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...

Harmonics

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