Symmetry And Cartesian Plane

Symmetry and transformations have been used for hundreds of year by various composers to create different styles of music. This method of making music is done on a musical plane and its dimensions are pitch space and time. Symmetry and transformations can be seen on both a smaller scale, when a measure is repeated upside down in the next measure and a larger scale where entire pieces are played forward and backward simultaneously in a crab canon or in a table canon where the notes are upside down as well as backwards. Other examples of symmetry and transformations is when notes are repeated, transposed, and in retrograde. Often times these transformations are used subconsciously but awareness of these symmetries gives composers control and
However the musical plane has an important difference from the Cartesian plane and that is that the two dimensions are not the same. Time is seen as the equivalent of the x-axis on the musical scale and pitch space is seen as the equivalent of the y-axis. Therefore, the musical plane therefore has less symmetry than the Cartesian plane. This is because the musical plane doesn’t have symmetry at any line but only has horizontal and vertical axes of symmetry. In the Cartesian plane there are three types of transformations: translations, reflections, and rotations. In the musical plane, all transformations treat two dimensions separately which leaves repetition (a horizontal translation), transposition (vertical translation), retrograde (horizontal reflection), inversion (vertical reflection), and combinations of these as the methods of transformation. A retrograde inversion is an example of a combination where notes are rotated 180°. Repetition and transposition can also be combined to repeat a measure in a different key. And another combination is the glide reflection which is a translation and reflection. These translations are done relative to
The hop is translational symmetry or in musical terms, repetition. Repeated parts of a song such as bass lines, choruses, etc. are so common that hops almost always apparent in all pieces of music. The step is a glide reflection which is a pattern and its inversion. Jump is basically a hop with a horizontal reflection which is when a part of the song and its inversion are played simultaneously and repeated. Jumps are much less common and are difficult to find in music. The sidle contains two vertical reflections which are made by alternating a section with its retrograde. The sidle is commonly found in compositions with a piano accompaniment. The dizzy hop also known as the spin hop has two points of 180° rotation. So in music that means the piece alternates a section with retrograde inversion. The dizzy sidle consists of one vertical reflection and one point of 180° rotation. It contains all four possible transformations of music which can be repeated indefinitely. The dizzy jump also has all possible symmetries and it uses one vertical and one horizontal reflection to create a point of rotation where they intersect. In order to create this transformation, the part of the song being transformed and its inversion are played simultaneously along with retrograde and retrograde inversion which must be played simultaneously. And similar to jumps this is less common unless the horizontal