1 April 2024 | Draft
Connecting the Multiple Voices of the Pattern that Connects
Metaphorical comprehension of complexity enabled by graph theory and polyhedra
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Introduction
Discernment capacity in organizing complexity memorably
Potential implications of mapping tone space correspondences
Symmetry-preserving operations between polyhedra
Polyhedral transformations and their potential correspondence to Tonnetz transformations?
Comparison of canon styles with basic symmetry preserving operations
Polyhedral configuration of logical operations and correspondence to Tonnetz transformations?
Mapping transformations and the correspondences exemplified in 36-fold sets
Mapping correspondences of 36-fold sets of strategems and dramatic situations
Mapping correspondences of 36-fold sets of games and katas
Mapping correspondences of 36-fold multi-voiced canon and Song of Songs
Polyhedral edge-mapping of correspondences of 36-fold sets
Comparability of cognitively fundamental sets
Recognition of "conceptual chords" -- as triadic combinations
References
Introduction
This is a further development of Eliciting a Pattern that Connects with AI? (2024) -- written as an experimental exchange with ChatGPT in quest of memorable integrative configuration. The argument here follows discussion of aesthetic insights for an integrative pattern of relevance to the times (Reimagining the Canon to the Sounds of Cannon Fire: challenging hegemony through the pattern that connects, 2024; The Song of Songs as indicative of the Pattern that Connects: implications for strategic cultivation of global division? 2024).
The previous argument highlighted the potential of aesthetic insights into an integrative pattern of relevance to the times, highlighting the potential cognitive and strategic implications associated with patterns of tones in the light of an understanding of the numbers by which they could be distinguished and organized -- as traditionally recognized. In contrast with the aesthetic focus, the concern in what follows is with the insights that may then be drawn from mathematics, especially graph theory and geometry, and visual representation in polyhedral mappings. The exploration is framed by the cognitive perspective articulated by George Lakoff and Rafael Nuñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001).
As noted in the previous exploration with respect to any new form of "integrative perspective", and as a focus of continuing comment, that framing derives from Gregory Bateson in clarifying the nature of a meta-pattern in the following context:
The pattern which connects is a meta-pattern. It is a pattern of patterns. It is that meta-pattern which defines the vast generalization that, indeed, it is patterns which connect. (Mind and Nature: a necessary unity, 1979)
And it is from this perspective that Bateson warned: Break the pattern which connects the items of learning and you necessarily destroy all quality (1979, pp. 8-11).
The preceding papers endeavoured to highlight patterns of transformation characteristic of the multi-voice musical canon, and their relevance to comprehension of the complexity of the Biblical Song of Songs -- both being variously upheld as exemplifying a higher order of integrative insight with fundamental implications. Both could be understood as distinctive instances indicative of a pattern that connects -- expressed dynamically rather than statically. Beyond their visual form, the challenge is necessarily how they might be variously represented otherwise to enable comprehension of that pattern and engagement with it -- for which game-playing may offer futher insight.
The focus on "multiple voices", exemplified previously in discussion of the 36-voice canon of Johannes Ockeghem (Indications of connectivity in a 36-voice canon, 2024) contrasts with the current preoccupation with ensuring the consensus of a "single voice" in response to strategic challenges (Ashwini Ashokkumar, et al, Censoring Political Opposition Online: who does it and why, Journal of Experimental Social Psychology, 91, 2020, 104031; Jesse Smith, Climate-Con and the Media-Censorship Complex, OffGuardian, 1 April 2024). The current quest for consensus at all costs could be considered comparable historically with the efforts of religious institutions regarding consensus on their sacred dogmas -- exemplified in musical terms by the challenge of the tritone -- a harmonic and melodic dissonance (diabolus in musica).
The approach here endeavours to highlight the insights that might be derived from mathematics, especially graph theory and the geometry of polyhedra. This is partly justified by the cognitive arguments of the George Lakoff and Rafael Nuñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001). In seeking correspondences between seemingly disparate frameworks, the approach is inspired by that of general systems research.
A particular concern is the comprehensibility and memorability of strategic and other articulations, as variously discussed previously (Higher Dimensional Reframing of Unity and Memorable Identity, 2024; Pattern connectivity and memorability, 2024; Memorable organization of strategic complexity in 3D, 2024). A related concern is the "discernment capacity" potentially indicated by preferences for between 7-fold and 108-fold articulations, for example, whether individual or collective. Given the reigning culture of global division, there would seem to be a strong case for exploration of related issues, however conventionally speculative (Time for Provocative Mnemonic Aids to Systemic Connectivity? 2018).
As with the earlier presentations on this theme, the following argument makes extensive use of ChatGPT (version 4, or its Scholar.ai plugin) as an experimental "cognitive prosthetic" -- anticipating future assistance from AI as it is developed. The responses have been placed in shaded areas for clarity -- and to enable them to be readily skipped by the reader. However when the documents are presented as PDFs, this shading may be lost (as with hyperlinks). Relevant precautions and reservations were previously indicated, including concerns with so-called "hallucinations" and the reinforcement of confirmation bias through what could be described as leading questions.
As might be expected, the responses to any AI prompt evolve with the development of such applications -- readily assumed to increase in quality over time. However the reverse may appear to occur, as recognized in the less significant responses in this part, in contrast to those in the parts which preceded it. When challenged on this apparent shift, ChatGPT simply framed any difference as an adaptation to the changing nature of the prompts, rather than a change in policy in response to marketing opportunities or for other reasons -- as might be readily suspected. Such qualitative shifts are clearly difficult to evaluate. This suggests that repeating the same question may engender a response of lower quality.
There is widespread concern regarding the potential dangers of AI -- a collective concern which merits attention in its own right (George Lakoff, Women, Fire, and Dangerous Things: what categories reveal about the mind, 1987). Curiously there is little corresponding concern with the "dangers", "hallucinations", "bias" and "leading questions" which could be recognized as characteristic of conventional global governance and its summits.
Discernment capacity in organizing complexity memorably
"Octave": With respect to preferences for organizing complexity, the contrasti in recognition of patterns remain unexplained -- for those deemed credible, especially with respect to strategc articulations. In the light of the manner in which tones and music have featured in the earlier discussions, a mnemonic device could perhaps be used for degrees of discernment, namely the notion of a set of "octaves", as illustrated by the following table.
In music, an octave or perfect octave (sometimes called the diapason) is a series of eight notes occupying the interval between (and including) two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems". The interval between the first and second harmonics of the harmonic series is an octave. In Western music notation, notes separated by an octave (or multiple octaves) have the same name and are of the same pitch class.
The challenge in ordering complexity in the light of such an array is evident from the following:
- choice of a "lower octave" incorporating a limited number of elements results in each implying the greater number of elements which are not articulated -- emphasizing the role of interpretation (and potential disagreement)
- choice of a "higher octave" incorporating a larger number of elements ensures a greater number of more precisely articulated elements -- increasing the challenge of remembering such details and comprehending the organization of the pattern as a whole (readily ignored or simplistically framed)
In making any such choice the question is the capacity to discern distinctions, rather than conflating (or confusing) the possibilities -- possibly deliberately -- with the respective strategic consequences. Both choices evoke contrasting uses of metaphor to facilitate comprehension. For "lower octaves" a metaphor may be required to frame implication and the "richness" associated with each element. For "higher octaves" a metaphor may be required to suggest the relation to other elements, otherwise easily ignored. Both forms of metaphor may be questionable. Memory "management" in relation to complexity is a focus of research on "chunking" -- a process by which small individual pieces of a set of information are bound together to create a meaningful whole in memory.
