Understanding Basic Music Theory, 2013a

140 CHAPTER 4. NOTES AND SCALES
• Inversions of major intervals are minor, and inversions of minor intervals are major.
• Inversions of augmented intervals are diminished, and inversions of diminished intervals are augmented.
4.6 Harmonic Series II: Harmonics, Intervals, and Instruments 70
4.6.1 Frequency and Interval
The names of the various intervals, and the way they are written on the sta, are mostly the result of a long
history of evolving musical notation and theory. But the actual intervals - the way the notes sound - are not
arbitrary accidents of history. Like octaves, the other intervals are also produced by the harmonic series.
Recall that the frequencies of any two pitches that are one octave (Section 4.1) apart have a 2:1 ratio. (See
Harmonic Series I (Section 3.3) to review this.) Every other interval (Section 4.5) that musicians talk about
can also be described as having a particular frequency ratio. To nd those ratios, look at a harmonic series
written in common notation (Section 1.1.1).
A Harmonic Series Written as Notes
Figure 4.40
Look at the third harmonic in Figure 4.40 (A Harmonic Series Written as Notes). Its frequency is three
times the frequency of the rst harmonic (ratio 3:1). Remember, the frequency of the second harmonic is
two times that of the rst harmonic (ratio 2:1). In other words, there are two waves of the higher C for every
one wave of the lower C, and three waves of the third-harmonic G for every one wave of the fundamental. So
the ratio 71 of the frequencies of the second to the third harmonics is 2:3. (In other words, two waves of the C
for every three of the G.) From the harmonic series shown above, you can see that the interval (Section 4.5)
between these two notes is a perfect fth (Section 4.5.3.1: Perfect Intervals). The ratio of the frequencies of
all perfect fths is 2:3.
Exercise 4.6.1 (Solution on p. 162.)
1. The interval between the fourth and sixth harmonics (frequency ratio 4:6) is also a fth. Can
you explain this?
2. What other harmonics have an interval of a fth?
3. Which harmonics have an interval of a fourth?
4. What is the frequency ratio for the interval of a fourth?
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