Sheet music has general timing information, but it’s not going to be absolute. And most people couldn’t look at something that says “play this at 132 beats per minute” and have an inherent, absolute idea of what that is and be able to do so perfectly.
The conductor thus uses the baton as a single point of focus for all of the musicians to play their tempo towards. Each wave is one beat, and it’s done in such a way that it will be consistent – the wave down is always the first beat of a measure for instance, and a 4/4 measure will go down, left, right, up for each of the four beats.
This also means that the conductor can control the tempo of the piece, if any part requires a slowdown or speedup, the conductor will convey the exact pace with the baton. Many pieces will have sustained notes that are held until the conductor cuts them off, which is conveyed by holding the baton steady, and then sweeping it horizontally for the cutoff. Other information can also be conveyed. Larger strokes can cue the musicians to play louder, and smaller strokes quieter. A flourish in the direction of a specific section or musician can function as a cue to resume playing after a longer rest. Etc.
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