During a band rehearsal the bandmaster may ask when playing a march: Let’s start from the trio!
a) What is meant by this term?
b) where does it come from?
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The trio in a march is a second contrasting section, often more lyrical in character and usually in the subdominant. Very often the first part consists of two sections or "strains", both repeated. The trio is the third strain.
Some marches end on the trio, some return to the first section. A classic example of a march that ends on the trio is Sousa's "The Stars and Stripes Forever". An example of a march returning to the first section is Johann Strauss Senior's "Radetzky March".
The term was originally used in baroque minuets. There the trio would often actually be played by three instruments:
Around Lully's time it became a common practice to score this middle section for a trio (such as two oboes and a bassoon, as is common in Lully). As a result, this middle section came to be called the minuet's trio, even when no trace of such an orchestration remains.
A minuet was often included as the third movement in a classical symphony. The minuet later developed into the scherzo which usually also has a trio.
The "trio" which is common in marches (and polkas) is generally a final section in a contrasting key. In major keys, it's usually in the subdominant of the opening key (not always.)
It's called a trio for historical reasons. In many baroque (and maybe earlier) suites, some dances had a part traditionally played by three instruments. The polacca from Bach's Brandenberg is a good example. There's a nice trio for two oboes and a bassoon; later there's another trio for two horns and three bassoons (in unison).
These trios were generally in A-B-A form with the B being the contrasting trio. Modern marches and polkas make the B the final. The "Stars and Stripes Forever" mentioned above is good; also see "Rosamunde" (known in English as the "Beer Barrel Polka").