Tutorial: The Harmonic Series

Posted on Aug 10, 2009 in Articles


Welcome to my first tutorial. Due to the many emails and messages I get, asking for certain things, I decided to start this little tutorial section on my webpage in which I will publish tutorials dealing with issues concering composition, orchestration and specifically writing music for movies. I will focus on purely musical issues in these tutorials which don’t require needing any special software skills etc. as there are still enough technical tutorials around online.

The first tutorial deals with the harmonic series from a musical and compositional point of view. There are tons of information on this issue available dealing with the acoustic phenomenon of this issue but I’ll try and focus on the things, the harmonic series can do for you in your music or what you can do for the harmonic series in order to not make it spoil your music.

Every tone an instrument produces consists 0f several frequencies. Apart from the base frequency, which we hear as “the pitch”, there is a series of frequencies above this. These frequencies are called “harmonics”, which always have the same distance to the base frequency the instrument plays. So if the player plays a note followed by a note a perfect fifth higher, the harmonics will also be a perfect fifth higher on the second note. The volume of the individual harmonics and their balance between each other determine what we hear as “instrument colour”, a “fingerprint” that makes the sound of an instrument individual and lets us recognize it. A very loud harmonic spectrum makes the instrument colour very rich and intense (think of a trumpet playing ff) while a soft harmonic spectrum makes a very warm, smooth sound (think of a french horn playing p).

Below you can see the harmonic series based on the note C. The numbers above indicate the tuning in cent compared to the tempered scale.

Harmonic_Series

These are the acoustic basics on this topic, if you want to read further about the acoustic details, you should check out the very good article on Wikipedia dealing with this topic.

To make things a bit clearer for you, here’s a small example:

This is an mp3 of a single tone, played by an oboe:

Audio clip: Adobe Flash Player (version 9 or above) is required to play this audio clip. Download the latest version here. You also need to have JavaScript enabled in your browser.

To make it’s harmonics series visible, have a look at this analyzing graphic:

harmonics_oboe1

The x-axis shows the time while the y-axis shows the frequencies. The brighter the colour, the louder the certain frequencies. As you can see there is the base frequency on the bottom of this image and quite a few loud harmonics (especially 1st & 2nd harmonic) which are actually louder than the base frequency. However, as our brain is used to the intervals in the harmonic series, it will still hear the base frequency as the actual tone. What you can also see is the vibrato, the player added as there are quite a few wavy lines in the spectrum :)

Now to make things even cooler, I used a tool which lets me delete frequencies in the spectrum as if i was cutting them out. Producing a file which looks like this in the analyzer:

harmonics_oboe2

As you can see, I left in the base frequencies and started cutting out the high harmonics letting them slowly come back in down to the base frequency.

The result sounds like this:

Audio clip: Adobe Flash Player (version 9 or above) is required to play this audio clip. Download the latest version here. You also need to have JavaScript enabled in your browser.

You can clearly hear the harmonics coming back in, especially at the end.

So far, this is only theoretical talk. However, understanding how this works is quite important in order to understand how to deal with this matter when writing music:

1. CHORD STRUCTURING

Having a look at the harmonic series you can see, that the lower ones up to the 6th harmonic produce a major triad when played together. However the spacing is somewhat different to how you would or COULD play it on a keyboard. Let’s stick to the harmonic scale based on C. The first harmonic which comes after the base C is the C again an octave above, followed by the fifth of the chord (G), the root again (C), the third (E) and the fifth again (G). When played together exactly as they appear in the scale, this triads sounds pretty full without getting overly muddy (as less muddy as you can get in this low register). Putting down the third one octave will dramatically change the sound to a undefinable mud while raising the lowest fifth up an octave will reduce the sonority of  the chord.

