Systematic Musicology

In the field of systematic musicology, Pareyon’s book On Musical Self-Similarity (Helsinki, 2011)^[9] predicts the role of analogy as one of the capital issues for future musicology and cognitive science, foreseeing conclusions of Hofstadter & Sander's Surfaces and Essences (2013). According to Curtis Roads (2015:316), On Musical Self-Similarity "is an intriguing treatise in which repetition is generalized to several modes of self-similarity that are ubiquitous in musical discourse.".^[10] The book is frequently referenced in monographs, journals and dissertations, mainly in the fields of representation of temporal groups and semigroups, machine learning and human-machine hybrid composition, non-linear cognitive studies of musical processes, neural dynamic programming, and self-repetition algorithmic modelling.

Textiles understood as Musical Patterns

Grandson of a textile worker from La Experiencia (Zapopan, Jalisco),^[11] Pareyon’s article “Traditional patterns and textures as values for meaningful automatization in music”, published in Finland, in 2010,^[12] is a seminal work proposing that textiles and traditional fabrics, generalized as frieze group patterns, may be and indeed are instructive as musical contents. This idea inspired a PhD dissertation from Durham University (2016),^[13] and contributed to a framework for systematizing the catalog of harmonic styles developed as an interactive inmersion by the University of Science and Technology of China USTC (Huawei Technologies Co Ltd).^[14] A clarifying chapter in these terms appears entitled “A matter of complementarity” within On Musical Self-Similarity, pages 458-461.

Music as an Ecology

Another pioneering writing is “The Ecologic Foundations of Stylistics in Music and in Language”, published by the Aristotle University and the University of Edinburgh, in 2009. There, the conclusions lead to conceiving culture as an intersection between the semiosphere and the ecological niche’s complexity:

“In the wake of considering style as a result of dynamic relations in music and in language, it might be questioned whether its cycles are involved into greater systems of biological complexity. (...) This would explain at least general aspects of attraction, repulsion and bifurcation in musical constraints, preferences and correctness rules.”

— G. Pareyon^[15]

The latter cannot be disengaged from the political and social dimensions of music, as Pareyon states at the end of another paper, “How Music Can Signify Politics in the Postmodern Era” (Helsinki, 2011):

“Musical nationalism, and postmodern corporatism equally play a role in colonisation and ecological deterioration, making trivial what was sacred or traditional in its original context. In musicology, a big question arises: can composers such as Chávez, Bartók, Sibelius, or Villa-Lobos, be judged as conquerors of local traditions, for the sake of the expansion of Classical music? Or are they rather cathartic agents of musical synthesis, attempting to save diversity in spite of an unavoidable, progressive unitarism as a process of cultural self-transformation?”

— G. Pareyon^[16]

Finally, this idea of diversity of music is developed in a later book, Resonancias del abismo como nación (in Spanish, 2021), as follows (page 372):

“Faced with the disaster of monoculture, the metaphor of bioecological disaster is concomitant: the rapid increase in the rate of destruction of languages, traditions and traditional ways of life, quickly replaced by a single way of existence, usually called “progress” or “development”, and inspired by the North American model of egotistical and irrational consumption of resources, in the midst of the cultural desolation that derives from the reproduction of this system, leads humanity towards a form of generalized poverty never seen before.”

— G. Pareyon^[17]

Polar semiotics

Probably Pareyon’s most important contribution, both to semiotics and musicology, is his construction of Polar semiotics (also Polar semiology) within the mathematical domain of Category theory.^[18] Thomas Sebeok’ famous statement the sign is bifacial (1976:117; with noticeable antecedent in Peircean semiotics) remained obscure in the context of interdisciplinary studies, until Pareyon’s formal generalization, in a fashion that makes possible harmonizing cultural semiotics within a range of Group theory. This theorization has an impact on the methods for social history, as a bond between the abstract and the socially real and the pathos, since, as Pareyon concludes:

“The more important a concept does appear in the human mind and in social history, the more heterogeneousness of thoughts around it. This, which is a problem for classical philosophy, is the greatest virtue for the concepts of philosophy behind polar semiotics. Yoneda lemma facilitates this recognition for philosophy in general, and for musicology in particular.”

— G. Pareyon^[19]

Intersemiotic continuum

Pareyon’s theorization on the intersemiotic continuum is an elaboration over Lotman (1984: 5-6) semiosphere and Sebeok semiotic continuum (${\displaystyle SC}$). The latter expressed that “no semiotic system can exist or function unless it is ‘immersed in the semiotic continuum’—which is what Lotman terms the semiosphere”. However, the ${\displaystyle SC}$ concept emphasizes the fact that there is no any gap along or across the sign network and its interpretant (of any sign). This is deeply related to the semiotic quiddity aliquid stat pro aliquo, conventionally translated and adapted to the terms: “[A sign is] everything that stands for something else”. Furthermore, Kotov and Kull (2011:183) specifies that (The) “semiosphere can be described as a semiotic continuum, a heterogeneous yet bounded space that is in constant interaction with other similar structures.”.^[20] Congruently, the intersemiotic continuum theory (${\displaystyle IC}$ theory), introduced in chapter 3.8.1. of Pareyon’s On Musical Self-Similarity, expands this notion to the principle that “there is no any gap along or across the semiotic dimensions and its interpretants”. Subsumed within the field of formal categories, this theorization adopted the rule of satisfying the Snake lemma. Subsequently, this theorization strengthened the complementary concepts of intersemiotic synecdoche and polar semiotics.^[21] Within the first years after the publication of these concepts in On Musical Self-Similarity (2011), the ${\displaystyle IC}$ theory was extended to several scientific disciplines, mainly in Eastern Europe and Russia.^[22]

Intersemiotic synecdoche

The classical concept of synecdoche, in which a term for a part of something is used to refer to the whole, or vice versa, here is embedded into a multidimensional semiotic depth. Thus, whether “classical synecdoche” dwells within rhetorics and speech theoretical contexts, the intersemiotic synecdoche is the analogous operation, transversal to ${\displaystyle n}$ number of semiotic dimensions. It is, also and necessarily, a subgroup of the intersemiotic continuum wholeness. Among other features, this framework expands the approach to abstract synesthesia in different conceptual domains, for instance, connecting partial codes or signs to complete codes or sign systems of potentially infinite semiotic varieties. A first order example would be as follows: let ${\displaystyle color}$_{${\displaystyle k}$} be part of pitch ${\displaystyle p}$_{${\displaystyle j}$} which in turn makes part of a chord ${\displaystyle ch}$_{${\displaystyle i}$} existent with specific timbre ${\displaystyle Fs}$_ω (i.e. Fourier spectrum) that represents specific combinatorics for a Dirichlet L-function, ${\displaystyle L}$. Thus, sumarizing:

${\displaystyle L(Fs)={\frac {ch(log)}{p(log)}}}$ ∝ ${\displaystyle color_{k}.}$

Although merely substituting a symbol by another symbol or a code by another parallel code obviously results trivial, when embedding this sort of relations as connected morphisms (see: Category theory), semiotics can be understood as the realm of signs, symbols and associated operations, characterizable as the ‘visible display’ (i.e. perception of the signs and signic processes: the ‘color’ in the previous example), in contrast with its transversal constraints (‘invisible’ or hidden to the senses). Nevertheless, both perceptible and imperceptible plots of signs integrate the same intersemiotic continuum (being ‘explicit’ the pars pro toto, and ‘implicit’ the toto pro pars).^[23]