On Musical Self-Similarity : Intersemiosis as Synecdoche and Analogy

Self-similarity, a concept taken from mathematics, is gradually becoming a keyword in musicology. Although a polysemic term, self-similarity often refers to the multi-scalar feature repetition in a set of relationships, and it is commonly valued as an indication for musical coherence and consistency . This investigation provides a theory of musical meaning formation in the context of intersemiosis, that is, the translation of meaning from one cognitive domain to another cognitive domain (e.g. from mathematics to music, or to speech or graphic forms). From this perspective, the degree of coherence of a musical system relies on a synecdochic intersemiosis: a system of related signs within other comparable and correlated systems. This research analyzes the modalities of such correlations, exploring their general and particular traits, and their operational bounds. Looking forward in this direction, the notion of analogy is used as a rich concept through its two definitions quoted by the Classical literature: proportion and paradigm, enormously valuable in establishing measurement, likeness and affinity criteria. Using quantitative qualitative methods, evidence is presented to justify a parallel study of different modalities of musical self-similarity. For this purpose, original arguments by Benoît B. Mandelbrot are revised, alongside a systematic critique of the literature on the subject. Furthermore, connecting Charles S. Peirce s synechism with Mandelbrot s fractality is one of the main developments of the present study. This study provides elements for explaining Bolognesi s (1983) conjecture, that states that the most primitive, intuitive and basic musical device is self-reference, extending its functions and operations to self-similar surfaces. In this sense, this research suggests that, with various modalities of self-similarity, synecdochic intersemiosis acts as system of systems in coordination with greater or lesser development of structural consistency, and with a greater or lesser contextual dependence.

'Self-similarity' an abstract concept borrowed from mathematics is gradually becoming a keyword in musicology. This is due to the fact that self-similarity often refers to the multi-feature repetition observed in a set of relationships, and it is commonly valued as an indication for musical coherence and consistency . In their own way, coherence and consistency are notions closely related to typical features of music, independently of any tradition or historical period: symmetry, similarity, and repetition. Most, or almost all aspects of music, including relationships of harmony, metre, rhythm, melody, timbre, and texture, depend on criteria of symmetry; and comparison between these relationships, essential for forming musical sense, necessarily demands for criteria of similarity and contrast also refered to as dissimilarity. Accordingly, it is quite hard finding a musical example avoiding any kind of repetition, because somehow music is a repetition of sounds in time and space, in agreement with the conventions of a certain tradition. Research on musical self-similarity usually provide rich information about essential aspects of symmetry, similarity, and repetition, crucial for establishing any kind of musical sense .