The Science Of Music

The other day I got into a discussion with one of my students about how the notes we use in music relate to each other. The guy is a scientist by profession & has a very analytical mind (as you’d expect). His curiosity went beyond the “this note is related to that one because they’re in the same key” kind of response that I’m used to giving – he wanted the physics behind what makes music work so I set about explaining thousands of years of musical evolution in a one hour lesson. This is the gist of what we covered…

Like many things… philosophy; democracy; mathematics; geometry and the donner kebab (all of which enrich our lives) we have the Greeks to thank. We can trace our current musical system back to the genius who was Pythagoras. When he wasn’t busy figuring out the ratios embedded in right-angled triangles, he set his mind to music. He figured out that a metal bar, when struck, would produce a note. His next discovery was that if you have an identical bar, but half the length of the original you’d get a new, higher, note which somehow sounded pleasing when heard alongside the original note. This works whether you’re hitting a metal bar (like on a glockenspiel) or plucking a string on the guitar – halve the length, and the note goes higher in a pleasing manner. He had discovered the relationship that we call the “octave”. This term probably makes you think of the number eight in some way (octagon, octopus etc.) and we’ll come to that later.

So, Pythagoras now knew that a ratio of 2:1 produced a pleasing combination of notes. Were there any other ratios which could produce a harmonious sound? Well, the story goes that one day as he was walking through the village he heard a pair of blacksmiths striking their anvils in a way that pleased him sonically. He somehow managed to persuade the two gents to let him weigh the anvils and by doing so he discovered a new ratio capable of producing a pleasant combination of notes. It turns out that anvil “A” was two-thirds the weight of anvil “B”. It is these two ratios – 2:1 and 3:2 which are the basis of all music.

Let’s start to put some meat on the bones of this theory. Take a note, let’s say the open 5th string on a standard tuned guitar. This is an A note, right? The actual frequency of this note is 110Hz, meaning that the sound it produces is actually a vibration in the air which is hitting your eardrum 110 times per second (click here to hear it as a “pure” sine wave). If we were to halve the length of the string, by playing it at the 12th fret, the new (higher) note you hear is double the original frequency – 220Hz. Because these two notes share such a fundamental relationship, we give them the same name, in this case “A”. As you can see by looking at the neck of the guitar, the distance between these two notes is divided into 12 steps, represented on the guitar as frets. How do we generate the notes found on the frets which lie in between our two A notes then?

Well it all comes down to that other ratio Pythagoras discovered, 3:2. If we take our original A note of 110Hz and multiply it by two-thirds, we get a frequency of 73.3Hz. Obviously, this is lower in pitch than our starting note, the open A string and we’re trying to head “upwards” to the 12th fret so this isn’t any good. However, if we use the 1st ratio Pythagoras discovered (2:1) and double the frequency of the 73.3Hz note we get 146.6Hz which IS inside the range of 110 to 220Hz defined by our two A notes (open string – 12th fret).

We can now multiply this new note by two-thirds to get yet another note, then do the same thing with whatever note that gives us to get another. If we keep on doing this, and remembering to use the 2:1 ratio to either double or halve the frequency of each new note to ensure it lies somewhere between our 110-220Hz open string/12th fret parameters, we get a progression of 12 notes, which if we put them in ascending order will sound like this (click here).

We now have the 12 notes which form the basis of our modern system of music. However, things aren’t quite that simple. If we take a look at the actual frequencies of the notes we’ve got, there is a problem. See if you can spot it…

 

Starting Note (open A string)

110Hz

1st step

115.88Hz

2nd step

122.08Hz

3rd step

130.37Hz

4th step

137.34Hz

5th step

146.67Hz

6th step

154.51Hz

7th step

162.78Hz

8th step

173.83Hz

9th step

183.13Hz

10th step

195.56Hz

11th step

206.02Hz

12th step

217.04Hz

 

See the problem? Going up 12 steps should take us to 220Hz, but it doesn’t. Twelve steps up the scale actually gives us a note with a frequency of around 217Hz. Does this matter? Well, actually, yes… here’s what 220Hz sounds like (click here), and here’s what 217.04 sounds like (click here).