So when you’re structuring your chords, you might want to have a look at the harmonic series in order to make them sound good. For major and minor triads, this principle is not of ultimate relevance but it can be extended quite alot, especially when writing complex harmonic situations, which should really be handled with alot of caution. Let’s say, you want to write a Cmaj7(#11/13) chord and you’re not sure where to put  which tones of this chord. The good thing is: jazz theory (of which this chord symbol originates from, it would be called quite differently in classical theory but that’s another chapter) has dealt with the issue of harmonic series already giving you a hint of where to put chord tones by their numbers (in modern jazz theory, even though, it’s the same note (in our case F#), it’s called #11 and not #4 etc.). What does this indicate? You can or rather should handle these chord tones as if you were organizing them in a kind of harmonic series, even though the tones don’t actually all belong to the natural harmonic series. However, you can still use what we’ve learned from it by structuring your chord like this from the bottom to the top:

C(Root) – G (5th) – E (3rd) – B (maj7) – D(9) – F#(#11) – A(13)

As you can see, the low chord functions (1,3,5)  are in the lower area while the higher chord tensions (9,#11,13)  are being put to the top. If you compare this structure to the harmonic series, there are quite a lot of similarities (leaving out several tones from the harmonic scale and altering a few others chromatically). Of course, this is not the only way you could put this chord but it’s a good guideline. Try fiddling around a bit with other positions and listen to how they sound, whether good or not so good. The most dramatic effects can be achieved by switching pretty low chord functions with the high ones, try exchanging the #11 with the 5 and play a F# low and a G high. It will totally alter the sound and make it sound less structured. However, switching around the 9, #11, 13 won’t alter the character of the sound much.

What is also to be noted is, how you can put a diatonic cluster (Which actually is this chord when you put all the notes as close together as possible (C, D,E,F#,G,A,B)) into a chord with structure by simply organizing it in the way of a harmonic series. Of course, one could go way more into detail on this topic but that’s far enough for now :)

2. MELODIC COMPOSITION

Let’s say you want to write a melody that has an uplifting feel, you decide to “definitely want to write something that goes up melodically”. But what sounds cool? How to get up there? Using even sized intervals like thirds to get from a low note to a higher, far apart note? Use small intervals first and use larger intervals later, shortly before reaching the goal note? There’s a thing called “ballistic melody curve”. Which sounds like martial art rather than music, actually originates in the harmonic series. The “most natural” way to get from that low note to that far apart high note would be to start off with larger intervals which get smaller and smaller before reaching the target note. As you can see in the harmonic series, it starts off with large intervals (octave, fifth, fourth…) while it later has very small intervalls (major and minor seconds). The ear is very much used to this structure and for the ear it will sound very natural if you structurize your rising melody like this, too.

This is by far no law, but something you should have in mind. You can say: ” I want to have a surprise there, I will start off with small intervals and then leap to the target.” which is perfectly fine, just as it is using even sized intervals, but it still won’t do you any harm knowing about that little trick, when composing something. By the way, the same principle can be applied to accompanying patterns (compare the famous Alberti-basses, especially Mozart used, structured mainly in the harmonic series) as well as to guide tones of your melody, meaning that you could try writing such a ballistic melody curve and then start embellishing it by adding notes in between but have the main gestus of the melody still follow this principle. All these things will make your melodic ideas sound more naturally flowing. However, as said before, it’s no rule, just a hint :)

3. INSTRUMENTATION

There are quite a few instruments which base their playing on the harmonic series. Especially brass instruments like trumpets, horns etc. To make complicated things short: by altering the tension of their lips, they can play the harmonics of the base frequency their instrument has (which depends on how long the tube is). By utilizing the valves and combinations of them they can make the tube longer in certain steps so they have harmonic series of several tones available which in combination fill in all the gaps between the low harmonics (like the fifth between 1.st and 2nd harmonic) and make it possible for them to play every chromatic note. If you want to know more about this issue, you should, once more, read Wikipedia :)

So, what to take care of here? Did you ever wonder why especially french horns, even in professional orchestras, tend to hit wrong notes, especially when they’re playing high? No? Well, then read on at the next part :)

