Lesson 2e - The Overtone Series

You may have heard the terms overtone series and harmonic series while discussing music, but unless you have studied them previously, you may not realize the importance of this concept in creating tonality. 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 can help you to understand:

• why we divide the octave into twelve parts.
• why we hear some intervals as consonant and others as dissonant.
• how we determine if an interval is in tune.
• why we use a harmonic system based on perfect intervals and thirds.

Bernstein on the importance of the overtone series

Let’s start by watching this video of Leonard Bernstein discussing an entertaining supposition as to how the overtone series helps to explain harmony’s evolution throughout the ages. Keep in mind that each evolutionary step he discusses adds another partial from the overtone series.

The Structure of the Overtone Series

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.

The Physics of Music

The division of the octave into twelve parts is our brains’ interpretations of a simple mathematical phenomenon, and this starts with the overtone series present in every sound. When a sound is created, any wavelength that can “line up” with the lowest pitch will create a frequency with greatly increased resonance. For example, the frequency of A2 is 110 Hertz (Hz), which means that it vibrates 110 times per second. If you double that frequency, you create a sound wavelength that will align with our original wavelenth every two times, creating our first overtone above the fundamental at the ratio of 2:1. If you triple the original frequency, you would get the next overtone at 330 Hz, and so on. 330 Hz is the pitch E3, a perfect 12th above our original pitch. It is this naturally occurring resonance above every pitch that creates the need for an octave to be divided into twelve. If you try to divde the octave any other way, you won’t create the naturally occurring resonances present in all sounds.

The influence of these ratios is among the most important concepts for understanding the way in which humans process sound. This concept not only explains why humans which intervals find certain intervals consonant or dissonant–simpler ratios are heard as consonant whereas complex ratios are heard as dissonant–but these importance of simpler ratios is 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 diminished 5th into the pattern. Try this quick exercise:

1. Choose any starting pitch, and then write it and the next six pitches from the circle of fifths on a separate piece of paper. (This will be easier if you choose a pitch that is flatted, because it will allow you to avoid dealing with double-sharps.)
   • If you’ve done this correctly, you should have one pitch for every letter name. e.g. one ‘A’, one ‘B’, etc.
2. Lower the final pitch by one half-step, thus creating the interval of a diminished 5th between it and the previous pitch.
3. If you re-arrange the seven pitches in ascending order now beginning with whatever pitch you chose to start, you will notice that you have created the major scale for that starting pitch.

In essence, the diminished fifth creates a new, smaller self-repeating loop inside of the circle of fiths. This slight change–the addition of one diminished 5th–creates the necessary tension for keys to function diatonically, so diatonic function could be described as an imperfection on an otherwise perfect series of intervals.

This can be further shown by looking at the naturally occurring 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.

• When a sharp is added to a key, it always raises the 7th scale degree in the new key, thereby creating the new ti.
• When a flat is added to a key, it always lowers the 4th scale degree in the new key, thereby creating the new fa.

Conclusions

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.

• Label the the first pitch of an overtone series the fundamental and every pitch above it is then numbered as an overtone.
  • Fundamental, 1st overtone, 2nd overtone, etc.
• Label each pitch as a numbered partial.
  • 1st partial, 2nd partial, 3rd partial, etc.

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.

Concepts derived from the overtone series

The overtone series is notable because:

• The overtone series is present above every note played by a standard musical instrument or the human voice.
  • When a person sings a pitch, we hear the fundamental as the pitch name, but the strength of each overtone above that fundamental determines many qualities that we associate with the pitch such as timbre.
• The overtone series is the basis for intonation.
  • Using the ratios from the overtone series, we can determine what is “in tune”. For example, an octave is a frequency that is exactly twice as high as the given pitch, and a true P5 is a frequency that is 1.5 times higher than the given pitch.
  • Fixed pitch instruments (e.g. piano) use a system of equal temperament to play in all twelve keys without having to re-tune the instrument every time the key changes. This means that we intentionally tune these instruments “out of tune” when compared to the overtone series ratios, because being always slightly out-of-tune is a far better compromise than being perfectly in-tune sometimes and grossly out-of-tune at others.
• The overtone series defines why the octave is divided into 12 parts, because it is from the overtone series that we derive the circle-of-fifths. Please refer to the discussion of the circle-of-fifths and tonality in Lesson 2c.
• The overtone series provides a visual example of why we find certain intervals more consonant or dissonant.
  • We hear simpler ratios of intervals as consonant and complex ratios as dissonant. So intervals at the beginning of the overtone series represent consonant intervals, but as the intervals decrease in size as the overtone series moves upward, the ratios become more complex and therefore more dissonant.
• The overtone series provides a good foundation for voicing chords.
  • Idealized voicing for chords often mimic the overtone series by placing each voice along the overtone series of the lowest voice.

  This video from the Score Circuit does a great job of making an entertaining summary of these concepts:

Remembering the overtone series

There are many ways to remember the overtone series.

• The first is to remember the intervals. This is easy at the beginning of the series, but becomes repetitive and easy to mess up as the series ascends.
  • P8-P5-P4-M3-m3-m3-M2-M2-M2-M2-m2.
• You can also think of them as a sequence of scale degrees above the fundamental.
  • 1 - 1 - 5 - 1 - 3 - 5 - b7 - 1 - 2 - 3 - #4 - 5
• And for some, it is easier to divide the series into groupings. For example, when looking at the first 12 partials
  • The first four pitches are a tripled fundamental and one fifth.
  • The second set of four pitches is a first inversion dominant triad built off of the fundamental.
  • The final three pitches represent 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:

• The first four pitches strongly imply a key that uses the fundamental as tonic.
• The next 4 partials imply a key with a tonic based on the subdominant of the fundamental, because the dominant seventh chord would be a V chord in the key of the subdominant.
• The third set of four paritals imply a key with a tonic based on the dominant of the fundamental as demonstrated by thinking of this pattern as sol-la-ti-do in the key of the dominant.