today, often being offered as an option on the artist level instruments.

  ARTICULATED G# KEY:
This allows you to keep the G# key depressed while playing any notes below it, similar to that on the saxophone. Those who play the saxophone will welcome it, many clarinet players don't like it since it has a pad operated by opposing springs, one that tends to stick a lot. (This can be cured by having a cork pad installed, by the way.)
The advantages are much more facility in extreme keys, combined with having a trill fingering for the G# (operated by the right hand) and (usually not mentioned, but very important in my eyes) an improved and correctly sized and sited G# tone hole that never collects any condensation. Disadvantages are that it is "complicated" (Compared to what? A bass drum perhaps...), that it adds "weight" to the instrument, and that it takes away one or two (depending on the horn) fingerings for notes up in the altissimo.

FORK Eb/Bb MECHANISM:
This is the added ring and vent key installed on the upper joint. It is a useful enough alternative fingering for a note for which a number of such fingerings already exist. I use it occasionally if an arpeggio will work better that way.
Plusses here are that the finger holes are now all set at the same height, and there's not the tiny but noticeable irregularity when dropping or lifting the third finger, and that you have another alternative fingering for Eb/Bb. Minuses are that "it adds weight" and that the pad on the vent key is hard to set.

LOW Eb KEY:
(Added again here for consistence in the comparison department)
As discussed above, it enables you to get by with one clarinet when in an orchestral setting. It also offers a less than stellar Bb in the staff.
One other advantage that I've not seen aired very much is that the emission of the B in the staff through the tone through the body of the instrument "regularizes" the timbre of that note. In other words, you still get the "through the bell" tone quality of the bottom-most note on the horn and its twelfth, but since it's not the E/B any longer, it tend to make "the break" not that big of a tonal issue.

Bad aspects are that it "adds weight". Not much wrong with it other than that.
Most clarinet players will advise against any of this, but they usually do so without any practical experience that would lead them to know one way or the other. Few horns to actually handle translates out to few chances to experience the difference one way or the other.
From my perspective, the weight is a no brainer. But, I'm the guy who can play an extended range bass off of my thumb for substantial periods of time, so I might be a bit biased.
As for the other factors, the plusses far outnumber the minuses. Added complexity is a crock; I've never had one go out of adjustment in the forty years that I've been playing them. (If oboe players can manage with their instruments, surely we can deal with another two adjustment screws.) The few lost altissimo fingerings are more than balanced out by the added facility in sharp and flat keys (for the articulated G#), and the left hand lever has zero negatives and a ton of positives.

Regarding availability:
At one time, you could order (from Selmer, Leblanc and Heckel, at least) soprano clarinets with a progressive installation of these features. The usual order was 1)Eb lever; 2) Eb lever, articulated G#, 3) Eb lever, articulated G#, fork EB, and 4) "full Boehm", with all of the above plus the low Eb extension. This sort of flexibility was available at least until the 1980's, when my latest such horn (a 10S in Bb) was produced.

However, I personally own a Series 9 A clarinet with only the fork Eb mechanism, and an old Selmer pro horn in metal that has only the fork Eb and the articulated G# (no side Eb lever), so there has been some variation over the years. And, while I know that these instruments have been produced in both A and Bb (as I own a pair of Selmer Series 9 sopranos of the "full Boehm" type), just how far the penetration of these options went is still a mystery.
In September, there was a Selmer Eb soprano with "full Boehm" keywork up for sale on eBay (from Italy, no less). So, it potentially was possible that such options were offered on the entire line of horns from Selmer.

Want to buy one now? Your options are pretty limited. The easy way out is to purchase an Amati "full Boehm" instrument, costing you about a thousand dollars new. You can have it in your hands in three days, and it will have all of the options thereon.

However, I'd advise against that route for a number of reasons. First off, the Amati instrument isn't up to the horns produced by Buffet, Leblanc or Selmer as far as tone and intonation are concerned. I've not owned an Amati instrument, but I have tested one at some length, and they just don't "get it", plain and simple.

Second, the worksmanship (fit and finish) on the Amati horns is less than optimal. I only eyeballed the "Full Boehm" one, but I recently had some modification work done on my Amati "Oehler" clarinet, and the following items were noted:

I) The tone holes were chipped where the seats were milled into the body of the instrument. You had to take the keys off to see the problem, but it was very clear to me that a dull tool or tools had been used. One chip was a major flaw, only fixed by wood dust and superglue.

II) As shipped from the factory/distributor, the instrument was completely unregulated. I got it into playing condition using my local repair guy, but for complete regulation (involving setting the rings at the proper height for my fat fingers, it took some big time work. Things are much better now. If the relatively simple Oehler system keywork had these problems, one wonders how they would do with the Boehm.

Also, a key post (where one of the long "clapper" key arms on the lower joint stops) was simply press-fitted into the horn, and when the key was removed to make the ring adjustments, the post literally fell out of the horn. (It's fixed now, however.) Based upon this fit and finish problem offered by an Amati horn, I'd be loathe to trust them to do it right for my main clarinet.

What to do? Well, you can't easily order a new one, but there are many hundreds (if not thousands) of them out there somewhere, and sooner or later they are going to come up on eBay. I've recently bought an "everything but the low Eb key" "full Boehm" in Bb (Selmer Series 10S), and have been more than satisfied with the result. I had it overhauled, regulated and silver plated, and it's now my "day to day" horn for non-classical stuff.

I have found that these instruments are more likely to be better cared for than a run of the mill R13 or Opus or Series 9, simply because the owner made an obvious effort to purchase something a bit more costly, and had a good reason for doing so. With the others, you're likely to get a student horn, but with one of these it almost certainly was in the hands of a professional musician, who took better care of the instrument overall.

So, if I were taken in by the descriptions that I and others have offered hereon as to why "full Boehm" instruments are a good thing, I would look to eBay to acquire one rather than trying to purchase a new one from the few suppliers left. My opinion, of course...

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Terry L. Stibal
Leader of Houston's Sounds Of The South Dance Orchestra
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Another fav of mine, Stephen Howard adds this:

Articulation describes a mechanism whereby a typically single key is broken into two parts that can act independently of each other. This means that the touchpiece ( the bit you press ) of the G# key is mechanically separated from the key cup. In normal use, when you press the G# touchpiece the G# key cup will open - giving you a G# ( or C#, depending on which octave you're in ).

However, when you finger an F ( or a B ) the F key has a linking mechanism connected to the G# key cup which brings it down ( as though you'd released the G# touchpiece ). This allows the player to keep their finger on the G# touchpiece and let the F link mech do the work of closing the G# key cup - so that instead of having to move two fingers to switch between F and G#, you need only move the F finger.

Standard Boehm clarinets have been produced with articulated G# keys - it requires no additional touchpieces to work.

It sounds ideal, but there are some drawbacks - the additional mechanism can make the G# feel less responsive, and there can be problems with the G# pad sticking. This can be alleviated to some degree by the use of a cork pad. A significant advantage of the mechanism is that the G# tone hole is place on the top of the bore - thus dispensing with moisture problems. (Note: Thanks to Carl Baron for pointing out this addition to me. Carl, you rock!)