They are noticeably different aren’t they? One sounds like a slightly out of tune version of the other. Does this mean that Pythagoras got it wrong? No, in a word. The ratio of 3:2 which his calculations are based upon is hard wired into the DNA of music – he didn’t invent it; he just discovered it. We can demonstrate this by playing some notes on the guitar…

Play an open top E (1st) string and (using as accurate a tuner as you can find) get it perfectly in tune, then do the same thing with the 5th (A) string. Now, play the harmonic found on the 5th string at the 7th fret (in theory, the same E note as the open 1st string) and measure this with your tuner. You should see, provided your tuner is accurate enough, that the harmonic (the note which is actually put there by nature due to the physics of how the string – or whatever is producing a note on any given instrument – vibrates) will read flat compared with the same note on the open top string. This discrepancy between the true “Pythagorean” note and the note we perceive as being “in tune” is known as the “Pythagorean Comma”. How did musicians & composers deal with this?

Well, to put it simply, they avoided it for as long as possible. If you just take the first five steps in that 3:2 progression of notes you get a 5-note (or “pentatonic”) scale. By only using the 1st five notes in the sequence, you avoid the comma altogether. This is why most folk music traditions across the world use the pentatonic scale as their basis: from Celtic folk jigs & reels (which would cross the Atlantic to become country & western and bluegrass) through to the indigenous musical traditions of Asia and Africa (the birthplace of what would become the blues), the pentatonic scale is the foundation for it all.

What about more sophisticated music which needs more than 5 notes though? Well, you can actually use the 1st seven steps of the 3:2 progression, which will give (if you include the note twice the frequency of the starting note) an eight note scale or “octave” – remember that from earlier? Doing this doesn’t give too many problems as you can fine tune the instrument in question to smooth out any obvious signs of the dreaded Pythagorean Comma. The problem really comes when you want to write music which needs more than eight notes… music which changes key…

Imagine the scenario… you’ve got an idea for a tune which starts in the key of (for example) A, so you tune your instrument so that it gives a pleasing sound in that key. The tune you’re hearing in your mind now wants to go to the key of D, which contains many of the same notes as the key of A, but using Pythagoras’ calculations, an E note in the key of A will be noticeably different from the E note in the key of D. Composers got round this by only writing music which moved from key to key for instruments which could be adjusted on the fly – the human voice, for example, or other instruments which didn’t have fixed “barriers” between notes: the violin springs to mind. Still, it was a problem and believe it or not, no practical solution was found to it from the time of Pythagoras (c.500BC) until the arrival on the scene of (arguably) the greatest musical genius ever to have lived… Johann Sebastian Bach.

It was Bach who figured out a method of practically tuning an instrument, based on a new ratio (the 12th root of 2), which ironed out all of the harsh dissonances caused by the Pythagorean Comma. In fairness, he didn’t invent “Equal Temperament” (as this method of tuning became known) – it had been around as a theory for quite some time. He was, however, the first person to demonstrate it could be made to work by successfully tuning an instrument using this system.

In 1722 he published a series of preludes and fugues, known as the “Well Tempered Clavier” which moved between all 12 major and minor keys without any need for re-tuning the instrument, something which had been impossible prior to this point. It was the musical equivalent of the apple falling on Newton’s head, or Einstein’s General Theory of Relativity, Thomas Newcomen’s steam engine, or the invention of the computer – it really is that significant. Every time you use a capo to move a song into a new key, the fact that you can do so without having to completely retune your guitar is thanks to Bach. Every time you learn a shape for a scale or barre chord which can be moved into any key you like, simply by shifting it up/down the frets, it’s thanks to Bach and his practical demonstration of Equal Temperament.

Also, Equal Temperament has affected the way we hear music. Have you ever found yourself listening to some form of non-western indigenous music (in a movie soundtrack or on holiday somewhere exotic, for example) and thought that it sounds somehow “out of tune” a little bit? This is because not all cultures across the world adopted Equal Temperament – some musical traditions still rely on the old Pythagorean system of note calculations. This is because the music in these cultures doesn’t need to change key all the time so there was no need for them to adopt the system which allows this to happen. Our modern western ears have been recalibrated to a new equally tempered standard which would probably sound odd & a little “off” to anyone arriving in a time machine from ancient Greece, or any other culture not accustomed to it. Some would (and indeed do) say that our modern system of equally tempered music is an artificial construct… an interloper which has corrupted our ears and given us a false sense of what “in tune” actually means. Never the less, Equal Temperament, for all it’s corruptions of the true Pythagorean “natural” system of tuning, made possible this piece of music & for that, I am eternally grateful…

 

So, there you have it… a whistle-stop tour of the past two thousand or so years of how our music came to be. Now… Any questions?

 

Until next time… Have fun!

John.

 

John Robson Guitar Tuition

The John Robson Jazz Project

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