Sound and colour: Verbal discussion of discernment capacity may be less meaningful than through the manner in which distinctions of sound or colour are made. In both cases a very wide variety of distinctions can potentially be recognized. A key question is whether and how a distinction can be said to have been discerned if it cannot be named. Of relevance to this argument is the number which can be unambiguously named -- with the associated cognitive, communication and strategic implications:
- sound: Discernment of pitch and issues relating to perfect pitch, framing a challenge of "strategic pitch" and the capacity to "hold a note" or any sense of "holding the line"
- ability to distinguish pitch is not fully understood; it involves some processing by the brain after a sound is perceived (Test Your Sense of Pitch, NIDCD, 31 July 2014)
- the ability for humans to discern changes in pitch as they listen to a speaker is essential for extracting meaning from words; pitch is so important to human language that the exact same words can have different meaning if the pitch is different (How people discern changes in pitch to extract meaning from language, American Association for the Advancement of Science, 24 August 2017) *** Varieties of Tone of Voice and Engagement with Global Strategy: alternating between a requisite variety of voices to engender coherence? 2020
- colour: Provocative issues are raised by the following in the light of the millions of colours which it is claimed humans can perceive:
- (Robert Kosara, You Only See Colors You Can Name, Eagereyes, 21 November 2011)
- wavelengths of colour that are easier to see also get names earlier in the evolution of a culture; colours familiar to one culture might not have names in another, suggesting different cultures indeed have distinct ways of understanding the world; colour names always seem to appear in a specific order of importance across cultures -- black, white, red, green, yellow and blue; the time needed for us to reach consensus on a colour name falls into a hierarchy that matches the human vision system's sensitivity to red over blue, and so on. (Charles Q. Choi, How Colors Get Their Names: It's in Our Vision, LiveScience, 17 April 2012; Hierarchy of Color Naming Matches the Limits of Our Vision System, Scientific American, 16 April 2012)
- there are countless named colours, but in terms of basic colour categories, there are 11 basic colour categories in the English language: red, blue, green, yellow, black, white, pink, orange, brown, purple, and gray. These basic colours can be combined and modified to create a wide variety of named colours. (How many colors have names? Quora)
- there are more colours than names; there are at least 3 million perceptually distinct colours. The Methuen Book of Colour lists, about 2400 names which means there are more than 1000 x more colours than names. This is because when we cite a colour name such as, for example, pink it evokes more than one colour (Stephen Westland.(How many colors have names? Quora)
Taste and smell: Similar points could be made with regard to the discernment of tastes -- as cultivated by those with "educated palates" with respect to wines (How can you tell if someone has an educated palate in wine tasting? Quora). Related insights are offered by odours -- as cultivated by those with an "educated nose" (Rachel Sarah Herz, The role of odor-evoked memory in psychological and physiological health, Brain Sciences, 6, 2016, 3; The Value of an Educated Nose, Vinesse, 26 September 2013). Recent research has shown that humans can identify more than 1 trillion smells -- although how many can be named is another matter.
The matter can also be explored through the number of people a person can "know", whatever this is variously held to mean. How "well" is another known? One upper limit for stable relationships is suggested by Dunbar's number of 150, although it is less evident how this may vary across cultures and comprehension of "knowing". A distinction in terms of "octaves" could be considered for mnemonic purposes.
Memory span: Another approach could be through exploration of the size of sets considered memorable in practice by those engaged with any such set -- as with species, music, books, celebrities, etc.How many "things" can be remembered and successfully communicated -- despite the challenge of aging (Lorea Lastiri, Does Your Brain Have a Memory Limit? Iris, 10 February 2023). More pertinent is the relevance to collective memory (Societal Learning and the Erosion of Collective Memory, 1980).
Such examples reframe the question of how values and strategic objectives are discerned and articulated. There is little consensus regarding discernment capacity in the examples indicated. The confusion is especially evident with respect to the pattern of tones. Despite widespread familiarity with music, supported by extensive research, this is most evident in contrasting appreciation of tuning systems. Of relevance is the challenge of the "colour blind" -- comparable to that of the "tone deaf" -- both with their global strategic implications (Group of 7 Dwarfs: Future-blind and Warning-deaf, 2018).
Set "completeness": The various examples frame the question as to whether a set is "complete" in any systemic sense, possibly to be understood as "well-formed" -- and how this may or may not be the case (as discussed below, for convenience). Issues in that respect include:
- lack of insight into what constitutes a complete set in a systemic sense
- a matter of questionable sensitivity (and/or preference), as indicated with respect to pitch, colour, and odours
- conscious or unconscious avoidance of the inherent challenge of comprehension of more complex sets
- "unsubstantiated" traditional articulations, highlighted for symbolic reasons -- possibly justified to a degree (post facto) through the criteria of sacred geometry (Embodiment of 108-foldness as ultimate spiritual challenge? 2024)
- the result of political compromise, as questionably claimed with regard to the UN's 17 Sustainable Development Goals (Systemic Coherence of the UN's 17 SDGs as a Global Dream, 2021)
- resulting either from errors of comission (Type I) or errors of ommission (Type II) in statistical terms (Type I and Type II errors)
- promoted as a matter of convenience -- in the absence of lesser-known alternatives (Thirty-six Dramatic Situations faced by Global Governance? 2022)
- incomplete research on a set, whether acknowledged or denied
- subject to ongoing further articulation with a higher degree of specialist expertise -- potentially rendering a set incomprehensible for many. The periodic table of chemical elements is in many respects well-formed as a set, but remains open ended in anticipation of the discovery of elements beyond the 118 known in 2024. This has implications for its insights in a general systems sense and as a metaphor (Periodic Pattern of Human Knowing, 2009)
- challenges to recognition and articulation, possibly constrained by popular marketing uptake (as with "games people play")
- degree of appreciation of symmetry and complementarity for mnemonic purposes, potentially associated with factoring the articulation
- a sense of "goodness of fit" (Yori Gidron, Goodness of Fit Hypothesis. Encyclopedia of Behavioral Medicine, 2013), namely the effectiveness of matching (fitting) a coping strategy to a situation's level of controllability, in relation to adaptation to stress. More generally it could be understood as the sense of an explanation that "works", whatever that may be held to mean.
- in addition to perception of the set by distinct senses are the potential issues implied by its perception by different senses, especially given lack of consensus on the set of intelligences as "ways of looking" (Multiple intelligences as pointers to comprehension of multi-dimensionality, 2006)
- notoriety of individuals or institutions affirming the size of a set -- deprecating alternative perspectives by the less renowned or by those associated with other cultures and traditions
Comprehensibility and memorability: In contrast to the preoccupation of mathematics and other disciplines, the approach adopted here is to focus on comprehensibility and memorability, guided by the apparent number preferences in any articulation, as previously explored (Patterns of N-foldness: Comparison of integrated multi-set concept schemes as forms of presentation, 1980). The approach is consistent with the preoccupations of general systems research and the recognition of correspondences ( Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007; Correspondences between Traditional Constellations and Pattern Languages, 2014).
Of particular concern in what follows is recognition of any insight -- from the articulation of disparate sets -- into the nature of transformations within the pattern constituted by a set. Such insights may take a variety of forms between which it is assumed that a degree of correspondence may potentially be found -- if only understood in mnemonic and metaphorical terms. The transformations explored in most detail seem to be those between polyhedra and those associated with the Tonnetz. From a general systems perspective, it is assumed that the sense of "octave" suggests a degree of correspondence between sets of different degrees of articulation.
It is intriguing to note the psychosocial dynamics associated with the articulation of relatively complex sets in a particular domain. The dynamics are perhaps best illustrated by the history of the evolution of the periodic table of chemical elements (D. Michael P. Mingos, The Periodic Table I: historical development and essential features, 2020; Geoff Rayner-Canham, The Periodic Table: Past, Present, and Future, 2020). Equivalent "struggles" and disputes over how elements should be grouped and organized are remarkably illustrated by the many efforts to represent the "table" otherwise -- and more appropriately (Types of Periodic Table, Wikipedia; Mark R. Leach, The INTERNET Database of Periodic Tables). Corresponding initiatives are evident with respect to the array of tones, as noted below -- complicated by contrasting capacities to distinguish between them according to particular conventions and preferences.
