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Tuning and temperaments

I'm currently snowed out of my workshop amidst unseasonably weird UK weather, so it's a great opportunity for me to put fingers to keyboard and share my thoughts on guitar tuning. I think this is a really fascinating topic because it is the bridge between the world of music (essentially a human invention) and laws of physics as found in nature. I'm going to use the guitar as my instrument of choice to explain these concepts. I hope you find it interesting.

Music today

Let's start with the typical Western tuning system in use today. This is called 'Equal Temperament'. What this means is that the interval between every fret on a string is the same. It's called a semitone. Musical instrument makers measure this gap in cents. In equal temperament one semitone is equal to 100 cents. You can think of this in a similar way as millimetres and centimetres, as a good analogy.

So the cents values of every interval along the 'E' string would be as follows:

The reason this system has been developed is entirely to do with the cultural development of western music. As musicians have written increasingly complex music that moves through different sets of keys, it has become necessary to have a system that plays equally in tune for every different key. But it wasn't always like this...


Lets go back to the fundamental stuff. Imagine a string tied at both ends (like the open string of a guitar):

Now imagine plucking that string, and visualising how it will vibrate:

Fundamentally, the string wobbles up and down. A good analogy is the idea of a skipping rope. The rate at which it does this is called it's frequency. This is the 'note' that you hear on an open string of the guitar. It's determined by all kinds of things, such as it's mass, materials and tension.

Next, let's imagine plucking the string but with the finger touching the string 1/2 of the way along.*

See that the string can't wobble up and down in the middle of the string anymore (because your finger is held there).

Now, try 1/3 of the way along**:

And a 1/4 of the way***:

If we were to tune our string to an 'E' note, like the high string on a guitar, the notes that we would hear in each of these instances would be approximately as follows: E, E (an octave higher), B(an octave and a fifth higher), and E (another octave higher) again. We can keep going in fractional divisions to get even more notes. This is called the harmonic series.

The (first) important bit!

The thing to point out about the harmonic series is that the waveforms are perfect, beautiful multiples of one another, which means when played together they are PERFECTLY harmonious. I don't mean that in any kind of subjective way, it's the physics working at it's most sublime and simple. Here are some of the waveforms overlapped and you might be able to see what I mean:

It's creates a visually symmetrical image, which basically means it will sound harmonious (check out this image for a much better representation, right up to 32 harmonics!). Also check out Mal Webb's incredible vocal representation of this (and buy his amazing music at the same time!).

The (second) important bit!

We used the harmonic series to find a new note (the 'B' note - or second harmonic - from the E string). This note is a 'perfect 5th' interval. We can then tune a second string to the 'B', and find another new note using the same second harmonic. This will result in an F#. We can then tune another new string to F# and get another new note, and so on. This is a 'cycle of fifths'. It creates the notes as follows:

E --> B --> F# --> C# --> G# --> D# --> A# --> F --> C --> G --> D --> A --> E

We now have our full 12 notes. So what's the problem?

The catch

That 'perfect 5th' interval is not 700 cents we might have been expecting. Let's check out the cents values of all these notes and compare them to the equal temperament system. I'll save explaining the maths**** and compare them here:

You can see that our perfect 5th (the G note) is 1.9 cents different to 'equal temperament'. Crikey... so following this through all the notes that means the 'equal' temperament tuning system is almost 20% out of tune in some places. A minor 7th interval (C to A#) is waaaay out! The human ear is supposed to be able to pick out differences of more than five cents. Does that surprise you? I didn't quite see it coming when I first crunched these numbers!


A few parting thoughts...

Essentially, everything we hear is out of tune. The reason we don't really hear this is mostly put down to subtlety and cultural conditioning - we have all grown up with the sound of equal temperament and take it as given that it is how music should sound. I find this out-of-tune-ness quite thought provoking in a time when many guitarists are constantly asking questions about how their instrument can be made perfectly 'in tune'. It is in essence, impossible.

I tend to tune my guitar by ear, by listening to the 'beating effect' between different strings. Rather than fretting a string at the 5th fret, I can normally hear the perfect fifth once the waveforms overlap perfectly, and this is good enough for equal temperament (there is only about 2 cents difference). However, using the same system I find the B string always ends up much too flat. This is because an equal temperament major third (G to B) is much wider than the harmonic one. So if you struggle to tune your B string, this could be one of the reasons why. Interesting stuff!

Historically there have been many tuning systems which attempted to find the middle ground between the disharmony resulting in playing across several keys, and pure harmony. One example is Bach's 'Well-Tempered Klavier', which moves through all 24 major and minor keys, for solo keyboard (typically a harpsichord or clavichord). It made use of 'Well temperament' (or possibly equal temperament), instead of 'Meantone temperament' which had been in use at the time. Go check it out and get yourself lost in the world of tuning and temperament!

Thanks for reading! Please do drop your comments (and corrections) below.

* Most of you reading this will be guitarists, and you'll likely be familiar with playing harmonics at the 12th fret. This is the same thing.

** The second harmonic can be found somewhere above the 7th fret.

*** The third harmonic can be found above the 5th fret.

**** Well, no I won't! You need the ratio of the note (eg the first harmonic is 1.5). Then find the logarithm of the note, and multiply it by 3986.

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