The guitar is a problem-haunted instrument. Because of its construction it can never be 100% in tune. As I wrote in my previous blog, the frets are positioned in a compromised way as to make it as good as possible under bad circumstances. But to make things even worse, Western music itself is never in tune! To explain why Western music is the reason of this predicament, we have to take a closer look at basic harmony, music theory and some math.
Each note can be played several times in different octaves. An octave means, quite simply, a note with frequency x, only doubled. So, let’s say we have an A tuned at 440 hertz. The next A, one octave higher, is 880, and the A below that one is tuned at 220, and so on and so on. The term octave has been given to this trait; we humans hear that doubling of frequencies and it sounds to us ‘similar,’ only higher in pitch.
Basic harmony teaches us that there are some intervals (the distance between the root and any other note) that we humans find pleasant to hear. The most important interval besides the octave is called the perfect fifth. The perfect fifth occupies a specific frequency relative to the root in a specific ratio to the root frequency. This ratio is 3:2. The perfect fifth is the second resting point, as it were, in the scale. Thats why playing a power chord doesn’t feel ‘empty.’ Try playing a power chord but using the third instead of the fifth. You’ll notice it just isn’t as fulfilling is if you were to use the fifth.
So, the root, the octave and the fifth are kind of set in stone, frequency-wise. The fourth is also a major player in harmony. Copying our previous test only using the fourth in stead of the third, we see that this comes much closer as to sounding ‘right.’ The fourth also has a fixed frequency ratio to the root: 4:3. These two notes give us a ratio between two sequential notes and determine the ratio for all other notes in one octave. By some easy calculation, we discover that the ratio between each note is . The number 12 denotes the amount of sub-notes between each octave.
But that doesn’t answer the question as to why there are 12 notes and not 11 or 13.
Actually, it does. The ratio between 4:3 and 3:2 is exactly that ratio, and you can’t do anything but count up and down towards the root and the octave. Of course other systems have been tried. This system, equal temperament, is only an invention of the 17th century and has many flaws. The reason why the fourth and fifth are called perfect is because the actual note we play is very close to the theoretical ratio that was calculated. All other notes, though, are not perfect and hence, by default, out of tune. The true temperament system of tuning allows for a minute difference between the actual note and the theoretical note.
Other systems have been suggested. For instance, Johann Sebastian Bach was fond of the well-tempered system. This required a retuning of the instrument for each key. His ‘Wohltemperierte Klavier’ was thus not intended to show the benefits of equal temperament (since each episode of ‘das Wohltemperierte Klavier’ can be played in equal temperament without the need of retuning), but to show the characteristics in tone and feel of each key in major and minor scales.
Unfortunately, this retuning proved to be a major nuisance and was abandoned quickly. Unfortunately, there is no better system that gives us the flexibility of the 12-note system in true temperament. The system allows us, the player, to change through the keys as we please yet retaining the same relative issues regarding out of tune. Other systems were not so easy, and by playing this way for hundreds of years, our collective hearing grew accustom to some of the issues.
It’s funny to note, though, that some scientists tried to improve the system by expanding the amount of notes that can be played in an octave. The Dutch scientist Christiaan Huygens even got as far as using a system with 31 notes in one octave! Extremely unworkable, but an interesting solution to the problem.
So, even as our guitars have some issues regarding intonation, western music isn’t really helpful either in solving the problem. Maybe in fact, Western music was the cause of the problem: if Western music had 26 notes in a scale, or even 31, would the problem have sorted itself out? I don’t think so, to be honest. That means more points that need intonation and more frets to be hammered in a fretboard and more frets where the relative intonation with the strings can go off… I think our (musical) ancestors got it right by using 12 notes in a scale after all!