Potential implications of mapping tone space correspondences
The focus of the first part of this exploration was on the role of number in defining articulations -- as introduced in the earlier document (Contrasting preferences for N-fold organization in disconnected patterns, 2024; Visualizing exponents of factors characterizing larger sets, 2024). The latter notably clarified the pattern of tones as presented in the traditional triangle of Nicomachus.
As might be expected, there has been considerable evolution of consideration of how an array of tones can be insightfully configured -- in the light of a variety of arguments. Most notable are those involving graph theory, as indicated by the following:
- Gilles Baroin: The Planet-4D Model: An Original Hypersymmetric Music Space Based on Graph Theory
- Giovanni Albini and M. P. Bernardi: Graph Theory and Music: A Mathematical Tool for Musicians (Advances in Intelligent Systems and Computing, 2018)
- Jordan B. L. Smith, Elaine Chew, and Gérard Assayag (Eds.): Mathemusical Conversations: mathematics and computation in music performance and composition (World Scientific Publishing, 2016)
- Jason Yust: Generalized Tonnetze and Zeitnetz, and the Topology of Music Concepts (Journal of Mathematics and Music, 2019)
Such preoccupation has been summarized in a compilation by Mariana Montiel and Robert W Peck:
Questions about variation, similarity, enumeration, and classification of musical structures have long intrigued both musicians and mathematicians. Mathematical models can be found from theoretical analysis to actual composition or sound production. Increasingly in the last few decades, musical scholarship has incorporated modern mathematical content. One example is the application of methods from Algebraic Combinatorics, or Topology and Graph Theory, to the classification of different musical objects. However, these applications of mathematics in the understanding of music have also led to interesting open problems in mathematics itself.The reach and depth of the contributions on mathematical music theory presented in this volume is significant. Each contribution is in a section within these subjects: (i) Algebraic and Combinatorial Approaches; (ii) Geometric, Topological, and Graph-Theoretical Approaches; and (iii) Distance and Similarity Measures in Music. (Mathematical music theory : algebraic, geometric, combinatorial, topological and applied approaches to understanding musical phenomena, World Scientific, 2018)
As implied in the previous discussion of the canon and the Song of Songs, the question here is how any configuration of tones in 3D (or more) might enable insight into the coherence of complex patterns -- as an exemplification of the "harmony" of which it may be held to be indicative. Beyond the 36-voice canon of Johannes Ockeghem, the point was made there with respect to George Polti's 36-fold articulation of "dramatic situations" and the traditional 36-fold articulation of Chinese stratagems (Thirty-six Dramatic Situations faced by Global Governance? 2022; Yuan Gao, Lure the Tiger Out of the Mountains: how to apply the 36 stratagems of Ancient China to the Modern World, 1993).
In contrasting such disparate patterns, the reference to "tone" is understood metaphorically in terms of "voice". This offers the implication that the challenge is one of comprehending a pattern of multiple voices as complementary insights into the "pattern that connects" (Interrelating Multiple Ways of Looking at a Crisis, 2021; Enactivating Multiversal Community: hearing a pattern of voices in the global wilderness, 2009).
Recent insights have been framed by Riemannian theory of harmony, characterized by a concept of harmonic functions and their transformations. This formed the basis for Neo-Riemannian theory, which has expanded the idea of transformations beyond the basic tonal triads with which the original theory was pimarily concerned. This is currently understood to be a loose collection of ideas among music theorists with a central commitment to relating harmonies directly to each other, without necessary reference to a tonic.
Under the heading The Tonal Universe and the Hunt for the Harmonic Higgs Boson, Kenneth Smith introduces the reviews of three books as follows:
Richard Cohn's Audacious Euphony [Chromatic Harmony and the Triad's Second Nature], Steven Rings's, Tonality and Transformation and Dmitri Tymoczko's A Geometry of Music [Harmony and Counterpoint in the Extended Common Practice] are three new technologies for astronavigation, each offering new ways of ofcharting the 'univese in tones'... three new alternative (or complementary) technologies for mapping the awe inspiring constellations of pilches and chords wbich form Western harmony. The wide scope: of harmonic theory, as established by Hugo Riemann and, later, David Lewin, has extended far beyond the boundaries of neo-Riemannian... and even neo-Lewinian space. (The Transformational Energetics of the Tonal Universe: Cohn, Rings and Tymoczko, Music Analysis, 33, 2014, 2)
Following an earlier review by Richard Cohn (Introduction to Neo-Riemannian Theory: a survey and a historical perspective, Journal of Music Theory, 42, 1998, 2), a valuable summary of the evolution of mapping is introduced as follows:
Richard Cohn, reviews graph-theoretic and geometric models of pitch and tonality. Graph-theoretic models are mathematical structures that model pairwise relationships between objects (such as pitches and chords) in terms of networks of vertices connected by edges. Geometric models are mathematical models based on the notion of space and concepts such as distance, neighborhood, and connectivity. Cohn begins in Ancient Greece with Crantor’s lambda, continues with Nicomachus’striangle, Torkesey’striangle, Euler’s Tonnetz and its various adaptations, Douthett’s Cube Dance, and finally concludes with Tymoczko’s orbifold. The juxtaposition of so many models in such a short space could have produced chaotic results, but Cohn successfully draws a common historical thread throughout these models, describing how each model was a natural response to contemporaneous developments in musical practice. This chapter could serve well as a first introduction to this area of music modelling, perhaps as part of a university course in music theory or music psychology. (Graph-theoretic and Geometric Models of Music(Empirical Musicology Review, 12, 2017, 1-2 )
An early representation of the tonal array (reproduced below left) by musicologist Ernest McClain is discussed separately (Requisite variety of "voices" for psychosocial wholth: 6, 8, 12, 20, 30? 2015). Current explorations centre on the so-called Tonnetz (or "tone net"). In musical tuning and harmony, this is a conceptual lattice diagram representing tonal space. Various visual representations of it can be used to show traditional harmonic relationships in European classical music.
| Contrasting representations of tonal space and the Tonnetz | |||
| 37 Tonal values in hexagonal lattice array |
Neo-Riemannian Tonnetz | Pachelbel's Canon in Harmony Space (path of chord roots) |
Neo-Riemannian Tonnetz (animation of a toroidal view) |
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| Reproduced from Ernest McClain (Meditations through the Quran: tonal images in an oral culture, 1981, p. 95) | "Watchduck", CC0, via Wikimedia Commons | Reproduced from Simon Holland, et al, Running up Blueberry Hill (2009) | By Davidwbulger - Own work, Public Domain, Link |
The challenge of 440 Hz versus 432 HZ as used by Ernest McClain) is discussed separately (Standard pitch imposition and its analogues: 440Hz vs 432Hz, 2024).
| Symmetric trigonometric functions within 432.00 -- referring to the hexagonal lattice (above left) | |
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| Since one angle of the 3:4:5 triangle is approximately 37° (consult any table of trigonometric functions) we need 36 smaller tone-ratios to carry the correlation back to 1°. The remarkable fact about the 37 tones of our hexagonal lattice is that their ratios, taken in order through one octave and part way through the second octave, generate Pythagorean triples in which angular values vary by an average of 1° all the way to 45°. Since complementary angles complete the correlation to 90°, our musical hexagon establishes the approximate arithmetic tools for correlation with a nascent astronomy of 360 degrees. The 37 tones of our lattice [above left] are defined in their base 10 numerical notation. The ratio between the reference D = 432,000, always represented by p and symbolizing the Unity | |
| Reproduced from Ernest McClain, The Pythagorean Plato: prelude to the song itself, 1978, pp. 120-121 and Meditations through the Quran: tonal images in an oral culture, 1981, p. 95) | |
Andrew Xu Canon in D - Visual Representation This is a visual representation of Johann Pachelbel's Canon in D generated with "The Music Animation Machine", which is available at http://www.musanim.com/player/ ****.
