Lesson 11: Harmonic Overtone Series |

So I got a message asking to keep writing these so here's one I had started a while ago and just finished. It may be a bit choppy reading so I apologize in advance for that.

Harmonic Overtone Series

So this will be the only other physics type lesson. To begin, let's review what we talked about when we discussed the production of sound.

Remember that sound is formed by waves, which are pulses of energy that vibrate through different media. Remember that a wave has a frequency, which is a characteristic of a wave, and that the frequency is directly related to the pitch (though this relationship is not linear). Now I want to introduce one more concept and that is that various waves can be added together to form one new, distinct wave that characterizes both of the original waves.

We come to an important question. If all sound is just these vibrating air patterns, then what makes the sound of a trumpet different from the sound of a violin? If the two play the same note (i.e. same frequency) why don't they sound the same?

The answer is that when we hear a note played on an instrument we don't just hear one wave, we hear a number of different ones that are a part of a series called the harmonic overtone series. Let's talk about the series and come back to this later.

The harmonic overtone series is a series of notes that form on top of one low note we call the fundamental frequency (or just the fundamental). The notes that we hear above the fundamental are called partials or harmonic overtones.

With C1 as the fundamental, the series would look like:

1 C1

2 C2

3 G2

4 C3

5 E3

6 G3

7 Bb3*

8 C4

9 D4

10 E4

11 F4

12 G4

A few things to notice:

For any partial, the partial that is twice that number is the same note at a different octave. Notice that all powers of two (i.e. 1, 2, 4, 8 ) are all C's and that partial 3 and partial 6 are both the same notes.

If we look at partials 1 through 8 we can see that the notes form a dominant seventh chord.

We've stopped at partial 12. If we graphed frequency versus partial number, we'd see that this series diverges to infinity, so there are in theory an infinite number of partials. The problem is that as we move up the series, the partials get closer until they are either horribly dissonant or indistinguishable. Other music theories, particularly non-western ones, use more partials and employ quarter tones (the interval of half a semitone) to make this possible.

Partial 7 has an asterisk. This is because the 7th partial is actually out of tune (31 cents flat of Bb, remember that each semitone is 100 cents so 31 is pretty significant). This is a naturally occurring problem that luthiers (instrument makers) and composers have struggled with for a long time. The dissonance from this pitch problem is significant, and there have been a number of solutions throughout history that all involve various ways of spreading this tuning problem out among other notes. Currently most instruments use a method called equal temperament where the dissonance is spread equally among all notes in the scale. There's a good post about intonation here: http://forum.bandamp.com/Music_Theory/3732.html.

For me, the most interesting thing about the harmonic overtone series is that it gives us some physical basis for our theory. The harmonic overtone series outlines the relative importance of various scale degrees. Look at the partials we've listed. Considering that notes will reappear every time their partial number doubles, we can say that the earlier a note appears as a partial, the more times it will occur in the series. We can see this in the first 12 we've listed. We have:

4 C's (tonic)

3 G's (dominant)

2 E's (mediant)

1 Bb (subtonic)

1 D (supertonic)

1 F (subdominant)

Notice:

The the first 3 distinct notes outline a major triad, first 4 outline a dominant seventh chord, the first 5 a ninth, and the first 6 an eleventh.

The tonic occurs every 2^(n) times at the nth octave above the fundamental, this is more than any other note.

Moreover we can use the overtone series to determine the ratio of the frequencies of the upper and lower notes of an interval. The ratio of an octave is 2:1, and notice every time a partial number is doubled the same note appears an octave higher. The ratio of a perfect fifth is 3:2 and the ratio of a perfect fourth is 3:4.

The series of overtones that an instrument produces is a characteristic of that instrument and accounts for its distinctive sound or "timbre". Some instruments, like cymbals, don't have a tone - these are called indefinite pitch as we can't clearly distinguish the fundamentals as overtones produce too much noise. This website has a good explanation of timbre and waveforms. http://www.soundpiper.com/elements/timbre.htm

See:

http://en.wikipedia.org/wiki/Harmonic_series_(music) for more information

Harmonic Overtone Series

So this will be the only other physics type lesson. To begin, let's review what we talked about when we discussed the production of sound.

