Math Research

How is trigonometry used in music?

1.     Sound consists of waves, and that sin(x) and cos(x) are waves.

2.     A wave might be representable as a combination such as  1.4 cos x + 2.8 sin x - 1.103 cos 2x + 3.32 cos 3x + 5.6 sin 3x.

3.     Musically, when an instrument plays a note, the basic note can be represented by A.cos Lt + B.sin Lt. The number L has to do with how high the note is (pitch), and A and B have to do with how loud it is; t is time. But that function is a "pure tone" that seems to sound like a flute.

4.     The unique sounds of oboes and clarinets and so forth are due to almost inaudible notes at the octave, octave+fifth, etc.

5.     Every note (pitch) in music is determined by the size of its sine wave---that is, it is determined by its frequency.

6.     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.

7.     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.

8.     The simplest sounds, called pure tones are represented by functions of the form

f(t) = A sin(2 pi w t)