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.
A harmonic series written as notes
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.