Remember that sound is formed by waves, which are pulses of energy that vibrate through different media. Remember that a wave has a frequency, which is a characteristic of a wave, and that the frequency is directly related to the pitch (though this relationship is not linear). Now I want to introduce one more concept and that is that various waves can be added together to form one new, distinct wave that characterizes both of the original waves.

We come to an important question. If all sound is just these vibrating air patterns, then what makes the sound of a trumpet different from the sound of a violin? If the two play the same note (i.e. same frequency) why don't they sound the same?

The answer is that when we hear a note played on an instrument we don't just hear one wave, we hear a number of different ones that are a part of a series called the harmonic overtone series. Let's talk about the series and come back to this later.

The harmonic overtone series is a series of notes that form on top of one low note we call the fundamental frequency (or just the fundamental). The notes that we hear above the fundamental are called partials or harmonic overtones.

With C1 as the fundamental, the series would look like:

1 C1

2 C2

3 G2

4 C3

5 E3

6 G3

7 Bb3*

8 C4

9 D4

10 E4

11 F4

12 G4

A few things to notice:

For any partial, the partial that is twice that number is the same note at a different octave. Notice that all powers of two (i.e. 1, 2, 4, 8 ) are all C's and that partial 3 and partial 6 are both the same notes.

If we look at partials 1 through 8 we can see that the notes form a dominant seventh chord.

We've stopped at partial 12. If we graphed frequency versus partial number, we'd see that this series diverges to infinity, so there are in theory an infinite number of partials. The problem is that as we move up the series, the partials get closer until they are either horribly dissonant or indistinguishable. Other music theories, particularly non-western ones, use more partials and employ quarter tones (the interval of half a semitone) to make this possible.

Partial 7 has an asterisk. This is because the 7th partial is actually out of tune (31 cents flat of Bb, remember that each semitone is 100 cents so 31 is pretty significant). This is a naturally occurring problem that luthiers (instrument makers) and composers have struggled with for a long time. The dissonance from this pitch problem is significant, and there have been a number of solutions throughout history that all involve various ways of spreading this tuning problem out among other notes. Currently most instruments use a method called equal temperament where the dissonance is spread equally among all notes in the scale. There's a good post about intonation here: http://forum.bandamp.com/Music_Theory/3732.html.

For me, the most interesting thing about the harmonic overtone series is that it gives us some physical basis for our theory. The harmonic overtone series outlines the relative importance of various scale degrees. Look at the partials we've listed. Considering that notes will reappear every time their partial number doubles, we can say that the earlier a note appears as a partial, the more times it will occur in the series. We can see this in the first 12 we've listed. We have:

4 C's (tonic)

3 G's (dominant)

2 E's (mediant)

1 Bb (subtonic)

1 D (supertonic)

1 F (subdominant)

Notice:

The the first 3 distinct notes outline a major triad, first 4 outline a dominant seventh chord, the first 5 a ninth, and the first 6 an eleventh.

The tonic occurs every 2^(n) times at the nth octave above the fundamental, this is more than any other note.

Moreover we can use the overtone series to determine the ratio of the frequencies of the upper and lower notes of an interval. The ratio of an octave is 2:1, and notice every time a partial number is doubled the same note appears an octave higher. The ratio of a perfect fifth is 3:2 and the ratio of a perfect fourth is 3:4.

The series of overtones that an instrument produces is a characteristic of that instrument and accounts for its distinctive sound or "timbre". Some instruments, like cymbals, don't have a tone - these are called indefinite pitch as we can't clearly distinguish the fundamentals as overtones produce too much noise. This website has a good explanation of timbre and waveforms. http://www.soundpiper.com/elements/timbre.htm

See:

http://en.wikipedia.org/wiki/Harmonic_series_(music) for more information

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