Topic 4 Reflection/Symmetry

In plane geometry, a reflection is an isometric transformation that “mirrors” all points with respect to a point or a line (called the center and axis of symmetry, respectively). 

Therefore, axial and central symmetry are reflections. 

AXIAL SYMMETRY 

Axial symmetry is a particular isometry that “alters” the image by inverting it, as if we were seeing it reflected in a mirror. 

In geometry, axial symmetry is defined as that transformation which, given a straight line in the plane, makes each point correspond to a second point on the opposite side with respect to the axis, such that they have the same distance from the straight line and both belong to a perpendicular to the axis. Here is an example of vertical axis symmetry. 

As you can clearly see, the figure to the left of straight line is perfectly reversed. 
Before moving on to analyze musical examples that exploit these techniques, let’s make some examples with real “word games”. 

If we are faced with a vertical axis symmetry of this type: 

We could almost say that a translation has taken place, since the two figures do not seem reversed. This is due to the fact that the figure to which we applied a vertical axis symmetry is a regular polygon. When this happens in the English language instead of geometry, it is called a palindrome. Do you know what a palindrome is? It is a word that remains the same even whet you read it backwards: Anna is, for example, a palindrome name; the same goes for the adjective “civic” or the noun “racecar”…or a typical expression ov your generation such as “LOL”. Let’s imagine to overturn this expression with a vertical symmetry with respect to line r and this will be the result: 

If we take the straight line r as the axis of symmetry, we obtain a palindrome melody. 

But if we take the word “war” and apply a vertical axis symmetry to it we will get this: 

War                  raw 

In other words, in overturning it we obtain another word with its own meaning, that is “raw”.

In the English language this game is called anagram: recombining the letters of a word to form another one with its own meaning. In this case, the anagram coincides with the reverse reading of the word “war”. 

In music, this procedure is more interesting from a creative point of view, because it generates a melody that is also very different from the original. It has a name: retrograde. The retrograde of a melody is the rewriting of that melody, but starting from the last note, retracing the sequence of notes backwards in the opposite direction. 

Let’s see an example: 

As you can see the two melodies are reversed but their musical sense is different just like the example of the anagram war = raw. 

If the axis of reflection is a vertical straight line we have said that in music we obtain a Retrogradation. 

If, on the other hand, we take a horizontal straight line as the axis of reflection, the result of the transformation will be an inversion: if the original melody rises by a semitone, the inverse will fall by a semitone. 

Let’s explain better by first visualizing what happens in geometry when we apply a symmetry of this type. Here is an example: 

As you can see, for example, if the AC side has a “descending” direction, its reflection A’C’ has an “ascending” direction; if the side CB has an “upward” direction its reflection C’B’ has a “downward” direction; and so on. 
This type of transformation does not apply to the wordplay seen above, neither to palindromes nor to anagrams, since the reading direction of the letters would also be reversed. 
In music, on the other hand, it is a very interesting technique which, like retrograde, generates new melodies. 
Let’s proceed with an example. Let’s take the following musical phrase as the original melody: 

Creating a horizontal axis symmetry means establishing first of all the position on the staff of this axis of symmetry. Let’s assume that it coincides with the second line of the staff and therefore with the note of G. In an horizontal reversal, the note of C will become D (under the first line) because if the axis of our reflection is the second line, then you must think to this: how many notes do I have to go down from C to get to G? Three: B, A, G. Therefore, we go down three notes from the G line (symmetry axis): F, E, D. 

The same procedure will apply to the other notes and therefore the result will be this: 

And, as you can hear, it makes some musical sense. The only note that does not change is the G note, because it lies on the axis of symmetry and therefore coincides with its own symmetrical note.