Polyhedral configuration of logical operations and correspondence to transformations within the Tonnetz?

Connecting the Multiple Voices of the Pattern that Connects (Part #7)

The set of 16 Boolean logical connectives is now fundamental to the binary operation of computers -- and therefore to artificial intelligence. However, although "logical", few would be able to cite all those operations, or offer examples of all their roles in discourse. The operations are also fundamental to analysis of the fundamental nature of the "opposition" which has become the defining characteristic of global dynamics. Naively it may be asked why no attempt is made to represent the challenges of logical opposition -- however irrational and divisive it may be held to be -- in terms of the visible tranformations encoded by polyhedral operations or the audible transpositions of musical key.

The potential relevance of the logic of opposition is the subject of separate discussion from a variety of perspectives:

Especially noteworthy is how "opposition" relates to the issue of negation -- given the manner in which denial is now a major feature of global strategic discourse. The possibility of depicting this otherwise is presented by Fabien Schang and Amirouche Moktefi (Depicting Negation in Diagrammatic Logic: legacy and prospects, 2008) in the following terms:

Here are considered the conditions under which the method of diagrams is liable to include non-classical logics, among which the spatial representation of non-bivalent negation. This will be done with two intended purposes, namely: a review of the main concepts involved in the definition of logical negation; an explanation of the epistemological obstacles against the introduction of non-classical negations within diagrammatic logic

A key polyhedron used to map the 16 Boolean logical connectives in that approach is the rhombic dodecahedron of 14 vertices (namely 16-2) with its 12 faces. The distorted mapping from 16 to 14 can be discussed as a "fudge" to avoid the challenges of 4D comprehension, as discussed separately (Governance beyond the logical focus on true vs false? 2019; Questionable confusion in configuring strategic frameworks: "fudging" self-reflexivity? 2019).

Hasse diagram or Tesseract	The Logic Alphabet Tesseract - a four-dimensional cube (see coding). by Shea Zellweger	Topologically faithful 4-statement Venn diagram is the graph of edges of a 4-dimensional cube as described by Tony Phillips	Organization of contingent bitstrings on a rhombic dodecahedron

	Diagram by Warren Tschantz (reproduced from the Institute of Figuring) .	A vertex is labeled by its coordinates (0 or 1) in the A, B, C and D directions; the 4-cube is drawn as projected into 3-space; edges going off in the 4th dimension are shown in green.	Adapted from Lorenz Demey and Hans Smessaert (2017)

A major challenge to exploration of potential corrspondences and equivalences has been highlighted by Daniel Costa JÃºnior (Intersemiotic relations through the bias of semi-symbolism and oppositional geometry : the nocturnal inspiration, Studies in Visual Arts and Communication, 4, 2017, 2):

As stated by ClÃüver (1997, p. 54), regarding interart studies, "[...] correspondence issues remain tangled." How does a stroke in a painting correspond to a melodic line? Or how does color correspond to timbre? If it was not for the sign mediation of some titles and descriptions, many intertextualities would not be perceived by inattentive onlookers. Nonverbal text titles and graphic symbols are means of intellectual appropriation, a way of providing intelligibility to that which is sensitive... One of the assumptions of the present study is that semi-symbolism provides binary relations similar to the genesis of the semiotic square. Therefore, at least one similarity relation between the planes 18 of content and expression must be enumerable between two inter- relatable texts, as in the case of interart pieces. The square was inspired by logic, although some logicists do not accept the semiotic version. In turn, Greimas & CourtÃ©s (1983) also had reservations regarding the use of the square in Logic 19 . One of the justifications for this discussion is that the field of Logic prefers universal truths, while the field of Semiotics prefers veridiction, with "truths" that are particular to a certain text. Several semioticians have used squares that have been amplified to hexagons.

Salience of the interart relation
Legend: (Lt) light, (Dk) dark, (Tp) plain tone, (Bm) boom, (Sl) silence, (A) audible, (C) continuous, (D) discontinuous, (G) generality, (P) particularity, (Â*) denial, (**) conjunction, (**) disjunction

Reproduced from Daniel Costa JÃºnior, Intersemiotic relations through the bias of semi-symbolism and oppositional geometry (2017)

Question to ChatGPT: With respect to the Conway notation, the 3 basic operations are the core elements in a more extended set of 13 (denoted by the letters abdegjkmoprst) -- further extend by 18 Wythoff-style constructions. It would then be of interest whether the more general sets had their correspondences in both Tonnetz transformations and logical operations

Exploring the extended set of 13 operations in Conway's notation (abdegjkmoprst) and the 18 Wythoff-style constructions for polyhedra could indeed find fascinating parallels in the more complex transformations within the Neo-Riemannian Tonnetz and the realm of Boolean logical operations. Such an investigation would likely reveal deeper systemic correspondences, potentially illuminating underlying universal principles that govern transformations across domains. This exploration could enrich our understanding of the interconnectedness of music theory, geometry, and logic, showcasing the versatility and depth of these transformational systems.

Question to ChatGPT: Do you have any trace of a set of more extended transformations within the Tonnetz or any corresponding transformations between elements of the set of Boolean connectives

Within the Neo-Riemannian Tonnetz, the three basic transformations (P, L, R) could potentially be part of a more extensive set if one considers further musico-theoretical extensions or analogues, such as those involving modulation, inversion, or more complex harmonic relationships. Similarly, the Boolean logical operations, while fundamentally defined by a set of operators (AND, OR, NOT), can be extended through combinations and permutations of these basic operations to express more complex logical conditions. The exploration of extended transformations in both the Tonnetz and Boolean logic would delve into a more nuanced understanding of their respective systems, potentially uncovering deeper structural parallels and correspondences.

Question to ChatGPT: From a general systems perspective again, it is intriguing to note the mappings of 14 Boolean logical connectives onto particular polyhedra, notably the rhombic dodecahedron, as studied by oppositional logic. With respect to musical transformation (as in forms of counterpoint), these suggest articulation in terms of both the Tonnetz and oppositional logic

Mapping the 14 Boolean logical connectives onto polyhedra, like the rhombic dodecahedron, as explored in oppositional logic, opens fascinating parallels with musical transformation and counterpoint. This connection suggests a shared foundational structure between logic, geometry, and music, where opposition and transformation play key roles. Viewing musical transformations through the lens of oppositional logic and geometric mappings could offer new insights into the structural underpinnings of musical harmony and counterpoint, aligning with general systems theory's emphasis on cross-domain analogies and the universal principles underlying complex systems.

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Polyhedral configuration of logical operations and correspondence to transformations within the Tonnetz?