Clarification was requested of ChatGPT with regard to the toroidal representation of the Tonnetz (above right) by David Bulger
The description you provided of David Bulger's toroidal representation of the Tonnetz suggests a model that intertwines both the cyclic nature of music theory and the spatial dynamics of a torus. Based on this visualization, it seems to incorporate two fundamental musical intervals: perfect fifths (toroidal direction) and major/minor thirds (poloidal direction). Here’s how this could conceptualize the representation of tones:
This model provides a rich visual and conceptual framework for understanding and exploring the interconnections within Western tonal harmony. By visualizing the tones and their relationships in both the toroidal and poloidal directions, it offers insights into the cyclical nature of music theory, the foundational role of triads, and the complex web of harmonic relationships that underpin Western music. |
Expressed otherwise:
While the classical Tonnetz has 12 nodes (representing the 12 pitches), the toroidal view expands upon this by creating a continuous loop around the torus. Let’s break it down:
So, in David Bulger’s toroidal animation, we indeed have 36 nodes, enhancing our understanding of harmonic motion and transformations |
A useful commentary on harmony space in the light of the Tonnetz is offered by Simon Holland, et al, (Running up Blueberry Hill: prototyping whole body interaction in Harmony Space Conference Paper, February 2009) following an earlier introduction (Simon Holland, Learning About Harmony with Harmony Space: An Overview, World Conference on Artificial Intelligence in Education, Edinburgh, 1993).
Potentially of particular interest is the work of Dmitri Tymoczko (The Generalized Tonnetz, Journal of Music Theory, 56, 2012, 1), notably his conclusion with regard to a three-note octahedral Tonnetz and tetrahedral models of four-note diatonic and chromatic chords. His work has been framed in terms of A Grand Unified Theory of Music (Princeton Alumni Weekley, 9 February 2011) by Steve Olson noting that chords don’t just have sound, they have shape recognized by Tymoczko as orbifolds:
Three-note chords occupy a twisted three-dimensional space, and four-note chords live in a corresponding but impossible-to-visualize four-dimensional space. In fact, it worked for any number of notes — each chord inhabited a multidimensional space that twisted back on itself in unusual ways -- a non-Euclidean space that does not adhere to the classical rules of geometry.
This has been remarkably presented and illustrated by Mattia G. Bergomi (Musical modeling through graphs and orbifolds, IRCAM, 20 December 2014).
Question to ChatGPT: The concept of "voice leading" in recent music theory [Dmitri Tymoczko, Scale Theory, Serial Theory and Voice Leading, Music Analysis, 27, 2008, 1] merits consideration with respect to multiple voices in democratic governance. Can you comment on potential metaphoric use of the expression
The concept of "voice leading" in music theory, particularly in the context of counterpoint and harmony, involves the smooth, logical progression of individual musical lines or voices. It's a fundamental aspect of composition and arrangement that ensures cohesiveness and aesthetic appeal by guiding each voice through transitions in a way that maintains harmonic integrity while allowing each to express its unique melodic contour. The aim is to create a complex, yet harmonious whole, where each voice contributes to the overall texture without any unnecessary jumps or dissonances unless artistically intended. When we metaphorically apply the concept of "voice leading" to the realm of democratic governance, we can draw parallels in several intriguing ways:
By viewing democratic governance through the lens of voice leading, we can appreciate the importance of balance, diversity, and careful progression in creating a system that is both dynamic and harmonious. This metaphor encourages us to consider how individual contributions, when thoughtfully integrated, can lead to a more cohesive and effective governance structure. |
Contrasting to a degree with that of Tymoczko, the approach of David Bulger and Richard Cohn is articulated as follows:
Dmitri Tymoczko describes the voice-leading space of N-note chords as the orbifold TN/SN , the N-torus modulo the Nth symmetric group action, "an N-dimensional prism whose simplicial faces are glued together with a twist, and whose remaining boundaries act like mirrors" (A Geometry of Music: harmony and counterpoint in the extended common practice, 2011). This quotient space TN/SN is produced from the space of all ordered sequences of N pitch classes TN by identifying each sequence with all its reorderings, indicating that we consider a chord unchanged under any permutation of its voices. Here instead we consider a polyphonic setting in which not all voices are free to move independently. Such constraints describe "power chords" in rock (bare fifths or fourths played on guitar) and can also be found in the classical repertoire. We present chord spaces describing excerpts from Bartók and Stravinsky (Constrained voice-leading spaces, Journal of Mathematics and Music, 10, 2016, 1)
An indication of the evolving possibilities of representation of the Tonnetz is the creation of a web platform (Corentin Guichaoua, José L Besada, Erica Bisesi, Moreno Andreatta, The Tonnetz Environment: a web platform for computer-aided "mathemusical" learning and research, 13th International Conference on Computer Supported Education, 2021, Prague). An example (left below) from the platform is the multiple visualizations of pentachord {C#, D, E, G, A} within the Tonnetz. The image below (right) is the representation of a subspace comprised of all the diatonic triads in the keys of C and G major for distances up to √6 by Marco Buongiorno Nardelli (Topology of Networks in Generalized Musical Spaces May 2019).
| Indicative representations of musical space | ||
| Multiple visualizations of pentachord {C#, D, E, G, A} within the Tonnetz. standard Tonnetz (left); circular representations of chord (right) |
Voice leading space lowest dimensional case: intervals |
Diatonic triads in the keys of C and G major ( thickness of edge lines proportional to weight) |
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| Reproduced from Corentin Guichaoua, et al, The Tonnetz Environment: a web platform for computer-aided "mathemusical" learning and research (2021) | Reproduced from Mattia Bergomi Musical modeling through graphs and orbifolds (2014) | Reproduced from Marco Buongiorno Nardelli (Topology of Networks in Generalized Musical Spaces 2019) |
https://www.laetusinpraesens.org/docs10s/toroid.php ***
It is intriguing why the obvious complexity of such approaches and representations is considered appropriate to the purportedly simple process of comprehension of music -- whilst held to be of little relevance to thinking about global governance. As noted previously, there is a strange cognitive disconnect within the policy community from the appreciation upheld for the complexities of the Ode to Joy in its use as the Anthem of Europe, for example (Reversing the Anthem of Europe to Signal Distress, 2016).
Symmetry-preserving operations between polyhedra
As noted above, Tymoczko makes use of polyhedra in one approach to visualization -- one that is intimately related to graph theory. One merit of polyhedra is they are readily visualized by those who are challenged by music. Distinctive polyhedra may be created through modification of a seed polyhedron by various prefix operations, as described by the Conway polyhedron notation, and discussed separately (Topological operations on polyhedra as indicative of cognitive operations, 2021; Memorability of cognitive implication in symmetry-preserving operations on polyhedra, 2021).
The following image shows how 11 new forms can be derived from the cube using 3 operations (named dual, ambo and kis). The new polyhedra are shown as maps on the surface of the cube so that the topological changes are more apparent. The three basic operations (when used successively) are sufficient for generation of the 5 Platonic and the 13 Archimedean polyhedra:
- dual: replaces each face with a vertex, and each vertex with a face
- ambo: creates degree-4 vertices (otherwise known as rectification)
- kis: raises a pyramid on each face
| Conway relational chart Showing 12 forms created by 3 operations on the cube |
Configuration of 16 Platonic and Archimedean polyhedra on a hypercube (not to scale) |
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| Tomruen at English Wikipedia, Public domain, via Wikimedia Commons | Reproduced from Implicate order through hypercube and drilled truncated cube?(2022) |
Other operations have been distinguished, together constituting a more conventional total of 13 and denoted by the letters abdegjkmoprst (see table below). These have been extended in the Antiprism polyhedron modelling software application of Adrian Rossiter to a further set of 18 -- although how they might together be understood as a set remains unclear (Conway Notation Transformation, Antiprism; Wythoff-style constructions, Antiprism).
