Capital-M Music

Music Theory Guides

Harmonic Series & Timbre

by Richard Bruner

This is a guide to the harmonic series and its role in musical timbre. The harmonic series shows up all over the place in music, so knowing the pitch sequence of the first several harmonics is useful. Timbre refers to the sound quality that separates the sound of different instruments, and these days from different sounds since electronic synthesizers can produce a very wide range of sounds (up to any conceivable sound, as we’ll explore a little bit in this guide). The harmonic series is also used by a variety of instruments in different ways beyond timbre as well - brass instruments work by activating different harmonics of a particular length of tube (the length of which can be varied either with valves that add tubing, or a slide in the case of a trombone). String instruments can play different harmonics by touching at particular points on the string to stop certain harmonics from vibrating and allowing others to come through instead (stopping at 1/2 the length, 1/3 the length, 1/4 the length etc. - see below for audio demonstrating this). Woodwind instruments get additional pitches by activating various harmonics of the base length of the tube, which is set by covering or uncovering holes drilled at specific locations in the body of the instruments.

So let’s look in the abstract at the concept of the harmonic series. At the most basic level, a harmonic is a sine wave at a specific frequency, which is in a particular relationship with other sine waves. By themselves, these sine waves are referred to as partials, but when you have sine waves at integer multiples of a specific frequency, they are said to be harmonics of that fundamental frequency. The sine wave at that fundamental frequency will then be referred to as the fundamental, or the first harmonic. If you multiply that frequency by 2, you’ll get the 2nd harmonic, if you multiply the fundamental frequency by 3 you’ll get the 3rd harmonic, and so on. If we start with a fundamental of 100 hz (hertz, the unit of frequency - going through 100 cycles per second), then the 2nd harmonic is at 200 hz, the 3rd harmonic is at 300 hz, etc.

The more harmonics we add in to a given sound, the brighter that sound is said to be. Here is an audio example of a sine wave at a fundamental frequency of approximately 100 hz - G at the bottom of the bass staff, so actually about 97 hz (on some speakers you might not even hear this, though I think most speakers today should be able to reproduce this sound), followed by an audio example of a sawtooth wave with a fundamental at the same frequency (warning - the sawtooth sound is much louder!). A sawtooth wave has all the harmonics, decreasing in volume (amplitude) at a rate of the inverse of the harmonic number (the second harmonic is 1/2 as loud as the first, the third harmonic is 1/3 as loud as the first, etc).

As you can hear, there is a big difference in the sound between these two audio examples. That difference is what we mean when we say a sound is bright. In subtractive synthesis, we start with bright sounds and then apply filters to selectively reduce some of the frequencies using filters. Here is an audio example of a sawtooth wave with a low pass filter applied that will reduce all the higher harmonics to effectively 0, and then it will “sweep” across all the other frequencies slowly letting in the higher harmonics and going from a dark to a bright sound:

On this sound, I’m going to add in resonance, which boosts the level of the sound right before the cutoff of the filter, which will emphasize each harmonic as it is added in. Notice that the pitch gets higher for each harmonic, and eventually we stop perceiving the pitch and the sound just starts to get “buzzier”.

Here is approximate notation for the pitches of the first 16 harmonics starting on the G at the bottom of the bass staff. This set of intervals is the harmonic series, and it will be the same sequence of intervals whichever pitch you start on. I’d recommend memorizing the sequence of pitches of the first 16 harmonics, as this comes in handy in all sorts of ways when thinking about and making music.

As you can see, the intervals start off far apart, and get progressively closer together. Above the 16th harmonic our notation system breaks down and all the intervals wind up being less than a half-step apart. This is also around the area where harmonics stop sounding like pitches when you pick them out individually and start just adding buzz to most sounds. Actually, these intervals are approximate, as every single harmonic is closer in pitch to the previous one, even if the notated intervals are the same. The most obviously “out of tune” one in the lower part of the series is harmonic 7, which appears to be a minor third above the 6th harmonic, which is a minor 3rd above the 5th harmonic. But the 7th is a closer minor third, so it sounds flat compared to our “tempered” scale we use for tuning in playing music. You can clearly hear this below in my viola demonstration. The next most obviously out of tune harmonic is the 11th harmonic, which is actually halfway between a major and minor second relative to the previous note. It’s usually written as a major 2nd above the 10th (so as a C# in this example), to which it is flat, but it could also be a sharp minor second (C natural). Note that the audio here is from the notation, so these tuning effects will not be audible in this audio file - look at my filter examples above or my viola examples below for the actual harmonic sound.

