Engineering
The sounds that we hear every day, including music, reach our ear as sound waves. These sound waves travel through the air at different angles from the original sound source. The sound then bounces off whatever is nearby, such as people or the walls of a concert hall. If a building is designed in such a way that the sound does not bounce back to the listener’s ear well, then the music can be hard to hear or it can sound unbalanced. Engineers use trigonometry to figure out the angles of the sound waves and how to design a room or hall so that the waves bounce to the listener in a balanced and direct manner. Studio producers or hall managers sometimes install panels that hang from the ceiling of the room—these panels can be adjusted at specific angles to get the sound waves to bounce correctly.
Production
Studio producers have the job of making a musical recording sound balanced. They use many different computer programs in order to do this. The computer programs can allow a producer to see the sound waves that have been recorded as different types of graphs. These graphs are produced as the program uses trigonometric equations to quickly calculate how the graph should appear based on individual points—the sine wave of a singer’s voice, for instance, can be viewed visually using this process. The producer then can tweak things like pan and volume based on the visual cues in the graphs.
Harmonics
Every note (pitch) in music is determined by the size of its sine wave—that is, it is determined by its frequency. Notes with wide sine waves are lower in pitch and have fewer cycles per second, while notes that have narrow sine waves are higher in pitch and have more cycles per second. Musicians can manipulate their timbre by manipulating the sine waves produced. For instance, if a player plays a note with a frequency of 512 hertz, then a harmonic or partial is produced above it at 1,024 hertz, and you may hear a base note with the same note an octave higher. Violin players use knowledge of harmonics frequently, and tuning is related to how the base frequencies and harmonics interact.