Examples of local symmetry-preserving operations on a cube ordered by increasing inflation rate, as generated by the Antiprism application, are presented separately (Operations transforming polyhedra as modelling transformations of dialogue coherence, 2021). That tabular presentation of operations has been sorted in ascending order of complexity of transformation, namely to indicate how much greater is the complexity of the polyhedron after the transformation. The table makes provision for indication of the degree to which such a transformation might apply in a dialogue mapped as a graph. The terms used to describe the operations are potentially somewhat suggestive of this in metaphorical terms -- inviting metaphorical interpretation of the operations as transformation of graphical mapping of a dialogue on a polyhedron.
In the presentation of these variants by Adrian Rossiter a reference is made to the fact that other operations have been recognized, as noted by B. R. S Recht (Notes on operations on polyhedra, Antitile, 2017) referring specifically to one formalization (Gunnar Brinkmann, Pieter Goetschalckx and Stan Schein, Goldberg, Fuller, Caspar, Klug and Coxeter and a general approach to local symmetry-preserving operations, arXiv:1705.02848, 2017):
This text is an attempt to use Brinkmann’s work to find ways to quantify, analyze, and expand these operators. In particular, it focuses on operators on that can be described in terms of a linear operator on the counts of vertices, edges, and faces. These linear operators can be used to examine the composition and decomposition of operations on polyhedra. Such operators do not constitute all possible operations on polyhedra, or even all those that can be represented by wythoff in [Antiprism], but they are an interesting subset of those operators with many nice aesthetic and geometric qualities.
In suggesting that there may be an infinite number of "prime" operators, Rossiter notes (personal communication, 2024) that:
There is also the case that some operators are defined as a class with numeric parameters (e.g. m, u), and you could consider that these are not really just one operator, but an infinite number. Also, these classes are not determined directly by the base operator, but are derived as a human perception of the essence of the base operator. In some cases there is more than one candidate for a class based on an operator....Wythoff patterns do not differentiate between the nature of vertex, edge and face elements, they are the same kind of thing, and their role in the pattern can be permuted. This permutation can be seen as an operation that acts on operators, and could reduce the total number of operators. Patterns that are equivalent under permutation are seed/dual/ambo, truncate/zip/expand, kis/needle/subdivide. It is also possile to conceive other operations that can act on operators.
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Again it might be asked how the systemic insights into such "symmetry preserving operations" might be relevant to the variable geometry envisaged with respect to institutions of governance, as discussed separately (Alternation between Variable Geometries: a brokership style for the United Nations as a guarantee of its requisite variety, 1985).
Polyhedral transformations and their potential correspondence to Tonnetz transformations?
Since the transformations within music are widely appreciated, commented and studied, it might then be asked whether the seeming complexity of symmetry-preserving operations in polyhedra offer a "language" through which both could be understood and described otherwise.
Question to ChatGPT: Could you comment on any formal relation between the Conway polyhedral notation and transformation within the Neo-Riemannian Tonnetz -- possibly from a general systems perspective. For example I note that both seem to involve three basic transformations. A variant of the Conway notation has also been elaborated for knot theory [discussed below]
Conway's polyhedral notation and the Neo-Riemannian Tonnetz both involve transformations, but they operate in different domains: geometry and music theory, respectively. Conway's notation describes operations on polyhedra, focusing on symmetry and structure. The Tonnetz represents harmonic relationships and transformations within tonal music, illustrating connections between chords. A formal relation between them could be explored through general systems theory, which examines patterns and structures across different fields. Both systems emphasize the importance of transformations and symmetry, suggesting a conceptual link that highlights universal principles of change and interconnectedness across disciplines. |
Question to ChatGPT: With respect to Neo-Riemannian Tonnetz theory and the Conway notation, can you comment on any correspondence between the basic three transformations in each case
The Neo-Riemannian Tonnetz and Conway's polyhedral notation both utilize three primary transformations, albeit in distinct domains. In music, the Tonnetz's transformations include Parallel (P), Leading-Tone Exchange (L), and Relative (R), which describe shifts between chordal structures. In contrast, Conway's notation for polyhedra (and by extension, knot theory) involves operations like dual (d), ambo (a), and kis (k), affecting the polyhedra's structure. While from different fields, both sets of transformations reflect an underlying interest in how simple operations can generate complex structures or relationships, hinting at a conceptual similarity in the abstraction of transformations. |
Question to ChatGPT: Does the degree of correspondence suggest the possibility of mapping Tonnetz transformations onto particular polyhedra
The conceptual correspondence between Neo-Riemannian Tonnetz transformations and Conway's polyhedral operations indeed suggests an intriguing possibility of mapping musical transformations onto polyhedral structures. While direct mapping might not be straightforward due to the different nature of these systems, the abstract principles underlying transformations in both domains could facilitate a creative exploration of such mappings. This approach would require identifying analogous operations in both systems that share similar properties or effects, thereby establishing a theoretical bridge that enriches our understanding of geometric and musical structures through the lens of transformation and symmetry. |
Comparison of canon styles with basic symmetry preserving operations
Given the esteem in which the 36-voice canon of Johannes Ockeghem is held, could the intricacy of its pattern of transformations be framed otherwise -- for some -- through the lens of the polyhedral sysmmetry-preserving operations?.
Question to ChatGPT: How could the distinctive types of canon (as musical transformations) be usefully compared as a set with the set of symmetry-preserving operations identified for polyhedra by Conway
John Horton Conway's work on symmetry-preserving operations for polyhedra, known as the "Conway Polyhedron Notation," provides a systematic way to describe and transform polyhedral shapes based on their symmetries. This set of operations can indeed be intriguingly compared to the different types of musical canons, as both sets involve transformational processes—Conway's operations transform polyhedral shapes while maintaining certain symmetries, whereas types of musical canons involve transformations of a musical theme under specific constraints. Conway's Symmetry-Preserving Operations and Types of Musical Canons: To draw a parallel between these two sets, let's consider the essence of Conway's operations and compare them with the characteristics of various musical canons:
Conclusion: Comparing Conway's symmetry-preserving operations for polyhedra with the types of musical canons offers a fascinating cross-disciplinary perspective on transformation and symmetry. Both sets involve structured changes that adhere to certain rules while allowing for creativity and variation. This analogy not only highlights the mathematical beauty underlying music and geometry but also suggests a deeper, universal language of patterns and transformations that span different fields of human knowledge and creativity. |
Question to ChatGPT: Given the correspondences, can each set be considered complete, or are some variants missing from one or the other
When comparing the sets of Conway's symmetry-preserving operations for polyhedra with the types of musical canons, it's important to recognize that while there are intriguing correspondences between the two, they operate within fundamentally different domains -- geometry and music. These correspondences allow us to draw parallels and explore conceptual similarities, but the completeness of each set must be considered within the context of its respective domain.
Conclusion: While there are compelling correspondences between Conway's symmetry-preserving operations and the types of musical canons, each set's completeness is confined to its domain and the current state of knowledge and creativity within that domain. The potential for "missing" variants or new discoveries highlights the dynamic nature of both geometric and musical exploration, underscoring the limitless possibilities for innovation at the intersection of different fields of inquiry. |
Question to ChatGPT: Although in anticipation of future creativity, the correspondence between the 8-fold symmetry preserving operations and the 8 canon styles suggests that -- self-reflexively -- both might be mapped onto one or more 8-fold polyhedra for mnemonic purposes. Whether such a mapping would be indicative of transformational pathways between disparate modes of organization is to be determined
The intriguing correspondence between the eight-fold symmetry-preserving operations defined by John Horton Conway for polyhedra and the various styles of musical canons presents a unique opportunity for conceptual mapping onto an 8-fold polyhedron, serving both mnemonic and analytical purposes. This self-reflexive mapping could indeed facilitate understanding and memorability of these concepts and potentially uncover transformational pathways between disparate modes of organization. Let's explore how such a mapping might be conceptualized and its implications.
