Memorability implied by Euler's isomorphism of musical and polyhedral order?

Identifying Polyhedra Enabling Memorable Strategic Mapping (Part #10)

Degrees of agreeableness: In the quest for greater understanding of the memorability of order through polyhedral patterns, it is somewhat extraordinary to note the role of music in the thinking which framed the articulation of insight into the patterns highlighted above -- effectively prefiguring it. That key insight was formulated by Leonhard Euler (1707-1783) in what is now universally recognized as the Euler characteristic. This was the discovery relating the number of vertexes, edges and faces of a convex polyhedron, namely V-E+F=2.

Euler's early interest in music, which persisted throughout his life, culminated in a focus on what has been translated as "degree of agreeableness' -- a gradus-suavitalis function, as variously explained by the following:

Jozsef Sandor: Euler and Music: a forgotten arithmetic function by Euler (Octogon Mathematical Magazine, 17, 1, 2009)
Roger Mathew Grant: Leonhard Euler's Unfinished Theory of Rhythm (Journal of Music Theory, 57, 2013, 2).
Jordan Alexander Key: Euler, Cowell, Polyhedra and the Music Genome: Leonard Euler's theory of music and its anticipation of modern musical discourse (10 December 2018)
Euler's "Degree of Agreeableness" for Musical Chords (Thats Maths, 9 August 2018)

Although the relation between music and order has been a preoccupation since the Pythagoreans, it could be said that, despite Euler, mathematicians and musicians have essentially gone their separate ways with respect to the implications of the relationship between polyhedra and music. This disassociation would seem to extend to any understanding of memorability, despite the fascination of mathematicians with symmetry -- diffidently framed in terms of the quintessentially non-mathematical reference to "mathematical beauty".

Of particular interest is the manner in which Euler's preoccupation with music can be understood as "prefiguring" his approach to polyhedra. A similar phenomenon has been documented with respect to the philosopher Ludwig Wittgenstein by Susan Sterrett (Wittgenstein Flies a Kite: a story of models of wings and models of the world, 2005; Pictures, Models and Measures, Belgrade Philosophical Annual, 30, 2017). As the author shows in that case, the glimpse of a solution to the problem of language in 1914 had to do with experimental models which had been so crucial to the Wright brothers' solution to the problem of flight. An analogous prefiguration can be speculatively explored in relation to Albert Einstein's fundamental insight (Einstein's Implicit Theory of Relativity -- of Cognitive Property? Unexamined influence of patenting procedures, 2007).

The relation between Euler's insight into musical order and polyhedral order has been described in terms of isomorphism by Peter Pesic (Music and the Making of Modern Science, 2014, Euler's Musical Mathematics. The Mathematical Intelligencer 35, 2013, 2). As Pesic describes Euler's exploration of a "degree of agreeableness" in music:

The structure of this relation between vertices, edges, and faces is strikingly similar to the structure of the degree of agreeableness of musical intervals, s - n +1. Without intending any direct connection between polyhedra and Euler's hierarchy of musical intervals, as such, both these relations (V + F - E = 2 and s - n + 1) give the kind of general categorization we now think of as topological and which Euler thought of in terms of geometria situs. To be sure, these relations are very different, and not just in the objects they describe. Euler's formula is an equation describing a necessary and sufficient condition for closed, convex polyhedra; his formula for musical degree defines a hierarchy between different intervals. They both pose a general schematization that categorizes a vast domain, of polyhedra or of musical intervals, respectively, subsuming many different individuals under a larger genus.

... his criterion for setting up his degrees is freely chosen according to his notions of what would be more "intelligible" and hence more "agreeable" (suavis). 1n his musical work, Euler first devised the general classificatory strategy that he then applied to the bridge problem and later to polyhedra. To use a later mathematical term, his approaches in these cases were isomorphic, that is, they had the same essential structure. Because the musical example came first, it arguably was the arena in which he first found and applied the kind of approach that he later (and perhaps without realizing it) then found appropriate to bridges and polyhedra (2014, p. 148)

The reference to the "bridge problem" relates to Euler's fundamental contribution to graph theory -- with respect to the problem of the so-called Seven Bridges of Königsberg. .

