You have likely heard the terms overtone series and harmonic series used in discussing music, but unless you have studied them previously, you probably do not realize how important this concept is to tonality and sound production. The overtone series occurs naturally in all non-synthetic tone production. When a person sings, a harmonic series is present above every pitch. When any woodwind, string, or brass instrument creates a pitch, a harmonic series is present above every pitch. Perhaps even more importantly for our discussions, we can study the acoustics–or the math–behind this overtone series to explain the fundamentals of Western harmony.
The overtone series is a series of intervals, or a harmonic series, above a given pitch. We call the lowest pitch the fundamental, and every tone above it is considered an overtone. In the example below, C2
is the fundamental, C3
is the first overtone, G3
is the second overtone, and so on.
You may also describe the tones of the overtone series by labeling each overtone as a partial. In this system, the fundamental is considered equal to all other tones, so it is labeled as the first partial. In the example below, C2
is the first partial, C3
is the second partial, G3
is the third partial, and so on.
For the example below, determine the numbering for each of these notes as both overtones and partials. Then practice transposing the entire series to other pitches.
Next, please watch this wonderful video of Leonard Bernstein explaining how the overtone series explains harmony’s evolution throughout the ages. Keep in mind that each evolutionary step he discusses adds another partial from the overtone series.
The division of the octave into twelve parts is our brains’ interpretations of a simple mathematical phenomenon. When the frequency of a soundwave doubles, our brains hear those two frequencies as sharing some fundamental commonality, so it interprets those two pitches as the “same” but separated by an octave. Therefore, octaves always have a 2:1 ratio. (A110, A220, A440, and A880 are all A
separated by octaves.) The next two simplest ratios ares a 3:2 ratio and a 4:3 ratio, which create a perfect 5th and a perfect 4th respectively.
The importance of these ratios is most easily observed in the circle of fifths. If you begin on any pitch-class and begin moving by ascending perfect 5ths (or 4ths), you will find yourself back at the beginning after cycling through all twelve pitch-classes. We call this the circle of fifths.
C - G - D - A - E - B - F-sharp - C-sharp - G-sharp - D-sharp/E-flat - B-flat - F - C
Perhaps more important for our discussion, though, is what happens when we introduce a non-perfect 5th into the pattern. If we begin on a pitch-class and begin moving through ascending perfect 5ths, each new perfect 5th will move us to a new letter. After seven letters, we will begin repeating letters but adding accidentals to them as seen in the circle of fifths above. If, however, we alter the last perfect 5th by a half-step to create a diminished 5th, we can break the pattern and shortcut to the end.
C - G - D - A - E - B - F - C
This slight change creates the necessary tension for keys to function diatonically, so diatonic function could be described as a slight imperfection on an otherwise perfect series of intervals.
This can be further shown by looking at the naturally occuring intervals if we write diatonic 5ths above the notes of a major scale.
Because we explored harmonic function in Unit 6, the tension provided by the one non-perfect 5th and its subsequent release should now be obvious.
Key signatures reflect the importance of the one non-perfect interval. When studying the consecutive key signatures, you should focus on which scale degree is changed between consecutive keys.
ti
.fa
.This directly reflects how diatonic function works; if we change where the one non-perfect 5th occurs, we change the key. We did not need to spend time determining the order of sharps and flats, because the class was already familiar with this from previous courses. It did not surprise them that circle-of-fifths plays a critical role in defining diatonic function and key signatures. The two orders are simply the reverses of each other.
The overtone series, often referred to as the harmonic series, is a series of intervals built off of a pitch. There are two ways to label the overtone series.
The overtone series built off of C2
would be:
The overtone series occurs naturally and can be explained mathematically, so it is one of the few objective ways in which we can discuss the origin of music. Any interval can be viewed as a ratio comparing the frequncies of the two pitches that create the interval. The simplest ratio, other than 1:1, is 2:1. For example, C2
has a frequency of about 65.4 hertz (hz = vibrations per second), and C3
has a frequency of about 130.8: a ratio of 2:1. When we hear two frequencies that have a 2:1 ratio, our brains interpret this as “the same pitch separated by an octave” – an elegant solution to interpreting a physical phenomenon. This example demonstrates that all concepts associated with music, such as pitches, dividing octaves, intonation, etc., are human creations trying to organize and interpret the physical phenomenon of soundwaves entering our ear.
The overtone series orders intervals by decreasing size but increasing complexity. The first interval of the overtone series, a P8, is the “simplest” interval of 2:1. As the overtone series moves upward, each interval becomes smaller but more complex. A P5 has a ratio of 3:2, a P4 has a ratio of 4:3, a M3 has a ratio of 5:4, and onward.
The overtone series is notable because:
There are many ways to remember the overtone series.
re-mi-fi-sol
or the beginning of the Lydian mode of the fundamental.When looking at this final method for remembering the first 12 pitches of the overtone series, I find it fascinating that each group of four pitches implies a different key area:
sol-la-ti-do
in the key of the dominant.