Conclusion: Mapping the eight-fold symmetry-preserving operations and the styles of musical canons onto an 8-fold polyhedron offers a novel way to visualize and explore the relationships between these concepts. This self-reflexive exercise not only serves mnemonic purposes but also opens up avenues for discovering transformational pathways between modes of organization, encouraging interdisciplinary insights and creative exploration across the spectra of geometry, music, and beyond. |
Polyhedral configuration of logical operations and correspondence to transformations within the Tonnetz?
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:
- Oppositional Logic as Comprehensible Key to Sustainable Democracy: configuring patterns of anti-otherness (2018)
- Neglected recognition of logical patterns -- especially of opposition (2017)
- Unexplored cognitive implications of 17 Sustainable Development Goals -- through "magic squares"? (2017)
- Oppositional logic and its requisite polyhedral geometry (2019) *** rhombic
- Relevance of oppositional logic to relating virtues and sins (2022)
- Reframing forms of connectivity through the logic of oppositional geometry (2020) *** better
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 |
The Logic Alphabet Tesseract |
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 |
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| 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 Júnior (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 Clüver (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 & Courtés (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 |
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| Reproduced from Daniel Costa Júnior, 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. |
Mapping transformations and the correspondences exemplified in 36-fold sets
Whilst the focus here is on polyhedra potentially suitable for mapping 36-fold configurations, a wider range is considered separately (Identifying Polyhedra Enabling Memorable Strategic Mapping, 2020). The approach has previously been explored in the documents noted above with respect to sets from 12-fold to 108-fold.
The following representations are usefully understood as contrasting "projections" of an elusive pattern which is a challenge to visualization. This can be appropriatrely compared to the challenge of representing the global form of the Earth -- exemplified by the multitude of initiatives in that regard (List of Map Projections, Wikipedia). The polyhedral configurations are presented tentatively as experiments suggestive of ways of comprehending the coherence of the sets and the potential correspndence between them. Each can be understood as suggesting a contrasting perspective about the transformations. It is appropriate to recognize that, as in the case of the Earth, the static images obscure dimensions of the configuration, partially mitigated by access to their rotating animations.
The library of Stella 4D offers 5 possible polyhedra with 36 vertices of which 2 were set aside because of the limited ability to associate labels visibly with their vertices (4D 6-6 duoprism Hiddip, Augmented great rhombihexahedron). The 3 selected are presented below with solid and transparent faces. Given that labels are subsequently associated with their vertices, it is appropriate to recall that the duals of these polyhedra would allow labels to be associated with the corresponding number of faces -- potentially more clearly.
In the mapping exercise which follows, the elements of the set are associated arbitrarily with the vertices as a preliminary to consideration of comparability. Consideration of how such attributions might be more appropriately mapped are considered later (below).
| 36-vertex polyhedra of potential use in mapping 36-fold configurations | |||||
| Cuboctahedra 3 V=36, E=72, F=42 |
Leonardo octahedron V=36, E=96, F=48 |
Faceted rhombicuboctahedron V=36, E=72, F=18 |
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| solid faced | transparent faced (rotated) | solid faced | transparent faced (rotated) | solid faced | transparent faced (rotated) |
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| Images prepared using Stella 4D | |||||
In seeking to map the elements of a set as a configuration, in order to highlight possible transformations between them, each could be associated with the edges of a polyhedron rather than with th vertices (or faces). As an indication of the possibility, the following were selected as of potential use for mapping purposes from the greater number of 36-edged polyhedra.
| 36-edged polyhedra of potential use in mapping 36-fold configurations | |||||
| Truncated cube *** V=24, E=36, F=14 |
Stella octangula*** V=14, E=36, F=24 |
Truncated octahedron** V=24, E=36, F=14 |
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| solid faced | transparent faced (rotated) | solid faced | transparent faced (rotated) | solid faced | transparent faced (rotated) |
| Animations prepared using Stella 4D | |||||
Mapping correspondences of 36-fold sets of strategems and dramatic situations
As a first set of examples of such mapping, a correspondence is potentially suggested between contrasting uses of the Faceted rhombicuboctahedron for mapping George Polti's 36-fold articulation of "dramatic situations" and the traditional 36-fold articulation of Chinese stratagems (Thirty-six Dramatic Situations faced by Global Governance? 2022; Yuan Gao, Lure the Tiger Out of the Mountains: how to apply the 36 stratagems of Ancient China to the Modern World, 1993). Note that for purposes of illustration these sets could have been mapped onto any of the other 36-fold polyhedra.
It could be considered somewhat remarkable that the two sets with arguably related preoccupations -- but from different cultures -- should have been determined to be of the same size. In both cases it could be argued that there should be a fewer or greater number of elements. Clearly there is then the question of how comparable are "dramatic situations" and "stratagems", although it could be readily argued that a dramatic situation implies a stratagem (and vice versa). Curiously the stratagems are appreciated as proverbs in Chinese culture, whereas the dramatic situations are especially familiar in Western entertainment -- although few could recognize them as a set.
The labels could be presented otherwise. In 2017, screenwriter Mike Figgis produced a reworked version of Polti's list of situations (The Thirty-Six Dramatic Situations, 2017). Although identically titled, it combines several of the original dramatic situations and added two new ones to arrive at that number. Figgis also replaces Polti's dated examples with new ones drawn from films, offering a more expansive and poetic interpretation of the situations. Recognition is also now given to the questionably related set of Wycliff Aber Hill (Rebecca Onion, The 37 Basic Plots, According to a Screenwriter of the Silent-Film Era, Slate, 27 October 2015).
Western commentaries on the Chinese stratagems have been variously offered (John Barkai, The Chinese Strategies applied to Negotiation, Pepperdine Dispute Resolution Law Journal, 403, 2008; Davia Temin, Ancient Wisdom For The New Year: the 36 Chinese Stratagems for Psychological Warfare, Forbes, 2 January 2017; Military Strategy and War: Concepts and Chinese Stratagems for War, Faster than Normal, 21 March 2022). Western and Eastern appreciation of the strategems has been the subject of a study (Cheryl Rivers and Roger Volkema, East-West Differences in "Tricky" Tactics: a comparison of the tactical preferences of Chinese and Australian Negotiators, Journal of Business Ethics, 115, 2013, 1).
| Comparability of strategic modalities suggested by 36-vertex Faceted Rhombicuboctahedron? | |||
| Dramatic situations (Polti) | Stratagems (China) | ||
| solid faced | transparent faced | solid faced | transparent faced |
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| Rotating animation | Rotating animation | Rotating animation | Rotating animation |
The alternative mapping on the edges of a polyhedron is indicated below.
| Comparability of strategic modalities suggested by 36-edged ***? | |||
| Dramatic situations (Polti) | Stratagems (China) | ||
| solid faced | transparent faced | solid faced | transparent faced |
| Rotating animation | Rotating animation | Rotating animation | Rotating animation |
Mapping correspondences of 36-fold sets of games and katas
As a second set of examples, a correspondence is suggested between contrasting uses of the polyhedral compound Cuboctahedra-3 for mapping a set of 36 games seemngly identified by transactional analysis (Eric Berne, Games People Play, 1964) and a set of 29 "institutional katas" identified by Thierry Gaudin, published as an annex to a book titled: L'Ecoute des Silences: les institutions contre l'innovation (1978). An English translation of that 29-fold set is presented separately as The Institutional Katas (2016) in a discussion of Game-playing in Global Governance? (2016) -- framed in terms of Engaging an Opposing Ideology via Martial Arts Philosophy (2016).
