Topic 2 Transfer

In geometry, the translation of a figure is the geometric transformation which, when applied, makes a figure move in a plane without changing its shape and size. 

To translate a figure, it is necessary to fix a vector v of any magnitude, direction and orientation, and apply it to each point of the figure itself. 

Here is a graphic example:

The same thing also happens on the musical plane, with the only difference that the horizontal component of the vector (which indicates the horizontal translation, in time) must be positive. 

Let’s see a musical application with the well-known piece “Frère Jacques”: 

As you can see, the notes of the first two bars are repeated identical: it is as if we had dragged the notes of the first bar without altering their rhythm and melodic intervals. The same happens for the third and fourth, fifth and sixth and finally seventh and eighth. 

A composing technique widely used in music, with J. Sebastian Bach as one of its greatest examples, is the canon. A canon can be seen as a combination of translations. Let’s take up our Frére Jacques but with the addition of a second voice that reproduces exactly the same melody, but starting from the third bar: 

Starting from the third bar, the same melody generates a new harmonization due to this very same shift but, as you can hear, it makes sense musically. 

Up to now we have seen the simple horizontal translations and superimpositions of translations, which in music is called a canon. 

Let’s now take a look at vertical translation, which can greatly modify the melodic sense of a musical phrase. 

From a geometric point of view, when we translate a figure within a Cartesian plane, we need to indicate the coordinates of the translation vector v. This vector will accurately indicate the translation of the figure, both vertically and horizontally.

Try transposing the notes of the first bar at an interval of: 

  1. third 
  2. fourth 
  3. fifth 

In order for these exercises to be effective, it is necessary to be able to listen to them. Equip yourself with a music writing program that you can easily find online. A good free program is Musescore. At this point it will be easier and more effective to understand the concepts seen in this module.

At this point it will be possible to superimpose the translations both vertically and horizontally and we will have what in music is called canon on the second, third, fourth etc. 
let’s see an example of a canon on the fifth: 

In the second staff we start a melody that is generated from the original one but transposing it to its fifth, or in this case starting from G (fifth note starting from C). This new melody will proceed keeping the same interval ratios as the original melody. 

Also in this case the musical composition has been somehow manipulated from a mathematical/geometrical point of view. 

Given the horizontal and vertical translations, let’s now move on to the reflections.