The most common and most likely answer to this crossword clue is the 4 letter word TRAD
I've seen this crossword clue in The Guardian
T
R
A
D
More clues leading to the result TRAD
Buddy explains!

The straight part
We know that "Satchmo’s sort of music" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case satchmo’s sort of music = trad.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does TRAD mean?
TRAD noun- Traditional climbing.
- Irish traditional music
- A traditionalist.
- Anything traditional, such as a school or a model of car.
- Traditional