In this example, the set of games is less evident. Many references to the games identified by transactional analysis refer to far smaller numbers -- more readily recognized in the media in which they are described. A set of 36 does indeed appear to be identified in the "thesaurus of games" in the original study (1964). A much-cited review that study indicates that "Dr. Berne sketches 101 games" (Kurt Vonnegut Review of Games People Play, Life Magazine, 11 June 1965, included in the 40th anniversary edition of Games People Play, 2004) -- a view echoed by Vidhu Bhatnagar (Games and Games Analysis), but otherwise unconfirmed.
Such transactional games could be considered comparable to the katas (or forms) in Eastern martial arts. Katas are choreographed patterns of movements practiced in many martial arts. Each martial art has its own set of katas, which serve as a way to transmit and practice the art's techniques, principles, and philosophy. The number of katas varies widely across different styles and schools. The kata are executed as a specified series of approximately 20 to 70 moves, generally with stepping and turning, while attempting to maintain perfect form. Shotokan Karate, one of the most widely practiced styles, traditionally includes 26 katas, but other styles may have more or fewer depending on their lineage and emphasis.
The total number of katas across all Eastern martial arts is difficult to quantify due to the diversity of styles and the evolution of new forms within each discipline. Moreover, martial arts are dynamic, with new katas being developed and old ones modified or rarely practiced. Each martial art and school values its katas for their ability to teach the art's techniques, history, and philosophy, making them central to martial arts training.
With respect to Karate kata, the number 108 is present in a number of traditional katas and has mythological significance in Dharmic religions(Embodiment of 108-foldness as ultimate spiritual challenge? 2024). This number also figures prominently in the names of Karate kata. The advanced Gōjū-ryū kata, Suparinpei, literally translates in Fuzhounese to the number 108, while gojushi of Gojūshiho is the Japanese pronunciation of the number 54 (half of 108). The other Gōjū-ryū kata, Sanseru (meaning "36") and Seipai ("18") are factors of the number 108.
With respect to requisite variety, the number of katas in a system needs to balance the need for comprehensive training in techniques and principles against the practitioner's ability to learn and perfect them. Too few katas might not provide enough variety for all combat situations, while too many could overwhelm students.
The sets offer contrasting framings of the manner in which people engage with each other individually (and potentially collectively) in an effort to achieve competitive advantage. The similairty can be framed through understandings of the deprecated manner in which people "play" each other.They indivte interpetation in terms of game theory (Vandra Attila, The Link between the Mathematical Game Theory and the Transactional Analysis. a new kind of psychological game comes into being in new interpersonal relations, Europe's Journal of Psychology, 2, 2006, 2)
| Comparability of gaming modalities suggested by 36-vertex Cuboctahedra-3? | |||
| "Games People Play" (Transaction analysis) | "Katas" (Eastern martial arts) | ||
| solid faced | transparent faced | solid faced | transparent faced |
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| Rotating animation | Rotating animation | Rotating animation | Rotating animation |
The alternative mapping on the edges of a polyhedron is indicated below.
| Comparability of gaming modalities suggested by 36-edged ***? | |||
| "Games People Play" (Transaction analysis) | "Katas" (Eastern martial arts) | ||
| solid faced | transparent faced | solid faced | transparent faced |
| Rotating animation | Rotating animation | Rotating animation | Rotating animation |
Mapping correspondences of 36-fold multi-voiced canon and Song of Songs
It could be inferred that the "voices" in the 36-voice canon of Johannes Ockeghem could be distinguished in such a manner as to enable them to be individually named. As noted previously, this appears not to be the case. In the light of the assertion that the 36-voice canon is effectively not a single canon, but a four melody canon with 9 voices per part, for the purpose of this exercise these have been denoted as Voice 1-1 through Voice 4-9.
As the third set of examples, a correspondence is suggested between contrasting uses of the Leonardo octahedron for the 36-voice canon and a 36-fold articulation of the Song of Songs, as speculatively requested of ChatGPT for the purpose of this exercise (Recognizing 36-fold and 64-fold patterns of relationship with an other, 2024).
| Comparability of 36 aesthetic modalities suggested by 36-vertex Leonardo octahedron? | |||
| Multi-voiced canon (Johannes Ockeghem) | Song of Songs (Bible) | ||
| solid faced | transparent faced | solid faced | transparent faced |
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| Rotating animation | Rotating animation | Rotating animation | Rotating animation |
The alternative mapping on the edges of a polyhedron is indicated below.
| Comparability of aesthetic *** modalities suggested by 36-edged ***-3? | |||
| Multi-voiced canon (Johannes Ockeghem) | Song of Songs (Bible) | ||
| solid faced | transparent faced | solid faced | transparent faced |
| Rotating animation | Rotating animation | Rotating animation | Rotating animation |
Question to ChatGPT: This exchange focused earlier on Polti's 36-fold set of dramatic situations and the 36-fold set of Chinese stratagems. Arguably both might be understood as implying "transformations" between conditions -- potentially with correspondence to the Conway notation, the Neo-Riemannian Tonnetz and those of the Boolean connectives
The idea of mapping Polti's 36 dramatic situations and the 36 Chinese stratagems onto the frameworks of Conway's notation, Neo-Riemannian Tonnetz, and Boolean connectives is intellectually stimulating. These sets, which fundamentally represent transformations or shifts between states—whether in narrative contexts, strategic maneuvers, musical harmony, geometric shapes, or logical operations—suggest a universal structural underpinning across disciplines. This approach underscores the potential for interdisciplinary exploration, highlighting the rich, interconnected fabric of transformations that govern change, whether in storytelling, strategy, music, geometry, or logic. |
Question to ChatGPT: Given that possibility, it is puzzling that transformations have long been recognized between the 64-fold set of Yi Jing hexagrams and implied in the case of 72-fold sets of some Western esoteric traditions. It is far from clear how the smaller sets might be "nested" systemically within such larger sets
The notion of "nesting" smaller sets like Polti's 36 dramatic situations or the 36 Chinese stratagems within larger sets such as the 64 Yi Jing hexagrams or the 72 elements of Western esoteric traditions suggests a complex interplay of systems. This interrelation implies that smaller sets might represent specific instances or facets of broader, more encompassing frameworks. The challenge lies in identifying systematic correspondences that respect the integrity of each set while exploring potential mappings or transformations that reveal deeper, universal structures underlying diverse knowledge systems. |
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Polyhedral edge-mapping of correspondences of 36-fold sets
morphing --same in dual
36 edged
symmetry preserving operations and tonal array
Tonnetz: A Visual Representation Of Chords And Scales
| truncated octahedron V=24, E=36, F=14 |
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Space Harmony https://en.wikipedia.org/wiki/Space_Harmony
36 Must-Know Basic Yoga Poses For Beginners Vutru, 2022 Yogasanas 36 Wikipedia List of asanas (91)
Katerina El Raheb, et al Moving in the Cube: a motion-based playful experience for introducing Labanotation to beginners Journal on Computing and Cultural Heritage, 14, 2021, 2, 14; Victoria Watts, Benesh Movement Notation and Labanotation: From Inception to Establishment (1919–1977) Dance Chronicle, 38, 2015, 3; Egil Bakka and Siri Mæland, The Manipulation of Body Weight for Locomotion: Labanotation and the Svikt Analysis, 2020)
Chanel Smith and Shani Robison, Rudolf Laban’s Space Harmony and Sacred Geometry Principles Applied to Ballet Methodology Journal of Undergraduate Research, 15 Ausut 2013
truncated cube |
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| Tai Chi | Yoga | ||
| Forms 1-36 | |||
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| Rotating animation | Rotating animation | Rotating animation | Rotating animation |
The decans "[those] connected with work" are 36 groups of stars (small constellations) used in the ancient Egyptian astronomy to conveniently divide the 360 degree ecliptic into 36 parts of 10 degrees each, both for theurgical and heliacal horological purposes. After Hellenistic astrology arose in Alexandria, recorded principally in the work of Claudius Ptolemy and Vettius Valens, various systems attributing symbolic significance to decans arose. Decans are named in various Greco-Egyptian sources, many Hermetic writings, the Testament of Solomon, and the writings of Aristobulus of Paneas. Julius Firmicus Maternus, Cosmas of Maiuma, Joseph Justus Scaliger, and Athanasius Kircher.
