Allegro ma non troppo
“Fast, but not too much” — the most common qualifying addition.
What it means
Ma non troppo means literally “but not too much” and qualifies the preceding tempo word. Allegro ma non troppo thus calls for a fast tempo, but at the lower edge of what an Allegro allows. The addition names no figure; it warns against an exaggeration that the composer evidently thought likely.
It is thereby the counterpart to the intensifying additions assai, molto and vivace. The phrase is to be distinguished from Allegro moderato: there the tempo is moved toward the moderate by a tempo word of its own, whereas here Allegro stays the measure and is merely bounded. The difference is small, but the direction of view a different one — ma non troppo draws in an upper limit, moderato shifts the point of reference.
Similarly built are ma non tanto (“but not so much”) and ma con moto (“but with motion”), which make things precise in the same direction.
In the notation
The phrase stands above the first system of a movement or section, wholly in italics, and frequently becomes the name of the movement. It also appears with other base words, for instance as Adagio ma non troppo, where it warns conversely against too great a slowness.
Musical use
The addition stands typically where the notation would tempt the player to a high tempo but the movement is to keep weight: in broadly conceived opening movements, in themes with long note values that seem empty when played fast, and in polyphonic sections whose voices must stay audible. It is the most common qualifying addition of all.
Performance practice
Set the tempo not by the main theme but by the densest place of the movement — that is exactly what the addition warns against. Check whether the inner voices and the harmonic changes are still audible; if they blur, the tempo is too high. Think in long arcs, so that the holding-back does not come across as ponderousness. And resist the tendency to become faster as you go: the qualification holds for the whole movement, not only for the beginning.