Topic 8 A tridimensionalview

We should point out the fact that, as we have already seen, the notes of a chord must be thought of as arranged vertically in the sense that, for example, starting from the first C under the staff and proceeding upwards, the notes are gradually of higher frequency and therefore higher pitch. After all, if you try to sing the C scale or play it on the piano (all white keys) you will hear that the pitch rises from lower to higher. Therefore, on the staff, we would find the C chord written like this:

But, as we already said about scales, the three notes of a chord could also, in theory, repeat themselves indefinitely (both above and below):

It is a sort of cyclic pattern.

Thus, for example, passaging from C to the E above it the sound rises (that is, it reaches a higher pitch) and it is for this reason that the scale is often represented as follows.

If you remember, the representation of the major scale is an irregular heptagon like the one in the figure below:

So, this is what we see if we represent the major scale in our chromatic circle while also showing the heights of the pitches:

As you can see the irregular heptagon that we had built before to represent the major scale actually has a three-dimensional spiral shape, because the series of notes could repeat indefinitely and therefore the rubber band would trace the same path but overlapping vertically, and that is what we call a spiral.

As shown in the figure, we start from C at the foot of the nail at 12 o’clock and then proceed clockwise to the other notes of the scale. Our rubber band will gradually rise in correspondence with the notes of the scale and in relation to their interval with respect to the following note. We have already found out what the interval formula of the major scale is: T-T-sT-T-T-T-sT. To follow this formula vertically too, notches representing the 12 semitones of the chromatic scale were arbitrarily marked on each nail.

Starting from the first C we will therefore rise by the pitch corresponding to the correct interval expressed by the interval formula of the scale.

Let’s summarize them:

  • C to D 2 sT
  • D to E 2 sT
  • from E to F 1 sT
  • from F to G 2 sT
  • from G to A 2 sT
  • from A to B 2 sT
  • from B to C 1 sT

Thanks to the chromatic circle we can therefore better represent the major scale pattern since we can highlight at the same time:

1) The scale’s name and pitch
2) The circularity of the scale
3) Its geometric representation

This aspect of spiral-shaped three-dimensionality actually applies to all the figures we have already seen, but since it would be impossible to represent their vertical development indefinitely (due to the fact that the notes repeat themselves cyclically), the simple projection on a plane is enough to evidence any existing geometric relationship.