As we know, the harmonics tend to have smaller intervals between each other the higher you go in the scale. The French Horn is an instrument with a very wide range. Its base frequency is pretty low but it can play pretty high, too. It can achieve this by altering the lip tension this much, that it can play very high harmonics (talking about the 15th harmonic here). While the difference in lip tension is quite significant in the lower notes which makes it easier to hit the right notes, it gets more and more difficult the higher you go. Even the slightest change can make the player hit a different note. So, it’s very difficult to control for him or her, which harmonic he or she hits. Of course, horn player train their whole life to do this properly but even the condition of the air, the temperature of the instrument or a bad day can cause hitting a wrong note. What is even worse: let the horns start at a high note without perparation. For the player(s), this is close to lottery. In scoring or rehearsal situations where time is money, if you write such a passage, you might want to prepare for some extra time needed to nail this. Depending on the level of professionalism of the players, you might even want to count in for the possibility of “never getting it right.”

Why is this the problem especially with horns and not so much for example trumpets? Trumpets generally have a shorter tube as horns which makes their overall sound higher compared to the horns. Also, their playing range in the harmonics series is way lower than the one of the horn players. So the difference in lip tension between the tones is greater which gives more safety in hitting the right note.

There are even more issues concering the harmonic scale with players but I’ll go into detail on this when I do tutorials about certain orchestration things.

4. CHEATING THE EAR/BRAIN
This is one of my favourites. As I have mentioned before a few times, the ear and brain is very used to the harmonic series so it is able to complete it even though parts of it are missing. This is the only reason why we can understand people on the telephone, even though it transmits only a very small band of frequencies. The brain just completes the missing parts of the harmonic series of the voice we’re hearing so we still have the impression that our grand uncle with the deep voice still has his deep voice even tho not even a frequency close to the bass range is being transmitted. The cool thing is, you can use this principle for your music as well, even though, it doesn’t always work and you don’t might want to overuse it. The most common way is, to cheat an octave below a tone you play. If you add a well intonated perfect fifth above the tone you play, played by an instrument which doesn’t have a very bright harmonic series and has a sound colour close to the base instrument, you can cheat the ear into hearing the octave below your base tone. Have a look at the harmonic series above. The 2nd and 3rd harmonic are a perfect fifth apart. If you play them together, the brain presumes them as part of an harmonic scale and adds the base tone, which would be an octave below. This is called “residual tone”, which actually doesn’t exist, but we hear it. Now this doesn’t work every time and you don’t might want to use it at all. It works very well if played by an organ in pretty low registers, and adds an enourmos low frequency which you wouldn’t be able to hear that intense if it were actually played.

5. BUILDING YOUR OWN INSTRUMENTAL COLOURS

This is a kind of very experty thing to do but when done, it’s a very cool effect. Church organs work like this. The full and rich sound we all know as the organ sound actually consists of the sound of several pipes playing a harmonic series. A single organ pipe produces a sound close to a sine wave. So by adding pipes which play the harmonic series of the lowest pipe that is playing and balancing their volume, you can produce strange but yet attractive sound colours. This principle can also – to some extend – be transferred to orchestra or other quite large ensembles with a large range of instruments. You can actually try and experiment by adding harmonics from the series of harmonics to your melody. Of course they should be pretty high, so actually only instruments like flute, piccolo, celeste, glockenspiel etc. would be appropriate for this, but when done well, it results in a very interesting sound. Maurice Ravel’s famous “Bolero” does just this right in the middle if you want to have to listen how this sounds.

So, this got quite an extensive first tutorial. I hope you liked it and had some interesting insights and inspirations. Make sure to check by regularly for more tutorials to come.

All the best

Robin

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  1. Strings Part II : How to orchestrate the Harmony in the String Section | SCOREcast Online » The Home of the Professional Film Music Community - [...] article on how to voice a chord according to the harmonic series. I suggest that you give it a …
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