Spectrum musicum and Gradus suavitatis: As further clarified in the Wikipedia entry with respect to Euler's music theory:

A first point of Euler's musical theory is the definition of "genres", i.e. of possible divisions of the octave using the prime numbers 3 and 5. Euler describes 18 such genres, with the general definition 2^mA, where A is the "exponent" of the genre (i.e. the sum of the exponents of 3 and 5) and 2^m (where "m is an indefinite number, small or large, so long as the sounds are perceptible" (Leonhard Euler, Tentamen novae theoriae musicae, St Petersburg, 1739, p. 115), expresses that the relation holds independently of the number of octaves concerned... Genre 18 (2^m.3³.5²) is the "diatonico-chromatic", "used generally in all compositions"...Euler later envisaged the possibility of describing genres including the prime number 7. Euler devised a specific graph, the Speculum musicum, to illustrate the diatonico-chromatic genre, and discussed paths in this graph for specific intervals, recalling his interest in the Seven Bridges of Königsberg (Euler, 1739, p. 147; Euler, De harmoniae veris principiis, 1774, p. 350)... Euler further used the principle of the "exponent" to propose a derivation of the gradus suavitatis (degree of suavity, of agreeableness) of intervals and chords from their prime factors -- one must keep in mind that he considered just intonation, i.e. 1 and the prime numbers 3 and 5 only.

As articulated by Jordan Alexander Key:

...Euler used the principle of the "exponent" when comparing two similar musical structures (pitches in harmonies, rhythmic values in rhythmic successions, etc.) to propose a derivation of what he termed the "gradus suavitatis" or "degree of agreeableness" between these structures. This degree could then be used to compare seemingly similar or dissimilar structures in music (different harmonies, rhythms, melodies, meters, forms, etc.); through this mathematically derived comparison, deep similarities could be uncovered that might not lie at the surface of the music or which might be obscured due to historical aesthetic bias. (Euler, Cowell, Polyhedra and the Music Genome: Leonard Euler's theory of music and its anticipation of modern musical discourse, 10 December 2018)

Of particular relevance to the role of music in relation to polyhedra framed in terms of factoral exponents (as above), is the use of exponential analysis by Ernest McClain as a musicologist with an historical preoccupation of significance to governance (Myth of Invariance: the origins of the gods, mathematics and music from the Rg Veda to Plato, 1976; The Pythagorean Plato: prelude to the song itself, 1978; Meditations Through the Quran: tonal images in an oral culture, 1981; Music and Deep Memory: speculations in ancient mathematics, tuning, and tradition, 2018).

With respect to any insight into memorability enabled by either music or polyhedra, it is the relation between Euler's "agreeableness" and the "beauty" -- which mathematicians struggle to relate to symmetry -- that calls for further clarification, notably in the light of comprehensibility and communicability. Euler's effort to recognize a "degree of agreeableness" suggests the possibility of an insight into a "degree of memorability", as this might contrast with "degrees of beauty" -- especially given the controversy which such recognition would imply from a cross-cultural perspective (Leonhard Euler, On the true principles of harmony as presented through the Speculum Musicum, St. Petersburg Academy, 1773; translated by Larry Blaine and Douglas Kendall).

The potential for necessary (?) "controversy" with respect to any relevance to memorable strategic articulations is usefully implied in a thesis on consonance in music by Julíán Villegas (Local Consonance Maximization in Realtime, University of Aizu, 2006). Specifically discussing the gradus suavitatis, Villegas introduces his argument in the following terms:

There seems to be agreement that music is more interesting when passages of 'tension' and 'relaxation' alternate while it is performed. Interchangeable expressions for the same patterns include 'pleasant' and 'unpleasant,' 'rest' and 'motion,' 'euphony' and 'cacophony,' etc. Different explanations for the subjective perception of these patterns exist and depend on the context in which they are studied: physiology, music, psychology, genetics, etc. The psychoacoustical explanation is one of the most accepted. According to this theory, the separation in frequency of tones sounding concurrently determines directly the consonance; the more separated they are, the more 'relaxed' their interaction is perceived. In general, it's desirable that in the alternation of tension and relaxation sections, the sounds corresponding to the relaxation state have a maximum consonance, so the problem of achieving this can equivalently be considered the problem of maximizing the consonance of the interaction of simultaneous tones at a given time. These issues have been addressed for about six centuries, since musicians started to formalize the use of multiple sounds at the same time coherently. (p. 3, emphasis added)

Given the degree of controversy with regard to any strategy in practice, the insight into "harmony" offered by music -- in the light of the recognized role of dissonance -- suggests that such necessary "dissonance" could be usefully used to reframe simplistic understandings of "positive versus negative", "agreement versus disagreement" and "consensus versus dissent" (The Consensus Delusion: mysterious attractor undermining global civilization as currently imagined, 2011). Arguably some new form of alternation merits consideration to engage more fruitfully with the prevailing degree of divisive fragmentation (Development through Alternation, 1983). Such fragmentation is of course curiously echoed in the world of music.

