HARMONICS, EQUAL TEMPERAMENT, and everything really


Without Music, we could completely destroy the structure of the space time continuum!

– Dr Emmett, Back to the Future



When you strike a string, it resonates at a particular frequency. As does the air in a pipe, or around a reed. Which, in physical terms, covers most of everything an orchestra does. Those resonating waves cause the air particles between the object and your ear to vibrate, and that is how you hear the sound.



If you kick a sack of potatoes, you don’t expect to hear the hallelujah chorus. That you hear noise – as opposed to music – is due to the fact that the sack and potatoes are not sympathetic to resonating. Too many different and low frequencies of sound are being produced at the same time. What you hear is a dull thud.


Key change?

Key change?

Find an object that is free to resonate at suitably uniform nature, say a wine glass or a metal bar, and you produce a noise much closer to a single frequency. Which sounds more like a note to our ears.

So in order to make notes, we need materials that oscillate usefully. A string is ideal. When a string vibrates at a certain frequency (let’s say 220Hz) we hear a certain note (in this case the note ‘A’, two white keys below middle C on a piano).


However – and this is where harmonics gets involved – there’s a lot more that happens when that string vibrates. In fact it oscillates at a number of different but related frequencies, all at the same time. In other words, it produces a number of different notes, all at the same time. These parallel notes are known as ‘harmonics’, and their sounds are usually hidden from our ears by the amplitude – the loudness – of the principal note.

All of which takes us right back into ancient history, since as far as we know, Pythagoras was the first to discover these harmonics, some 500 years before Jesus was even born.




The story goes that one day as Pythagoras was out walking, he heard a sound of pure beauty coming from a nearby blacksmith. On running to the smithy, he saw the sound was produced by two different sized hammers hammers striking at the same time. Pythagoras’ stroke of genius, according to the tale, was to realise the hammers were mathematically proportional to each other in size. Harmonies produced by harmonious proportions.

medieval depiction 2000 years after event

medieval depiction 2000 years after event

The story is almost certainly apocryphal. You don’t just double the size of a hammer to create an exact harmonic. Much more likely Pythagoras would have discovered harmonic relationships by playing strings of different lengths but related lengths. He may then have observed that by dividing a string into exactly half its length, he produced a note that sounded the same, only higher. This interval, termed an ‘Octave‘, was already covered by us back here in Beethoven 1.



String Theory

So let’s just peek at the science of harmonics, and in particular making octaves.  If we halve the length of a string, we produce a note with a frequency exactly twice the size. The wavelength of sound is exactly doubled – literally. Which is why a string vibrating at 220Hz will sound exactly an octave apart from an another at 440Hz.


String vibration by Andrew Davidhazy

String vibration by Andrew Davidhazy

Now remember, we are talking about a harmonic here, not a separate note. When you play the note A on a violin (or in scientific terms when you vibrate that string at 220Hz) the string simultaneously resonates at different but sympathetic frequencies. Parts of the string are literally vibrating at twice the speed of the rest.

Which produces this primary resonance we term the octave, though of course the note itself is too indisctinct to hear from the original principal note.


But it doesn’t stop there, for within that string, other sympathetic vibrations cause more notes, or harmonics. Something Pythagoras discovered thousands of years ago, as he proceeded to extract further secrets of harmony from nature.


The Fifth Dimension

Producing harmonics with a string

Producing harmonics with a string

The next harmony after the octave that Pythagoras must have discovered is the interval we now call the fifth (aka the perfect fifth or the dominant). This is produced when a string is divided by into 1/3rd of its length.

The sound this note produces lies at the exact centre of an octave. We call it a ‘fifth‘ because, in a major scale, it is the fifth note (for the same reason, the eight notes in an ordinary major scale is the octave).

In frequency terms this note lies exactly between the wavelengths of the octave. An octave running from 220hz to 440hz would have a perfect fifth at 330hz (on a keyboard, that would be the note ‘E’, between two As).

Here’s a demonstration of that interval:



Harmonics of note, octave, fifth and fourth

Harmonics of note, octave, fifth and fourth

Like our octave, this fifth occurs naturally in the world. It isn’t man-made. Although the octave is clearly the principal harmonic (we’ve already seen how it divides up the universe of sound here in Beethoven 1.

But next comes the fifth interval, this is real primeval stuff in the world of music. Go anywhere on this planet, and hear any music being played, and you will somewhere find this fifth interval.

So strong is this relationship between the root key (termed the ‘Tonic’) and its fifth, that the fifth is termed the ‘Dominant’ in musical lexicon.



The Fourth

Subdominant is the same distance below the tonic as the dominant is above...

Subdominant is the same distance below the tonic as the dominant is above…

The next most important relationship, after the octave and the dominant, is the 4th note in the scale, what is termed the ‘Sub-Dominant’. You achieve this note harmonically by dividing a string into 1/4rts as opposed to 1/3rds. But this subdominant is harmonically related to the tonic by making a 5th interval counting downwards, as opposed to upwards.

