Series


(entry for 12/6/2024)


One important aspect of music we haven’t touched on yet is the notion of the Harmonic Series, or Overtone Series, or Series of Partials. (All of these refer to the same sonic phenomenon, though the definitions used in each are a bit different.)

We have needed to cover some other concepts first, like the idea of ‘intervals’ and the meaning of ‘chords’ and ‘triads,’ but we’re now (finally!) ready to get to one of the most fundamental ideas of all.

Every musical note you hear contains other notes that you aren’t aware of. (Well, there’s one exception to that fact, but we’ll get to it later.) It turns out that this fact is the entire basis of what we call ‘harmony,’ so it’s extremely important. High time, then, that we explore it in some depth.

Let’s take a very familiar pitch, for example: the note ‘middle C.’ When you hit that note on a piano, you are not hearing just the C sound for that pitch. You are also hearing the C an octave above that, the G above that, the C above that, the E above that, and the G above that, to name just a few of the other sounds you are hearing. (Note that all of these sounds are higher in pitch than the note you thought you were playing by itself. This series of unnoticed pitches continues to go ‘up’ until it leaves the range of human hearing, and it goes on higher and higher, forever!) You do not consciously hear these higher pitches. You only hear, or think you hear, the note you played. All of those other pitches are interpreted by our brains as part of the ‘Tone Quality’ of the sound. Some instruments, like the trumpet and piccolo, have a higher proportion of the highest pitches in the makeup of their basic pitch, and are therefore thought of as ‘bright’ or ‘piercing’ sounds. Some instruments, like the string bass and bass flute, have less of the higher pitches and more of the lower ones in their makeup, and are therefore thought of as being ‘mellow’ or even ‘bland.’ And some instruments, such as the clarinet, are totally missing some of these ‘extra pitches’ from their sound quality and therefore have a ‘hollow’ tone, especially when played in their lower registers.

If these upper pitches are ‘all over the place’ due to the materials and shape of the instrument, the extra pitches overwhelm the basic sound, and you get an ‘un-pitched’ sound, such as made by a cymbal or tam-tam. Or a drum.

The makeup of the total sound can be quite predictable, and you can imagine what an instrument would sound like even if it didn’t exist in real life, if you knew what its ‘overtones,’ as these ‘extra pitches’ are called, consisted of.

A bowed string instrument, such as a violin, viola, or ‘cello,’ has the most predictable sound makeup of all, so we’ll start with one of those to explain the whole idea.

You learned in the Temperament post that the only pitch that is ‘pure’ on a keyboard or fretted instrument is the note ‘A.’ It was also mentioned that A440 was the standard pitch from which all other pitches were defined. And you also learned in the same post that doubling the vibrations of a pitch made a sound an octave higher, or that cutting the vibrations in half lowered the sound by an octave.

So, we’re going to start with an ‘A’ pitch in this example, but we’re not going to start with A440, because the numbers would quickly get too big and unwieldy, and we wouldn’t be able to do the match in our heads.

So instead of using the ‘A’ string on a violin or viola (A440) or the one on a ‘cello (A220), we’re going to use the ‘A’ string on a string bass (contrabass, double bass, doghouse bass, etc., whatever you want to call it) which is A110. (See the bottom note on the example at the head of this post.)

When you consider the string vibrating as a whole, the vibration is 110 per second, which is why we call it A110. But that makes up only half of the total sound! The next 25% of the total sound is A220, because the string is also vibrating in half-lengths, each of which is half as long as the whole string, and each of which vibrates an octave higher, namely at double the speed.  (Shown as the top note on the bottom staff above.)

So the A110, which we call the ‘Fundamental,’ because it’s the pitch we thought we were playing, is 50% of the sound, while the first overtone (the half-lengths of the string) is 25% of the sound; guess what comes next? Right, dividing the string in thirds, which is 12.5% of the sound. But it isn’t an ‘A’! It’s an ‘E’! To get the first overtone, we multiplied the fundamental vibrations frequency (110) by 2, to get 220. For the third-lengths of string, we multiply by 3, to get 330, and that happens to be the sound ‘E.’ (Bottom note of the treble clef staff at the head of the post.)

