What's that circus music?

You know it. It says ‘Big Top!’ It’s the tune of the circus. IIRC it has ‘Centurions’ in the name.

Also, what is the name of the tune that is (allegedly) traditionally played if there is an emergency, which is a code to the performers but keeps the audience from panicking?

It doesn’t have a sample, but “circus music” in Wikipedia brought up Entrance of the Gladiators. Not sure about the other song you asked aboot.

Ah. Gladiators, Centurions… I was close!

And…Google is saying that the other songs are:

Stars and Stripes Forever for an emergency, and…
12th Street Rag for smaller problems

From here

Ahh…Centurions were members of the Roman military. Free men.
Gladiators were, for the most part, slaves.

If you called a Centurion a Gladiator, to his face, he’d of kicked yer fanny.
If he didn’t just kill you.

If I were a gladiator I’d kill the guy who gave us such fruity music!

I believe you’re referring to Julius Fucik (1873-1916), Czech composer and conductore of military bands.

There is no truth to rumor that a concert series was once planned around the music of Fucik in combination with that of his countryman Josef Suk.