How do valves, slides, and finger holes help brass instruments play chromatic pitches?

Brass instruments produce sound through the vibration of the player's lips against a mouthpiece, which naturally generates a limited harmonic series. To play a full chromatic scale, these instruments must access multiple harmonic series by altering the length of their tubing.12

• Valves: These devices divert the airstream into additional lengths of permanently attached tubing. By using valves singly or in combination, players can lower the instrument's fundamental pitch, effectively providing new harmonic series that fill in the missing chromatic notes.312
• Slides: Functioning similarly to valves, slides allow the player to physically extend the length of the instrument's tubing. On instruments like the trombone, moving the slide to different positions progressively lowers the fundamental pitch, making a complete chromatic scale available.241
• Finger Holes: Historically, instruments like the cornett used finger holes to shorten the sounding length of the tube. Later, makers covered these holes with keys—as seen on the keyed bugle and ophicleide—to enable the production of a complete chromatic scale before the widespread adoption of valve systems.15