https://www.eastrohelp.com/blog/importance-36-gunas-successful-marriage/
The 36 Tattvas and Their Significance, Saivism
| stella octangula V=14, E=36, F=24 |
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| Decans | Tattvas | ||
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| Rotating animation | Rotating animation | Rotating animation | Rotating animation |
truncated octahedron -- truncated cube -- stella octangula
Comparability of cognitively fundamental sets
In endeavouring to elicit insight into a "pattern that connects", the concern is necessarily with how a set is formed -- beyond its articulation as a linear sequence in a list. Exploration of the use of polyhedra as a means of mapping such sets is seen as a device to suggest the possibility of connectivity to which consideration may -- as yet -- not normally be accorded. It is further assumed that determination of the size may be governed to some degree by human preferences and cognitive processes, as implied by the arguments of George Lakoff and Rafael Nuñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001). It is further assumed that sets identified as being of a particular size may constitute a form of system within which a pattern of transformations may then be identifiable -- potentially comparable to other sets of the same size.
Question to ChatGPT: What constitutes a well-formed set of elements forming a pattern. Are there degrees of well-formedness and how is such a set distinguished from a series -- perhaps due to the degree of connectivity of the elements
A well-formed set of elements forming a pattern can be understood through various criteria, depending on the domain of application (e.g., mathematics, music, art, linguistics). Generally, a well-formed set exhibits internal coherence, consistency, and follows specific rules or principles that define its structure. Here are some factors that contribute to the well-formedness of a set and how such a set might be distinguished from a mere series:
Understanding what constitutes a well-formed set involves analyzing these characteristics in the context of specific domains, recognizing that the criteria may shift based on the patterns and structures valued within those fields. |
Question to ChatGPT: How is a well-formed set distinguished from a complete set.
Understanding the relation between a well-formed set and a complete set involves delving deeper into the characteristics that define both concepts and how they interrelate, often within specific disciplinary contexts:
Understanding both concepts requires considering their definitions within specific disciplinary frameworks and how these definitions impact the set's formation and perceived integrity or completeness. |
Question to ChatGPT: How do these concepts relate to systemic coherence with its dynamic implications
The concepts of well-formedness, completeness, and systemic coherence are deeply interconnected, especially when considering their dynamic implications within various systems. These relationships are pivotal in fields ranging from mathematics and science to music, literature, and beyond, each offering unique insights into how systems function, evolve, and maintain integrity over time.
Understanding the relationship between well-formedness, completeness, and systemic coherence underscores the importance of structured interrelations and adaptability in sustaining the dynamic integrity of systems across various domains. |
Question to ChatGPT: Could you comment on any distinction between the following "sets" (featuring in this exchange) in terms of completeness, well-formedness, and systemic coherence: array of tones (Tonnetz), Polti's 36 "dramatic situations", 36 Chinese "stratagems", symmetry-preserving operations on polyhedra, the games people play of transaction analysis, 36-voice musical canon, Boolean logic connectives, periodic table of chemical elements
The distinction between the mentioned "sets" in terms of completeness, well-formedness, and systemic coherence can illuminate the underlying structures and principles that govern each domain. Let's explore these sets briefly:
Each of these sets, despite their domain-specific differences, showcases the balance between completeness, well-formedness, and systemic coherence, reflecting the structured and interconnected nature of knowledge across diverse fields. |
This focus frames the question as to what is meant and recognized as "transformation" within a pattern. To what extent is it then implied by the following -- and the possibility of corrspondences between them despite the disparate nature of the domains:
- shifts between "dramatic situations" and "ploys" in governance -- or in courtship
- game-playing
- symmetry-preserving operations on polyhedra
- moves in a complex dance
- shifts in any musical articulation
- logical operations within the Boolean set
Question to ChatGPT: Taking the clarification further, could the sense of "systemic coherence" be contrasted with that of a "viable system" with its dynamic implications. Arguably the encoding of the Yi Jing and the transformations implied between 64 hexagram conditions constitutes an extreme example of viable systemic coherence.
The concepts of "systemic coherence" and a "viable system" offer intriguing perspectives on understanding complex systems, particularly when considering their dynamic implications and applications, such as in the Yi Jing (I Ching) with its 64 hexagrams.
In essence, the Yi Jing embodies an "extreme example of viable systemic coherence," merging the internal consistency of its symbolic system with the practical adaptability required to navigate life's complexities, thereby sustaining its relevance and utility across diverse contexts and through time. |
Question to ChatGPT: Given the preoccupation in this exchange with the "pattern that connects", in what way might "systemic coherence" and "viable system" apply to such a "meta-pattern"
The "pattern that connects" concept, especially as articulated by Gregory Bateson, refers to the fundamental, underlying patterns that connect disparate fields of knowledge, showing how similar principles of organization and interaction apply across biological, ecological, social, and cognitive systems. This meta-pattern perspective is crucial for understanding the interconnectedness of all things and the importance of systemic thinking. When relating this to "systemic coherence" and "viable system," we can see how these concepts illuminate aspects of such a meta-pattern in profound ways:
In essence, the "pattern that connects" serves as a meta-framework for understanding the universe's complexity, guided by principles of systemic coherence and the attributes of viable systems. It underscores the deep connections that unify diverse phenomena and the dynamic processes that sustain life and consciousness, offering a profound insight into the fabric of reality. |
Recognition of "conceptual chords" -- as triadic combinations
- Poly nesting / projections ***
- polyhedral combinatorics
- Isomorph
- Maths of 36
- Torah righteius
- rhinogame
- On s'occupe pas de 36, 36 s'occupe de vous
- Dimensions 6 stella
- Games berne katas *
- Strategies polti cinese *
- Canon song *
- Yoga laban yoga poses
- Conway tonnetz
- Gods -- astro/decans -- compatibility
Bruce Alderman Sex, Pronouns, and Prepositions: how an integral mathematics of perspectives can stop the AI apocalypse Integral Life, 18 March 2024
- integral grammatology,
- 16 grammar Esperanto -- Korsybski 16 ? ***
- array of prepositions / Thom / 150 English operators prefixes
jitterbug embodment -- cuboctahedra 3
cell splitting
- There is little agreement regarding attempts to find a chiastic structure. ***
- chiastic Chisstic structure and square of opposition
- https://en.m.wikipedia.org/wiki/Chiastic_structure
Opposition geometry and canon -- symmetry polyhedra -- synaestthesia???? *******************
*** no consideration to positioning
A comprehensive approach to cycles (but not to their representation or interrelationship) is produced by the Foundation for the Study of Cycles (Classic Cycles Library, 1987). It notably explores cycles of importance to business. More than 500 different phenomena in 36 different areas of knowledge have been found to fluctuate in rhythmic cycles.
move refs
References
Peter Gärdenfors:
- Conceptual Spaces: the geometry of thought. MIT Press, 2000.
- Conceptual spaces as a framework for knowledge representations. Mind and Matter, 2, 2004, 2
- Semantics, Conceptual Spaces and Music. Acta Philosophica Fennica, 43, 1988
G. Oliveri. Mathematics: a science of patterns? Synthese, 112, 1997, 3
J. Petitot. Morphodynamics and the categorical perception of phonological units. Theoretical Linguistics 15, 1989
Steven Schockaert. Knowledge Representation with Conceptual Spaces. Proceedings of the Third Workshop on Semantic Deep Learning (Association for Computational Linguistics), 2018 [text]
R. N. Shepard. Geometrical approximations to the structure of musical pitch. Psychological Review, 89, 1982, 4
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