Among unknown great saxophonists was one by the name of Sigurd Rascher, who in the '50s published a full set of the "secret" fingerings for extending the 2-1/2 octave range used by most players to the full 4 octaves that the instrument could use fairly easily.

A range of about 3 octaves, or a little more, was known from the beginnings of the instrument, but composers who wrote early music for it mostly used only the "obvious" first 2 or 2-1/2 octaves due to the difficulty of finding players who knew about the rest of the notes then known. Rascher apparently discovered (or you might say invented) about the last half octave through his own research, and they were a bit of a complex feat for some of the fingerings but produced crystal clear tones (if one can say that about a saxophone) for the full range.

Most of the best players, at least those recent enough that I've heard much of their playing, have been able to use the "top tones" with varying fluency, and they can be heard in many performances; but they're only occasionally used by most for long runs or licks. Knowing how to use them doesn't mean you have to use them if they don't fit into the music.

The saxophone is one of only a couple of instruments that wasn't "evolved" from earlier instruments. It was deliberately invented for a specific kind of music - the "marching band." Since the dominant instruments in bands of that kind were trumpet and clarinet, both in Bb (and Eb for alto and bass clarinets), the sax was commonly made in those keys. Bass, Tenor, and Soprano saxes in Bb, and Alto and Baritone instruments in Eb were (and are) commonly used.

The other commonly cited "invented" instrument is the Sousaphone, but the "invention" aspect is a little less defensible there since it's really another existing instrument "unwrapped and wound up differently" to make it easier to carry while marching.

The "C" saxophones were built to "popularize" the instrument - i.e. were mostly a sales trick for people who couldn't quite get the key changes, but were fairly popular in the "jazz age" and a little later.

In the US, at least in my time, C-Melody usually meant a soprano; but the tenor range may have been the more common one elsewhere. C-saxes are still available (or available again) but aren't particularly popular so far as I know.

The saxophone, with the "classical" mouthpiece, allows (requires) you to change the pitch of the free part of the reed, using lip pressure, to match the note you intend to play, and for good tone it's also necessary to match the "head volume" in the mouth and throat to get a similar match. This allows a player to "bend" almost any note over about a semitone in either direction quite easily (but also allows a less gifted player to play everything a little out of tune).

Since the "back volume" (air volume inside the player) can be used to help both pitch and tone, and must be used properly in order to get a good performance, Mulligan played mostly baritone because he had a "big head" and large lung volume, while Desmond was "built for alto." Almost nobody has enough "body space" to play a bass or contrabass saxophone "properly" so mostly they sound like series of controlled farts when anyone tries. That may explain why neither was ever very commonly used.

John