One thing that interests me is the phenomenon of a tonal center. The fact that, when listening to music, we all have a strong sense of key. In other words; we know when we are “home” or “away” when listening to a tune or a song. We feel safe and the song feels complete (and predictable) when it stops on the tonic. I.e. the root of the scale, or key. We can create surprising and sometimes funny effects when going in another direction than the expected note or chord. We all have the ability to decide which key is being the center or root of a song. We can do this from just a few notes in a melody. One example is the second half of “Twinkle, twinkle, little star”, where the melody goes; G, F, E, D. From this we can deduct that the key is C.
This is a very universal and strong force. Scientists claim that we are born with this ability.
Why does this matter? Well, we all like to have a sense of bearing when listening to music. We don’t like it when we are lost. Also, with this strong sense of tonal gravity, we are amused when the music takes an unexpected turn. This keeps our brains entertained. If the music is too predictable, we get bored. If the music is too unpredictable and totally lacking any “flirting” with the tonal center, we get lost. Finding the right balance between the predictable and the surprising is a never ending quest for any composer, songwriter and music listener. The “right formula” is, of course, a matter of personal taste and depending on the context in which the music will be used.
The standard way today, of depicting the relationship between notes and chords is either by a circle of fifths or ascending scales with corresponding chords. (see pictures below)
The Swiss mathematician Leonard Euler made one of the earliest attempts of mapping the relationship between notes with pure intervals. His map is from 1739 and is organized in columns of perfect fifths and rows of major thirds. This gives a two dimensional map of the notes and their relationship. It is interesting to see that the major triads of a key is represented by a L-shape in the map and that the key itself creates a bigger, sort of L-shape in the map (see picture below).
In the picture, the triad box is marking the triad of a C major chord, that consists of the notes C, G and E. If we would move the triad box one column to the left, it would highlight a F major chord that consists of the notes F, A and C. You may have noticed that within each “scale box” (the fat L-shape), the tonic, in this case C, has the shortest average distance from all other notes. British mathematician Christopher Longuet-Higgins suggested that this may be one of the reasons for our strong sense of percieved tonal gravity.
I think that Euler’s map gives a picture of a wide open musical landscape opposed to the more closed circle of fifths. It truthfully displays that the note G is closer or more related to C than C#. It also gives a good graphic representation of modulation between different keys. It shows us that the walls between different keys aren’t that thick. If you move the triad box in the picture above, one row up, you’ll get an E major chord (the notes E, G# and B). E major is an alien chord in the key of C, but it still has two of it’s notes (E and B) within the boundries of the key of C. Thus, the chord E is related to the key of C. Not in the nearest family, but maybe like a cousin? An example of this can be heard in the song “Sitting On The Dock Of The Bay”. If it was to be played in the key of C, the beginning of the chord progression would be: C, E, F. This sounds perfectly natural, even though it is not entirely in the key of C. A pleasant balance between the unexpected and predictable.
Euler’s map is a beautiful and fluent graphic representation of the relationship between notes, chords and keys. Now, let’s make some music!