A mathematical way to say this is "if two notes are an octave apart, the ratio of their frequencies is two to one (2:1)". Although the notes themselves can be any frequency, the 2:1 ratio is the same for all octaves. And all the other intervals that musicians talk about can also be described as being particular ratios of frequencies.
Take the third harmonic, for example. Its frequency is three times the first harmonic (ratio 3:1). Remember, the frequency of the second harmonic is two times that of the first harmonic. So the ratio of the frequencies of the second to the third harmonics is 2:3. From the harmonic series shown above, you can see that the interval between these two notes is a perfect fifth . The ratio of the frequencies of all perfect fifths is 2:3.
- The interval between the fourth and sixth harmonics (frequency ratio 4:6) is also a fifth. Can you explain this?
- What other harmonics have an interval of a fifth?
- Which harmonics have an interval of a fourth?
- What is the frequency ratio for the interval of a fourth?
- The ratio 4:6 reduced to lowest terms is 2:3. (If you are more comfortable with fractions than with ratios, think of all the ratios as fractions instead. 2:3 is just two-thirds, and 4:6 is four-sixths. Four-sixths reduces to two-thirds.)
- Six and nine (6:9 also reduces to 2:3); eight and twelve; ten and fifteen; and any other combination that can be reduced to 2:3 (12:18, 14:21 and so on).
- Harmonics three and four; six and eight; nine and twelve; twelve and sixteen; and so on.
- 3:4
Brass instruments
The harmonic series is particularly important for brass instruments. A pianist or xylophone player only gets one note from each key. A string player who wants a different note from a string holds the string tightly in a different place. This basically makes a vibrating string of a new length, with a new fundamental.