Theory Top-Up Harmonics: an introduction to the mysterious overtones in our music


Piping Today #82, 2016.

When you come across the word “harmonics”, do mysteriously inconspicuous high notes come to mind, the ones that some people claim they can hear embedded in the sound of drones, and that are apparently used to more finely tune their instrument?  If so, you’re not too far off.  They are strange things, these harmonics, and you may doubt their existence.  But I’m here to tell you they do exist, and not only that, but they are a beautiful and essential natural phenomenon in music and beyond.  

Harmonics are ubiquitous: every one of us hears, creates and responds to them many times every single day.  They are embedded in every voice, every musical instrument, the ring of a church bell, the ringtones of your phone, the song of a sparrow, the ‘moo’ of a cow, the hum of your refrigerator, the honk of a car horn and even the buzz of a Cessna flying overhead.  If you enjoy music and want to get better at listening to it and creating it, harmonics are worth getting to know.

Those of you following the Theory Top-Up series and any of the preceding articles I’ve written on musical keys will have encountered a number of analogies relating music to food: comparing musical keys to a bowl of flavoured ice cream, B-minor tunes to salted dark chocolate, and so forth.  The worlds of culinary taste and musical sound have a surprising amount in common, particularly in the way that we discern particular flavours and timbres.  Sometimes these worlds overlap, such as when someone describes a pipe chanter as sounding “sweet”.  For this article, I’m hoping you’ll tolerate an imperfect analogy and a simplified physics lesson in order to help you better understand the world of harmonics.

The harmonics of beer

When young people sip their very first ‘adult’ beverage, such as coffee or beer, their initial reaction is rarely one of immense pleasure.  But over varying amounts of time, palates mature, and the same individuals may find themselves not only enjoying these beverages, but craving them and developing strong preferences for specific types and brands.  Sure, part of that craving has to do with the physiological effects of caffeine and alcohol, but people also begin to distinguish why they like one variety over another, singling out subtle differences in aroma, taste, aftertaste, the feel of the drink in the mouth and even appearance.

With this in mind, read the following blurb, plucked from the website of the Stone Corral Brewery, owned by Bret Hamilton, a piper who also hosts the annual Càirdeas gathering (a.k.a. The Vermont Bellowspipe & Fiddle School).  This is a description of his Stampede double-IPA:

“Meridian, Sterling and Nugget hops showcased against a robust blend of 6 malts. Notes of stone fruit, citrus, honey and resin with a big malt background and assertive hop bitterness.”

Bret, a talented and experienced brewer, has the ability to distinguish a remarkable range of specific flavours effervescing in his frequently blissed-out mouth.  Nobody who describes or reviews a beverage such as this ever seems to say that it just tastes like beer.  Of course it does taste like beer but that is to speak very broadly.  Obviously there’s much more happening in addition to that fundamental beer taste.

The same is true for many other beverages and foods such as coffee, tea, whisky, cheese and so on.  We all recognise that on a generic, fundamental level, each of these types of drinks or foods tastes like coffee, tea, whisky, cheese, etc., and that variations in their native soils, climates, source ingredients, production, ageing and packaging all contribute various flavour ‘overtones’, some of which are so slight that only the most gifted sommeliers can detect them by engaging in peculiar tasting rituals.

You may know that flavours are essentially combinations of chemicals which create a variety of sensory impressions via the taste buds on our tongues.  We can distinguish between any number of foods and beverages simply because our taste buds are sensitive to a seemingly infinite combination of these chemicals.  If someone were to blindfold you and give you a glass of apple juice and a glass of pear juice, you could immediately distinguish between the two, thanks to your ability to perceive differences in the chemical make-up of each of those juices.

Likewise, if someone were to blindfold you and play a low-A on a Highland pipe chanter and then the same exact note on a clarinet, you could immediately distinguish between the two.  But how can your ears ‘taste’ the difference?  After all, the chanter and clarinet are both wind instruments made of similar wood, and which produce sound from a cane reed vibrating at the same exact number of beats per second.  And they both have vertically arranged sound-holes and a relatively similar shape.  So how is it that we hear them as being so different?  And how is it that some pipers can hear the difference between a set of Lawrie drones and a set of Henderson drones, even if the pipes had happened to have been made from the same tree, in the same year, reeded with the same reeds, tied into the same type of bag and played by the same player?

The answers to those questions have a lot to do with the harmonics that are naturally embedded within most of the sounds we hear.  The way we distinguish harmonics with our ears is a lot like how we distinguish flavours with our tongues.  The way we hear harmonics helps us hear the difference between the words “peer” and “poor”.  It is the variations in ever-present harmonics that help us distinguish the voices of different people, or different instruments, or the difference between good chanter tone and bad.  We pipers, whether consciously or not, also tune our pipes by the way we hear particular harmonics.  And just like with food and drink, we each have different harmonic ‘tastes’, whether or not we’re even aware that those harmonics exist.  It could easily be argued that understanding harmonics, and being able to hear and distinguish them, is paramount to becoming a successful piper.

The physics of a vibrating string

Disclaimer: The following is a greatly-simplified physics discussion.  If you wish to understand this phenomenon at a more deeply scientific level, I encourage you to talk with an actual physicist and/or study the wealth of material available online.

The simplest and most accessible way to introduce harmonics to you may involve a string.  The image below represents a string with nodes at either end.  Let’s pretend it’s the A string on a fiddle, with a bridge at one end, and the nut at the other.  If you were to bow or pluck the string, it will vibrate, oscillating back and forth along the whole of its length.

