Geometry in the world of sound
Hello my dear bloggers! We know that geometry is intrinsically linked to artistic creation in the visual arts, but what if it were also linked to music and sound?
Sound is produced when a body vibrates in a regular and periodic manner; if this vibration is not regular, it is considered noise. An author named William Hogarth argues in his book An Analysis of Beauty (1753) that beauty can only be found by studying nature, and he finds it in the natural curvature of the human body, defining the line he calls the serpentine as the basis of beauty. Hogarth seeks examples in nature, in art and in everyday objects, and this enables him to evaluate painting, sculpture and any other design according to their degree of conformity to the predominance of the serpentine curve. The author also considers that this principle governs dance and theatre, arts governed by movement. Important artists such as Michelangelo used the serpentine line in their works, and the musical wave of sound could be associated with the serpentine. (Trigueros, 2022)
Figure 1
Canon of Polykleitos
To make this connection, we need to explore the concept of the golden ratio in greater depth. The golden ratio is an irrational number, represented by the Greek letter ϕ. Studies have shown that the perception of beauty lies in the golden ratio. Anything that mathematically comes closest to the number ϕ will be perceived as more beautiful and perfect. In violins, the placement of the f-holes (the ‘ears’ or openings in the soundboard) relates to the golden ratio. It also appears in the relationships between height and width of the objects and figures depicted in the works of Michelangelo, Dürer and Leonardo da Vinci, amongst others. It is found in the formal structures of Mozart’s sonatas and even in Beethoven’s Fifth Symphony.
The relationship between geometry and music is fundamental and structural, based on numerical proportions, symmetry and spatial patterns. Since Pythagoras, it has been known that harmonic musical intervals correspond to simple geometric proportions in the length of the strings (1/2, 2/3). Music uses geometry to organise chords and scales in forms such as the circle of fifths or the ‘Tonnetz’ (tonal network). The circle of fifths is a geometric tool, a circle, which allows the 12 notes of Western music to be organised and the relationship between scales and keys to be understood.
Figure 2
Circle of fifths
Note: Recovered from https://flashradio.es/flashblog/que-es-el-codigo-de-quintas-y-como-se-usa/
Finally, I will discuss another key element in which we also find a connection between geometry and sound: cymatics. The ‘hidden geometry of sound’ studies how sound vibrations create tangible geometric shapes in materials such as sand or water, demonstrating that sound physically organises matter. This field shows that sound is not only heard, but also has a physical geometric structure capable of organising materials such as sand, water or dough (Arce, 2011).
Figure 3
Cymatics.
Note: Recovered from https://lamusainstrumentos.es/blog/cimatica-convertir-el-sonido-en-arte/?srsltid=AfmBOoo6OtlatBd0UAX-coSMfzCNouoNbLGly7Wt3-lQxWZHzgl4j1JS
References
Arce, M. (2011). Cimática: las formas e imágenes del sonido, una aproximación desde la experiencia personal. Bilbao, España. Recuperado de https://www. researchgate. net/publication/303547983_Cimatica_las_formas_e_imagenes_del_sonido_u.
Trigueros, J. A. (2022). Ritmos, arte y geometría. Pensamiento Matemático, 12(1), 5.
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