Geometry in music?? Part 3

More about geometry in music

Music finds one of its main pillars compositional, present in almost any piece of music, which gives rise to work in the classroom with the geometric concepts from the musical perspective:

 The analysis of the musical structures of the scores develops the analytical capacity geometry and its transformations.

 Dance helps to consolidate geometric concepts and helps to calculate areas and perimeters. 

 Musical instruments are a tool in the understanding and real application of calculations of areas and volumes.

 The concepts of frequency and waves, together with the geometry and design of instruments, develop the ability to relate different mathematical-physical concepts with the real manipulation through the construction of instruments (technology subject).

 Computer-composed music works on the concept of fractals and algorithms.

Musical mathematics draws heavily on the branch of geometry, a veryused in musical compositions. The combination of symmetry and asymmetry is a principle musical, like repetition, widely used.

Within this field we can find geometric parameters such as repetition, translation, reflection (notes, chords, time, intensity), palindromes, rotation, dilation, change of scale, permutation... These geometric transformations, musically, preserve the intervals or their proportions, and can occur both in the notes, as in the tempo, in the intensity of sound, in sentences or between different voices. Any movement like translation, rotation, or reflection are interval-preserving transformations, while any dilation of enlargement or reduction preserves its proportion.

An example of symmetry is the palindrome, a resource that we can also locate in some pieces like El palíndromo (Symphony No. 47 by Haydn) or in a more modern Spanish work, Flamenco palindromy, by Antonio Rueda Peco.

We can see several examples such as the 'Mirror Duet', by Mozart, composed in such a way that it can be performed by two musicians, each reading it in one direction, one beinning with the first measure and another for the last one. The harmony is maintained throughout the piece because it has been used a geometric resource, the rotation of intervals. Each geometric transformation is associated with its musical equivalent. For example, a horizontal reflection in musical terms would be a inversion, or a vertical translation corresponds to transport.

The use of geometric transformations is a widely used resource that appears frequently in any document that relates music and mathematics. For this reason it has not been considered necessary to influence this, and we will treat musical geometry from other points of view.