There are good reasons for learning to play diminished 7th chords (broken chords, or arpeggios) and using them in improvisations.
They are useful over some of the commonest chords (as well as those pesky diminished ones, of course), and best of all, you only need 3 of them to cover every key, instead of the usual 12. That saves the time taken to practice 9 chords, so you can spend more time down the pub and still astound with your virtuosity at the workshop on Saturday!
On the sax, I find the diminished 7th chord that goes D-F-Ab-B
the easiest to
play. I really need to practice starting on each note, going up and going
down, so I can start anywhere without thinking. At the moment there's a bit of
a delay while I find the note, and then it's too late - we're already on the
next chord! I also need to practice the other two chords: the one that goes
C-Eb-F#-A and the one that I find trickiest, E-G-Bb-C#.
Anyway, concentrating on the first one, it can be played over the diminished chord indications Do, Fo, G#o, Abo and Bo, which are all the same chord, of course. It's quite handy to be able to do this, because it can be hard to "hear" your way through diminished chords sometimes. But the main thing I want to point out is that it can also often be used over E7, G7, Bb7 and C#7.
If you think of it over a G7, the four notes are the 3rd (B) , 5th (D), 7th (F) and b9th (Ab). So it is ideal over a G7b9 chord. Even if the G7 chord isn't actually notated as a b9 chord, it is often OK to play it as if it were a b9. The only problem will be a clash if the rhythm section is playing a natural 9, or if they're augmenting the 5th. Don't worry, I'm sure no-one will notice. And a sensitive rhythm section will soon adapt themselves to your harmonic ideas (!)
You can use these chords in Autumn Leaves wherever you see a 7 chord, for example. Considering Autumn Leaves in the key of A minor (the usual key for Bb instruments), the chord in bar 2 is G7, which can be interpreted as G7b9, and played over with notes from the diminished 7th D-F-Ab-B. You can use that same diminished 7th over all the G7 and E7 chords. Not only is this easy to do, but it also sounds interesting because of the "colour" added by the b9.
Once you've mastered playing the 4 chord notes as broken chords, you can add approach notes to them from above and below. Each chord note can be approached from 1 tone above or from 1 semitone below. So you could play Bb-B, G-Ab, E-F, C#-D, using the lower approach notes, or E-D, G-F, Bb-Ab, C#-B using the upper approach notes. You'll be fascinated to hear that these 8 notes make up the "diminished scale"!
I'm proposing a very "vertical" approach to building a solo here. The vertical approach consists of taking each chord (or key centre) as it comes along, and playing something that is harmonically acceptable and interesting with that chord, rather than the horizontal approach of building a melody (e.g. by ear) that spans the chords and makes sense over several bars. Of course, it is also possible to create a good melody using the vertical approach, and I think anyone using a vertical approach is going to be thinking horizontally as well to some extent. The big advantage of vertical thinking comes when you are unfamiliar with the changes, or when the changes are too weird to hear your way through. Another benefit of the vertical approach is that you start to play things that may not have occurred to you, which helps to develop your repertoire of "hearable" lines.