The physics behind musical instruments is beautifully simple. The sounds are associated with standing waves, which come from constructive interference between waves traveling in both directions along a string or a tube.
Why do some notes, when played together, sound good, while others do not go together well at all? To answer that requires some understanding of the scale.
Twelve notes make up one octave, covering a factor of two in frequency. Each note on the scale is a factor of 21/12 times the previous one. Let's look at one octave starting at a frequency of 440 Hz, which happens to be an A.
| Note | 2n/12 = ? | Fraction | Frequency (Hz) |
|---|---|---|---|
| A | 20/12 = 1.00 | - | 440 |
| A# | 21/12 = 1.06 | - | 466 |
| B | 22/12 = 1.12 | - | 494 |
| C | 23/12 = 1.19 | 6/5 | 523 |
| C# | 24/12 = 1.26 | 5/4 | 554 |
| D | 25/12 = 1.33 | 4/3 | 587 |
| D# | 26/12 = 1.41 | - | 622 |
| E | 27/12 = 1.50 | 3/2 | 659 |
| F | 28/12 = 1.59 | - | 698 |
| F# | 29/12 = 1.68 | 5/3 | 740 |
| G | 210/12 = 1.78 | - | 784 |
| G# | 211/12 = 1.89 | - | 831 |
| A | 212/12 = 2.00 | 2/1 | 880 |
When notes are played together, such as when a string instrument is played, they sound best to our ears when the frequencies of the sounds are in integer ratios. As shown in the table, this happens for several of the notes on the scale.
A major chord on the guitar, for instance, consists of the notes 4 steps and 7 steps away from the base note. These have frequencies that are about 5/4 and 3/2 times the frequency of the base note, respectively.