Alto Clarinet/Bassett Horn
The alto-clarinet and the modern form of the bassett horn are looking very similar. They are alto voices, about half an octave below the "normal" B flat clarinet. They transpose to E flat or F. Today both have the characteristic bend in the barrel, with a barrel that is usually made of metal, and they usually have a metal funnel like a bass clarinet. The bent barrel is both an aesthetically feature but also allows the player to hold head and mouth in a normal position while the instrument is fully vertical (while B-flat clarinets are held in 45 degree to the body). Because the funnel is bent, too, the whole clarinet can be built in roughly the same hight as a Bb clarinet and it is usually played just holding it with your hands (with an optional neck strap or a long metal thorn), while bass clarinets really need to have a thorn to be placed standing on the ground. A wooden bass clarinet is too heavy for holding it while you play.
Alto Clarinet in E-flat
You find alto clarinets in harmony and wind bands, large symphonic bands (like high school bands) or clarinet choirs, but hardly ever in classical symphony orchestras (there you'd rather find the basset horn). The beautiful and "sonore" sound combined with a practical size and good looks makes it an attractive instrument. But since it is produced in rather small numbers, excellent instruments are rather expensive.
In the larger wind orchestras there is always a need for an alto clarinet, and the more demanding pieces usually will have a voice for this instrument. Nevertheless, an alto clarinettist will always have a B-flat clarinet that can be used optionally, so that one does not get bored in pieces in which there is no alto voice. Alto players don't usually have extensive solo parts. But the clever alto clarinettist knows how to solve this: The instrument is notated in E-flat, so you can play anything that an alto saxophone can play. The clarinet's tone range is generally larger than that of similar height saxophones, and technically an alto clarinet is more agile than an alto saxophone. The alto clarinet sounds neutral enough that you can play alto saxophone voices without attracting unpleasant attention. If you are good, this is you chance for a solo every now and then.
Bassett Horn in F - Mozart's favourite instrument
The modern basset horn in F - one full tone higher than the Alto in E-flat - is hardly ever used in contemporary music, and you won't find it very often in symphonic wind bands. And there is only a limited number of classical compositions for bassett horn, too, but then the composer was mainly Wolfgang Amadeus Mozart, because the Bassett Horn was his favourite instrument, the one Anton Stadler played. Mozart wrote some concertos for it, including the famous concerto in A and that alone will make it very likely that this instrument will stay with us forever. The most famous clarinet concerto in A, KV622, was - as far as we know today - originally composed for a bassett horn or a bassett clarinet (see below) and not for an A clarinet as it is used today. That instrument might have looked a little like the Bassetthorn, maybe without the funnel. Since such instruments don't exist any more, the musicians use an A clarinet and changed some of the lowest notes to be able to play that parts. There are original hand-written notes by Mozart and since he precisely knew the instruments and their tone range we can tell that something like this must have existed.
A bassett horn enthusiast I played with a lot in Hamburg in the nineties wrote as a comment to my page:
"Without bassett horn you would miss something in the "Zauberflöte" (Mozart's Magic Flute) as well as in the Requiem! Mozart had a special relationship to these clarinet instruments, as can be seen from the "Maurerische Trauermusik" and the "Notturni for 3 singers and 3 bassett horns", which are somewhat rustic in text. By the way, basset horns were available in the tunings in F and in G. These instruments became very fashionable at the end of the 18th/beginning of the 19th century, mainly in chamber music, but the orchestral parts were sparse (Beethoven: Ballet music "Prometheus' Creatures", Mendelssohn, Dvorak). Later they were forgotten again, until Richard Strauss "rediscovered" the basset horn."
The famous clarinet concerto KV 622 in A was - as far as we know today - originally written for a historic basset clarinet (probably similar to a basset horn, and not to be confused with the modern bassett clarinet) in A and not for the A-clarinet, with which it was played regularly until the middle of the 20th century. Some passages went down so deep in the original that it is impossible to play with the A clarinet. Therefore the play was rewritten afterwards. Mozart, of course, knew exactly what the range of the voices for which he wrote was, and what was possible with it - in this respect, this instrument must have already been very useful at that time, not long after the invention of the clarinet.
Historical bassett horns
Today there still are some very old bassett horns in museums (e.g. in Hamburg and Berlin - see picture) and they all do have the typical bend and a box-like connecting piece (the "book") - and with some the bell looks like a french horn. This is probably the reason why it was named bassett "horn". The crooked form was due to the technical problems building long and still effective keys. The player had to reach and cover nearly all tone holes with his fingers - in Mozart's days the modern key was not yet invented.
Bassett clarinet is no basset horn
The modern bassett clarinet is less a Bassett Horn but more an extended version of an A- or B-flat Soprano clarinet. Today professional clarinet players (like Sharon Kam and Sabine Meyer) use this type of instrument to play Mozart's famous concerto KV 622 in its original form.