In chord terms, these three keys are therefore Tonic, Subdominant, Dominant, or I – IV – V. If any of you have experience with pop music, you will know that those three chords cover the majority of the music made.



Building the Triad

Beethoven himself uses the harmony of the fifth (but in the key of A minor) as the opening for this very symphony. He uses it to represent some kind of proto-musical state, you might describe like an orchestra tuning up:

Opening of 1st movement

Can you hear how that harmony sounds hollow? Listen to it again on its own (although in Dm):



The reason the fifth harmonic sounds hollow to our ears (as opposed to the wondrous harmony Pythagoras presumably heard) is because it is hollow.

To our ears at any rate.

We are used to adding a third in the middle of this first and fifth. If we add a third, we fill out the sound, and get the major chord of D:

Major Triad 

And if we add a minor third, we have D minor, which sounds thus:

Minor Triad 


There is a strong degree of what we might term ‘harmonic tension’ to do with this third interval. The fact we have major and minor versions of a chord by changing the third reveals something of this tension (as covered back here in Bach 4). It took a long while for this third interval to be included in western music

To understand why, we have to return to Pythagoras.



Back to Samos

Samos today

Samos today

In his exploration of nature, Pythagoras believed he was unlocking secrets of the fundamental workings of the universe. The community he established on the island of Samos really was more of a religious sect than an educational establishment. It’s easy to understand why. Pythagoras did pioneering work in geometry and harmonics at the very dawn of science. He had no scientific method to rely on, no body of academia to research or test his theories. He was seeking hitherto hidden relationships within nature. For him, there would never have been a split between scientific discovery and spiritual revelation.

With respect to harmonics, Pythagoras thought the sun moon and planets orbited the earth within spheres, and these spheres chimed with natural harmonics beyond the ken of humans. In establishing proportional relationships between sounds, he believed he had divined something of the inner-nature of reality. The mystical properties of such belief persist to this day, as any search for ‘Pythagorean fifths’ on YouTube will show you.

And yet, when inspect matters more closely, we soon see that nature is not as simple as some mystical healers may wish to believe.



Cycles of Circles



We can find every note of our 12-note scale (the black and white keys between an octave on a keyboard) by taking using our middle interval, our ‘fifth’, and continuing to split each new octave in half.  This process is known as ‘The Cycle of Fifths‘. The fifth of C is G, then the fifth of G is D and so on, until you come back to C.

Or that’s the theory.

The problem is if we actually continue to find notes this way – as Pythagoras would have done – we in fact move a fractional interval higher each time. And that shift, known to this day as the ‘Pythagorean Comma‘, becomes more audible with each step we take. Nature turns out to be more elusive than she may at first appear to transcendental lovers of Pythagorean harmonies.




Let’s put this in practical terms. If a piano tuner begins on a C, and works his way through thirteen steps of fifths (C to G to B to etc), he doesn’t arrive back at a C, but a note some quarter of a tone higher. This is an audible dissonance. Nature, it seems, doesn’t want to play ball. Or rather, she’ll play, but not in the geometry of a ball. Harmonics, in the actual world, doesn’t work as a neat circle, but rather a spiral (as illustrated right).

If you wish to find a our ‘third’ interval from C, we need to cycle through G – D – A – to get to our E. By this time, due to Pythagoras’s comma, the note has become ever so slightly off what we would term a ‘just’ intonation of E, that it doesn’t sound quite right.

For this reason, throughout the dark ages and early medieval periods, the third interval was regarded with suspicion, and was generally avoided in church music of the time.



Agreeing a solution

The great discovery of western music was that if we re-pitch this third a fraction lower, and then adjust other notes within the octave, we can create the major scale, the major chord and now we can create the entire major key.

Of course this is a man-made solution, a bodge. In fixing exactly proportional distances between the notes of the scale, some notes are fractionally off their harmonic equivalents. But it’s really hard to even tell the difference, and a pittance to pay in exchange for the masses of Bach, the symphonies of Beethoven, and the rest of the huge heritage of classical music this simple convention allowed.

Now, an orchestra could be formed, comprising of instruments like the horn or clarinet or valve trumpet which employ fixed notes, and it could play together in any key and remain – relatively at least – in tune.

Beethoven Excerpt 


Listen again to this Beethoven extract. It employs the very simple harmonies of church music, so create a sound with spiritual depth.

But there is more to it: underneath the singing, are horns, clarinets, flutes, oboes, all playing through different key changes. This would not have been possible until the system of equal temprement had been adopted. The depth of the sound, in other words, is only possible through a slight warping of harmonics.


It’s fascinating to think that our civilization has achieved the sound of this music, not just by unlocking the secrets of nature, but by cheating the results. However divine the music may sound, it has very much been created by the hand of man.