Logan is an active Jazz Guitarist, and classically trained composer with an affinity for contemporary musical styles.
Harmonics are a widely used musical technique, but where do they come from? This lesson will focus on what a harmonic is, where they come from, and how this all relates back to musical pitch.
When a musical instrument is playing a note, what we are actually hearing is the fundamental pitch, which is the pitch being played by the instrument, accompanied by a series of frequencies that are usually heard as a single composite tone. Those frequencies that are integer multiples of the fundamental pitch's frequency are called harmonics. If a musician causes one of these harmonics to sound, without sounding its fundamental frequency, it is called playing a harmonic. This can be a little bit confusing, so let's backtrack for a second. First off, we need to understand frequency.
Frequency is the rate at which a vibration occurs. This is measured in hertz (Hz), which is calculated by finding the number of vibrations per second. For example, a frequency that is vibrating 100 times per second would be described as having a frequency of 100Hz. When a pitch is produced, it creates a sound wave that vibrates at a specific frequency, the fundamental frequency, but it also causes a variety of other, higher frequencies to vibrate. These vibrations will be referred to as composite frequencies because they are a result of the vibrations of the fundamental frequency.
When the fundamental frequency and all of its composite frequencies are perceived by a listener, they are rarely heard as separate pitches. A listener will more likely perceive all of the frequencies wrapped together to form what we refer to as a composite tone. Any time an instrument produces a pitch, it will inherently produce a range of composite frequencies that add to the richness of the tone, and allow us to differentiate sound qualities, such as the difference between the way a violin sounds, and the way a guitar sounds. Ok, now that we've established a bit about how a pitch is heard, let's make it even more complicated!
In order to discuss harmonics, we need to add one more component to the mix . . . MATH! Mathematics plays a big part in discussing harmonics, but lucky for us, none of it will get overly complex. For a composite frequency to be considered a harmonic, its frequency must be an integer multiple of the fundamental frequency. Don't worry if that came on a little strong, we're going to elaborate a bit on it now.
Let's start with a hypothetical fundamental frequency of 100Hz. If we were to multiply it by any integer, our result would be considered an integer multiple of the fundamental frequency. In contrast, if we have a composite frequency, divide it by the fundamental frequency, and the result is an integer, then that composite frequency is an integer multiple. This is elaborated on a bit in the table.
Hypothetical Composite Frequency
Is It an Integer Multiple?
200 / 100 = 2
250 / 100 = 2.5
100 / 100 = 1
An overtone is any composite frequency that vibrates at a higher frequency than the fundamental frequency regardless of whether or not it is a harmonic. Most of the time, all of the overtones an instrument will also be harmonics, and because of this, the two terms are often used interchangeably. There are however, some instruments that will produce overtones that are not harmonics, most notably, percussion instruments.
Playing a Harmonic
Don't worry if you didn't catch everything right away, these topics can be very confusing at first, and sometimes they require a bit of time to grasp. For now, let's talk about why, as musicians, we care about harmonics in the first place! Musicians have found ways to isolate certain overtones and make them sound without the use of a fundamental frequency. The resulting sound is called a harmonic. Wait a minute! I thought a harmonic was the frequency itself? Well, it's both! It's confusing at first, but just remember that anytime a pitch is produced, it comes with composite frequencies, some of which are harmonics. However, if a performer is isolating one of these harmonic frequencies without the fundamental, that is called playing a harmonic, and the sound heard is also referred to as a harmonic.
The technique used to play a harmonic varies greatly from instrument to instrument; however, they are much easier to produce, and thus much more commonly used on stringed instruments. When used tastefully, playing a harmonic can be a beautifully expressive technique in music. Their sound is so distinctly different that they tend to take on an almost 'other worldly' quality.
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The term harmonic derives from its use in physics where it is used to describe a wave that is added to a fundamental wave in a specific pattern. These component waves vibrate in a predictable manner that relates back to the fundamental frequency (the rate at which a vibration occurs). Unfortunately, to discuss the nuts and bolts of harmonics, we have to get a bit mathematical and scientific again. Harmonic frequencies are integer multiples of the fundamental frequency. Another way of saying this is to say that harmonic frequencies are equally spaced by the width of the fundamental frequency. To find subsequent harmonics, we simply add the frequency to the previous harmonic.
For example, if our first, fundamental frequency is 10hz, then the second harmonic would be 20hz, and the third harmonic would be 30hz, etcetera. If for some reason, you'd like to find, say, the 7th harmonic in a series, you could do this by using the following formula. If 'f' stands for the fundamental frequency, then all subsequent harmonics in the series can be described as 2f (2nd harmonic), 3f (3rd harmonic), 4f (4th harmonic), etc. Using this formula, if we wanted to find the 7th harmonic of a fundamental frequency of 10hz, we would need to find the value of 7f, which happens to be 70hz. This pattern is demonstrated visually on screen.
A visual representation of the wave lengths of harmonics
A pitch at its most basic is a sound wave, and like any other wave, a sound wave will create harmonics. If we were to order all of the harmonic frequencies in ascending order, we would get a list of frequencies that looks something like this
etc . . .
etc . . .
Ok, now what? Well, what we need to do is a bit of conversion. Because musical pitches vibrate at specific frequencies, we can convert our list of harmonics above into musical pitches.
etc . . .
etc . . .
etc . . .
This ordered list of notes is referred to as a harmonic series. Don't worry if you don't know the exact frequency of every pitch, because thankfully, as with most things musical, the pattern is the same regardless of the starting pitch. In this image, you will see a harmonic series that has been created starting on C natural. This series can be transposed to any starting note, and it will still be the same.
The numbers above the harmonic indicate the difference in cents from equal temperament, rounded to the nearest cent.
Due to the nature of our contemporary tuning system, some of the upper harmonics will be a little out of tune, but if we hear them as a composite sound, this is not often bothersome. The harmonics are also gradually quieter than the fundamental, so the higher we go in the harmonic series, the less apparent a note will be. If, however, we are playing a harmonic in the series that is out of tune, then that note will be more audible and it's tuning should be considered.
Let's try and experience a little bit of this for ourselves. If you are able to find an instrument, ideally a piano, play the pitches from the image, then play only the bottom note and try very carefully to listen to the upper harmonics. With a bit of focus, they can begin to become quite recognizable.
Harmonics exist any time a note is played, they are heard together with the fundamental, and they are usually perceived as a composite sound representing the fundamental pitch. The order of harmonics is derived from multiples of the frequency, measured in hertz (Hz) of the fundamental pitch, and in music, these corresponding harmonics can be represented as pitches which make up a harmonic series.
A harmonic may also be referred to as an overtone. To play a harmonic, we isolate a specific harmonic from the composite frequencies of the fundamental pitch through various techniques, depending on the instrument, and they have a very unique sound quality.
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