To hear a clarinet player doing just what a traditional jazz clarinet player should, CLICK ON THIS VIDEO. The clarinet listens well to the trumpet lead and harmonises beautifully. It is a great demonstration of what can be achieved even with very limited resources.
My friend Jonathan Graham - a fine guitarist and a trumpet player - told me he has been listening to lots of jazz recordings from the 1920s and has come to the conclusion that the clarinet is usually the most important instrument in the band.
The situation can be particularly bad if a band has both a clarinet and a saxophone playing, probably in addition to trumpet and trombone. If the reed players do not play as team members, with a high level of musical awareness, the result can be excruciating.
For an example of a modern clarinet-player and saxophone player getting things absolutely right, CLICK ON THIS PERFORMANCE. The clarinet player is John Doyle and the saxophonist is Ewan Bleach. It is also a joyous example of traditional jazz teamwork at its best.
The book Playing Traditional Jazz, by Pops Coffee, is available from Amazon.