Today's Blogging A-to-Z challenge entry examines the physics of music. Specifically, when a musician looks at a note on a page, what tone does she actually produce?
Most people today have passing familiarity with the piano, which has one key per note. This means the frequency of each note remains the same no matter what key a pianist plays in. If she hits the A above middle C, the piano strings vibrate at 440 Hertz (cycles per second). The A below middle C is 220 Hz, the A below that is 110 Hz, and so on. All of the notes in between have fixed frequencies as well.
This system, dating from about the beginning of the 20th century, is called equal tempering. It has some pretty interesting consequences, first among them that only the octaves are perfectly in tune. Every other interval is slightly out of tune—sometimes in two ways.
Equal tempering is a compromise, driven in part by the popularity of the piano, because they're so hard to tune. Other instruments don't have this limitation, so in some circumstances (i.e., string quartets), you might hear well-tempering instead.
In well-tempered tuning, some intervals actually do retain their proper mathematical relationships. But only some. Well-tempering is another compromise, resulting in different keys having wildly different tone colors. Bach promoted well-tempering with his two-volume set of preludes and fugues called The Well-Tempered Clavier. (Cue the irony that most people today have never heard it played on a well-tempered instrument. I found a good demonstration of the differences between equal- and well-tempering that's worth 12 minutes of your time.)
Before well-tempering, musicians used Pythagorean tempering (based on perfect 5ths) and meantone tempering (imperfect 5ths to get better 3rds).
For more about this topic, Nathan Nokes has a good video about the physics of these earlier tuning systems, with pure-tone examples. Notice how just and Pythagorean temperaments sound out of tune. Except they're not; they're just different.