This vibration produces sound waves, and the most prominent wave produced by the A string on a fiddle produces, unsurprisingly, an A.  If it is a standard, well-tuned fiddle, that string will vibrate at a frequency 440 times per second, or 440 hertz (Hz).  Any note that is produced from the vibration of the full length of the string, a “standing wave”, is called the fundamental frequency (or fundamental for short).  The fundamental of your Highland pipes’ bass drone is most likely a B-flat in the 120Hz ballpark.  The fundamental of your tenor drones would also be a B-flat, but an octave higher, near 240Hz1.

It’s nothing too complex; but look beyond that fundamental vibration and things start to get a little more interesting.  That string can’t leave well enough alone and it’s going to do this crazy thing where it also decides to vibrate in whole-number ratios of its fundamental frequency.  At the very same time that string is vibrating along the whole of its length, it is also vibrating at twice the frequency of the fundamental:

Being twice the frequency of 440Hz, we therefore have a simultaneous vibration of 880 times per second.  As it turns out, 880Hz is also an A, precisely one octave higher than the fundamental.  And what that means for you, the listener, is that you’re actually hearing two notes at once, from the very same string: the fundamental A (440Hz) and the A an octave above that (880Hz).  The fundamental A, however, is so much stronger that it mostly eclipses the other, which you really have to concentrate to hear.  In fact you may not hear it at all at first, or you might think you’re imagining it.  But it’s really there, I promise.  That secondary A, the much quieter one, is generally called a harmonic (or partial, or overtone).

I can tell this is starting to make you thirsty, so let’s look back to our beer analogy, and imagine that the fundamental 440Hz A (or the string vibrating along its whole length) is the basic, fundamental beer flavour.  The 880Hz A, on the other hand, could be analogous to the more specific “stone fruit” presence in the Stampede double-IPA.  It’s not the main flavour, but it is palpable and definitely contributes to the overall impression of that particular beer.

Put down your pint glass for another moment and let me introduce to you the next harmonic that would naturally occur with the aforementioned vibrating string.  Not only will this string, when bowed or plucked, vibrate at 440Hz and 880Hz, but it will also be vibrating at three times the fundamental frequency.  The harmonic produced from the corresponding 1320Hz is the E above the A at 880Hz (an important interval in music theory called a “perfect fifth”):

Let’s arbitrarily assign this third partial the “citrus notes” of the delectable India Pale Ale.  So now we have the fundamental beer taste, and within that we can just barely pick out stone fruit and citrusy flavours (overtones).  Make sense?

But wait, there’s more!  Our humble fiddle string is busier than you might think.  It is also vibrating at four times the fundamental frequency (1760Hz, an A two octaves above), and five times (2200Hz, a C#), and six times (2640Hz, another E), and seven times (3080Hz, a G)… and this pattern goes on, in theory, infinitum:

With all those new overtones embedded in our ‘beer’, we’re now getting the whiffs of honey, resin and the various malts and hops.  These flavours are super faint — most people can’t detect them without having practised the art of tasting — but they are there.  And some people can sense more of the embedded flavours than others.  The same is true for just about any musical note you’ve ever heard.

Bagpipes, of course, are not stringed instruments.  But the same idea of fundamentals and vibrating multiples applies to wind instruments.  Take the above explanation, replace the vibrating string with a column of air — like those found inside our chanters and drones — and voilà, you’re on your way to understanding how all this works on your bagpipe2.

The intervallic pattern

As an aside, you may also be interested in thinking about the “harmonic series” in terms of intervals, that is, the way we measure the distance between any two notes.  As briefly mentioned over the page, for example, the distance between A=440 and A=880 is an “octave” (technically a “perfect octave”), and the distance between A=880 and E=1320 is a “perfect fifth”.  Understanding this natural pattern in terms of intervals may help some of you transfer this knowledge to other keys, such as when playing a pipe chanter in a non-standard key:

It’s important to remember that pretty much any musical note or spoken vowel you are subject to is full of these harmonics, and that the pattern of intervals is essentially the same in every case, regardless of instrument or key.  Hearing these harmonics takes practice but if you can detect the difference between the sound of a clarinet and a pipe chanter, then you can hear plenty well enough to distinguish some of the more prominent individual harmonics.

Stay tuned…

But now you may be wondering about a question posed earlier in this article: how is it that we can distinguish between the sound of the clarinet and pipe chanter, even when they’re playing the same note; or distinguish between someone saying the words “peer” and “poor”?  The answer largely involves harmonics, of course.  Unfortunately the answer also involves another fairly in-depth explanation, and seeing that my pint glass is now in need of a refill, I will have to leave the second part of this story for the next issue of Piping Today.  In the meantime, I urge you to ask a guitarist, fiddler, or other string player to demonstrate harmonics to you on their instrument, especially in regards to the various fractions of a vibrating string.  You may also enjoying searching online for recordings of “throat-singers” and Jew’s harp players who are masters of manipulating harmonics with their mouths — sometimes to such a degree that you can hear the harmonics at least as prominently as the fundamental pitches themselves.

1. These are the sounding pitches, based on the assumption you are playing a standard Highland pipe that tunes at competitive pitches.

2. That’s once again a great oversimplification (like much of this article), as things get much more complicated in regards to open-ended columns of air, or conically-shaped columns of air, and so forth.

Tim Cummings plays, teaches, writes and publishes bagpipe musicHis Theory Top-Up series ran in Piping Today magazine for more than five years.

Theory Top-Up articles published on Bagpipe.News so far:

  1. Tunes in the key of D-Major
  2. Tunes in the key of A-Mixolydian
  3. Tunes in the key of A-Major
  4. Tunes based on a ‘gapped’ A scale
  5. Tunes based in A-pentatonic major
  6. Tunes in B-minor
  7. Double Tonic Tunes
  8. Tunes in the Dorian mode
  9. Tunes in G-Major
  10. Exotic tunes and tunes that change key
  11. Compressing tunes with low F-sharp notes
  12. Compressing tunes with high-B notes