Arguably it’s actually our system which is out of tune, but there are other physical reasons why we needed some kind of tempered system to make certain musical effects possible - notably allowing fixed pitch instruments like keyboards to modulate to all (or even most) keys without being physically retuned to do so. This was the goal of developing “equal temperament”, the system we usually use today. Two books I liked if you want to pursue that story are “How Equal Temperament Ruined Harmony (and Why You Should Care)” by Ross W. Duffin and “Temperament: How Music Became a Battleground for the Great Minds of Western Civilization” by Stuart Isacoff.

Another thing to notice here is that a doubling of harmonic number produces an octave compared to the lower pitch. This is clear in the sequence of the fundamental. The second harmonic is an octave above the first, the 4th is an octave above the 2nd, etc (1,2,4,8,16,etc). But it’s also true for all the others - the 6th harmonic is an octave above the third, the 14th is an octave above the 7th, etc. This corresponds to a doubling of the frequency of the sine waves in question, which is how an octave is defined in technical terms.

One more practical point I will make about this series is that this is a good rough system to use when voicing chords in several notes. You should use larger intervals in the lower part of your voicing, and then you can use smaller intervals the higher you get. In the bottom of the range of the orchestra (or the piano, for that matter), using octaves for the bass is useful. 5ths and 4ths can come in the upper bass range or tenor range, then thirds come in the upper tenor, alto, and soprano range, and 2nds should probably normally stick to the middle of the instrumental range and above. You don’t have to stick to this ordering, but it will produce the best blend and resonance in the sound, and if you vary the voicing from this (approximate) pattern the effect will be striking, so you should make sure you know why you want to vary it. By the way, the harmonic series is the reason why this voicing produces the best blend and resonance, and it is a basic fact of human anatomy and the physics of sound, not just a cultural or historical accident.

In our specific culture of music (so called “Western Music”), you can trace out a path through music history using the harmonic series. Leonard Bernstein has a magificent lecture where he did just that, going through “10 million” years of music history in about 5 minutes by adding harmony based on the harmonic series. He admits that it’s a “very high overview”, but here it is below if you are interested:

I mentioned that string instruments can play harmonics (and also that the sounds are made of harmonics, as all pitched sounds are). Here’s a demonstration of that with me playing my viola. The first (left) file is me playing just the open C string for several seconds. I took this recording and loaded it into Arturia’s Pigments synthesizer as a sample, and then applied a bandpass filter with a very narrow bandwidth (which “band” of frequencies it allows through - here almost only one). You can hear the filter pick out each harmonic as it sweeps, demonstrating that they were all there to begin with. This is the 2nd file here.

Then I played the first 8 natural harmonics on the viola itself, as a playing technique. It gets progressively more difficult to isolate the harmonics as you get higher (the finger spacing between them gets smaller and smaller). 8 harmonics is about as high as you can reasonably get, and I can’t always get that many to sound decent. We are rarely asked to play above the 5th harmonic in actual music with natural harmonics on violin and viola, and most of the time it’s really the first 4 (on each string). We can also use artificial harmonics, where we finger one note, lightly touch a fourth higher, and get a sound two octaves above the fingered note (that would be the fourth harmonic for people keeping track at home). These are just the first 8 natural harmonics of the C string. The final audio file (the right one) is a demonstration of a technique called “harmonic glissando”, which is where I lightly run my finger up and down the C string to pull out the different harmonics in turn, used mostly as a textural sound effect in orchestral music with a section playing this together. I’ve included a spectral analysis image below the audio for the open C string audio and the 8 harmonics audio with additional analysis of the audio / images (see Figs. 1 and 2).

Given what we now know about how (pitched) sounds are made of sine waves in harmonic relationship with each other, and given that we can electronically generate sine waves, you may be wondering if we can create sounds by layering sine waves together directly. The answer is yes, as I hinted at above. Additive synthesis is the name of this technique, and an early version of the technique as an electronic technique (or really an electro-acoustic technique) was found in a machine called the Telharmonium as a proof of concept experiment in the 1890s. This was effectively later miniaturized and commercialized in what became the Hammond organ, most famously the Hammond B3, using tonewheels in both cases to create nearly perfect sine waves at specific frequencies.

These days, you can certainly still find the Hammond organ in studios, churches, and bands all over the world, but as a concept, additive synthesis is usually thought of as a technique found in software synthesizers. The basic idea is that we can generate hundreds of sine waves, give them all an initial frequency, amplitude, and phase in relation to each other, and then use envelopes to alter the amplitude and frequency of each one over time to generate a sound that changes over time. 