Now we come to fourth-lengths, which is 440, and another ‘A,’ which makes up 6.25% of the total sound. Each additional overtone gets a percentage of the total sound that is half as much as the percentage before. So 550 vibrations per second on that A110 string bass open string will get 3.125% of the total sound. And the next overtone, 660, (another ‘E’) will be 1.5625% of the total sound. If we add all the percentages we have so far, we get 98.4. In other words, the fundamental plus the first five overtones accounts for almost 99% of the total sound. In ‘Western’ music, that is, the music of Europe and the Americas, that’s all we worry about. (We’ll come back to other parts of world geography in a later post.)

And so far, everything makes sense. We have three A’s (110, 220, and 440) two E’s (330, and 660), and one frequency we haven’t defined yet. 550.

But hold on! What is 550? It isn’t an ‘A’ and it isn’t an ‘E.’

Well, it turns out that it’s a ‘C-sharp’!

Now your first reaction to that fact is very likely to be, “So what!” But stop and think: if you stack up the third, fourth, and fifth overtones, that is, A440, C-sharp 550, and E 660, what does that make? Answer: it makes a major triad! In this case, an A-major triad. (Top three notes of the example at the head of the post.)

But remember, we’re not actually playing an A-major chord. All of this is contained in that single note A110 that we started out with.

Moral of story:  Every time you play a single note, you’re also hearing a major triad contained within that note, in its overtones!

Of all the sentences you’ve read in all of these posts so far, that is probably the most important one. Think about it. Every note contains a major triad within it!  For future reference, we’ll call that ‘Statement One.’

This explains so many things: When we’re playing a piece in the key of A major, why does an A major triad sound like ‘coming home’? Answer: Statement One! Why do pieces written in minor keys sound more complete when they end on a major chord? Answer:  Statement One! Why do our ears demand that the V chords (the chord built on the 5thdegree of the scale we’re in), in both major and minor keys, be major triads or dominant 7 chords? Answer: Statement One! We could go on and on, but you get the idea. The foundational nature of Statement One is responsible for countless aspects of music theory and related musical subjects. Our ears expect to hear major chords in important places because that’s what they’ve been hearing all along!

(I earlier promised you one exception. Here it is: a wooden flute played softly produces only the fundamental note that is being played. No overtones. That’s why it sounds so ‘pure.’ It has no ‘tone color.’ But that’s it. In all other cases, and in the human voice and in all other instruments, the proportions of the various overtones may be different, but they’re always there.)

Which explains something important: most audiences react negatively to so-called atonal music, not just because it ‘sounds different,’ but because it ignores major triads, and the ear knows better! It isn’t just a matter of ‘educating the listener’s ear,’ as some pedagogues have averred. It’s also a matter of educating composers to the reality of the science of sound. The major chord is the most common chord we ever hear, and the most ‘natural sounding,’ because it’s already in every note we ever listen to!

By the way, when you stack up all those percentages, and look at the resulting wave shape on an oscilloscope, you see something that resembles the teeth on a handsaw.  The total waveform therefore is called a 'sawtooth' wave. It looks like this:



You don't have to have a string instrument to hear the sound. You can ask a synthesizer to produce a sawtooth wave sound on command. Not too surprisingly, the result sounds exactly like a string instrument!

Bottom line: a group of string instruments sounds 'best,' that is, most 'at rest,' when the final chord is a major triad. Since that sound is contained in every note already.

Yes, I'm aware that minor chords can be very beautiful, and some people even prefer them.  I was of that school of thought for many years myself.  But facts are facts. A minor chord occurs nowhere in the overtone series.  But the major chord is in every note.

The famous atonal composer Anton Webern once said something like, “Music is not about the listener.”

Well, with all due respect, sir, I beg to differ. If music isn’t about the listener, that what the heck is it about?


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To see an index of all of Len’s Music Blog posts, please click HERE.

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