In principle, this technique can be used to generate any sound (at least any pitched or semi-pitched sound) that can be created acoustically or by any other synthesis technique. But it’s cumbersome enough that it’s rare outside of experimental contexts to work directly at the level of the additive parameters. This would be something like the equivalent of working at the machine-code level for computer programming vs. higher-level languages like C (and variants), Python, or even BASIC. 

The hammond organ featured 9 tonewheels per note, which can be thought of as nine harmonics per note, and pretty much any sound you can come up with at that level of additive synthesis will sound more or less like a hammond organ. To get much more complex sounds requires manipulating dozens or hundreds of harmonics, and so nowadays we tend to leave the low level aspect to computers and use it as the back end to something that looks like another kind of synthesis on the front end that we manipulate as sound designers or musicians.

Additive synthesis is often going to be the back end sound generator for a physical modeling synthesizer, which will set the values for the additive parameters from its physical model. I can’t verify it completely, but I’ve worked out how every parameter of the GUI in Pianoteq can be mapped to additive parameters, and I suspect that’s more or less what they are doing for the pitched components of the sound of Pianoteq. Additive synthesis is the back end for what looks like a variant of subtractive synthesis with Native Instruments’ Razor synthesizer. It used to be the main component of the sound generation in a plug-in called WIVI by Wallander Instruments, which has since become a component of Wallander Note Performer for notation software (the woodwind and brass sounds are additive, the strings and some of the other sounds are sampled).

Going the other direction, a company called VirSyn has worked out ways to use a version of additive synthesis as the front end of a system in plug-ins and iPad synths like Cube Synth, MicroTera, and parts of Tera Synth, sometimes using wavetable synthesis as the backend of those plug-ins. Xfer Records Serum and Arturia Pigments both feature different implementations of Additive Synthesis as one of several possible techniques in those plug-ins. Of those two, Serum is closer to the low-level version of additive synthesis, used as a technique to generate single frames of wavescanning wavetable synthesis, which you can then morph between to generate a full sound.

The closest that I’ve found to pure additive synthesis in a commercial synthesizer is Alchemy, which is now included in Logic Pro (and limited to Logic Pro). You can use it as one of several techniques in that synthesizer, and either hand code the parameters or have the synthesizer analyze an audio sample that you provide (called additive resynthesis). It will then create a sound that is as close as it can to the audio sample, but now you can manipulate it using the additive parameters rather than sample playback parameters.

I will use some of these plug-ins to demonstrate some more aspects of additive synthesis and the harmonic series below (particularly in the section labeled “When ‘Harmonics’ Aren’t Harmonic” - coming soon!).

Sonic Illusions are fascinating. You’ve probably run across optical illusions before, where you see things that aren’t really there in certain images (or you can see an image in multiple ways depending on how you look at it). But have you ever thought about a sonic illusion, where you hear things in different ways, or you hear things that aren’t really there? I took a class at Berklee in Psychoacoustics, which is the study of how our hearing works. In that class we focused on perception, so how sound goes from acoustic air pressure waves to electrical signals that can be interpreted by your brain. They had another class I didn’t get to take that looked at how sound goes from electrical signals in your brain to your conscious experience of sound (things like association - those sine wave combinations are the sound of a piano, and then that links into your lived experience and triggers memories of the significance to you of the piece of music they are playing, etc.)

We looked at several sonic illusions in the class I took, and one illusion I liked a lot was the “Missing Fundamental”. Here are two ways to look at this.

The Missing Fundamental

Ex. 1 - Below, I have an audio file playing two sets of tones. Listen to this file a couple of times and decide whether you think the second sound you hear is going up or down in pitch compared to the first sound (the two sounds are separated by a second in time, and then there’s a couple seconds between 4 repetitions of the pair of sounds). I’ll put what’s actually happening in an accordion tab below, which you can click or tap when you are ready to see the answer.

Ex. 2 - Here is another way of exploring the same phenomenon of the missing fundamental. I have five audio files below. All have the same structure, a set of tones played for two seconds and then with a two second gap before the next repeat of the same tone, played 4 times total. The first file has just one sine wave, at 1000 hz. We are trying to create a 200 hz tone here, so this will be the 5th harmonic. Then the next file has both 1000 hz and 800 hz combined as the 5th and 4th harmonics. Then we add the third harmonic at 600 hz, then the 2nd harmonic at 400 hz, and finally I add in the missing fundamental (so it’s not missing anymore). Somewhere between the 2nd and 4th files, you should begin to hear a lower tone (possibly faintly), and most likely by the 4th file you will hear the 200 hz tone. Only the final file actually has it in the audio itself (and it will be much more obvious - it shouldn’t sound that prominent in the other files).