Of particular interest, given the reference above to 17 parabolic orbifiolds, it is appropriate to note the role that orbifold structure has been held to play in the organization of music as articulated by Dmitri Tymoczko, who models musical chords as points in orbifold space (The Geometry of Music, 2011). As discussed separately (Musical implications of orbifolds for comprehension of questioning dynamics, 2014), there is the possibility that the distinctive cognitive feel for logical distinctions and connectivity might be associated with chords.

Dissonance and memorability: This emphasis on the requisite interrelationship between "pleasant" and "unpleasant" merits consideration in imagining how any global strategy might be rendered memorable through song or music, as can be otherwise considered (A Singable Earth Charter, EU Constitution or Global Ethic? 2006; Aesthetics of Governance in the Year 2490, 1990; Reversing the Anthem of Europe to Signal Distress, 2016).

To whatever degree dissonance is to be be considered essential to memorability, in his review of dissonance Villegas notes (p. 16):

A successful single theory to explain consonance and dissonance remains elusive, and some theoreticians and researchers argue that there's no single theory that could explain it but a set of them....:

frequency ratio: the auditory 'preference' for small integer ratios, because of the resulting periodicity of the stimuli,

harmonic relationship: the expected dissonance when the harmonic relationships of a composition doesn't follow the classical canons of western harmony,

temporal dissonance: related to the beating of a pair of sounds when the difference of their frequencies is small enough to partially cancel the effect of each other (amplitude modulation),

tonal fusion: the perceived euphony of simultaneous sounds that can be perceived as a single tone,

tonotopic dissonance: the perceived dissonance of a pair of sinusoidal waves when their frequency difference is less than one critical bandwidth,

virtual pitch: the component of dissonance that arises from competing (unclear) virtual pitches,

expectation dissonance: alterations on learned harmonic patterns, as in 'cadences' where a leading tone resolves to a different note than the tonic or its equivalent,

interval category: or the difficulty to classify the formed interval of a pair of sounds into a learned category of intervals,

absolute pitch category: the perceived dissonance of a tone by a person with absolute pitch when it's impossible to classify it into one of the learned pitches due to the ambiguity of its frequency

stream incoherence theory: the component of dissonance that arises due to confusion regarding streaming, and

relative dissonance: the contextual relative consonance of a sonority when it is preceded by other sonorities of contrasting dissonance. This effect is related to the sensation of rest and peace experienced when the most dense and dark dissonant composition, listened as loud can be stood, finishes

The metaphor of a "memory theatre", as mentioned above, could be usefully extended to that of a "concert hall", even an open-air concert hall -- given the manner in which any performance might be rendered "memorable". The concert metaphor has however already been borrowed in reference to a "concert of democracies" as an alternative form of international organization -- but without any consideration of memorability (Ivo H. Daalder, Who and Why: The Concert of Democracies, Brookings, 15 December 2006; Beyond a "Concert of Democracies"? 2011).

Poetry-making and policy-making: The question of what makes music memorable, in contrast to the kind of strategic laundry list exemplified by the UN's Sustainable Development Goals, is presumably a matter of great concern to the music industry in competitively marketing its products (Eleanor Crane, et al, Musical Hit Detection, 12 December 2008).

With respect to governance, a potentially useful approach is through recognition of the extent to which iconic leaders -- of contrasting political persuasion -- have claimed a particular interest in poetry (Poetic Engagement with Afghanistan, Caucasus and Iran: an unexplored strategic opportunity? 2009). From that perspective it may be asked whether the relations between the elements of a poem, most obviously through "rhyme", complement and enhance the strategic thinking otherwise associated with "reason". How the articulation of strategic elements "rhyme" then invites investigation -- as a complement to any understanding of cybernetic feedback loops between them through "reason" (Poetry-making and Policy-making: arranging a Marriage between Beauty and the Beast, 1993).

The argument can be reinforced by the much-publicized engagement with haiku of Dag Hammarsköld, an early Secretary-General of the UN, and of Herman Van Rompuy, as a recent President of the European Council. As noted above, curiously the 17-fold pattern of the UN's SDGs has been celebrated in the 17-fold organization of haiku poetry, itself supported by UNDP (Inspired by Nature: Celebrating Biodiversity with Haikus, UNDP, 22 May 2017). The notable subtlety of haiku merits reflection on the insights it offers into strategic resilience, as argued separately (Ensuring Strategic Resilience through Haiku Patterns: reframing the scope of the "martial arts" in response to strategic threats, 2006).

Poetry can be especially valuable in interrelating into a coherent pattern themes evoked in successive parts of a poem -- effectively feeding back and forward in a manner reminiscent of the positive and negative feedback valued from a cybernetic perspective. This ability may be reflected in the thematic content, perhaps most succinctly indicated by the poet John Keats as;

... Negative Capability, that is, when a man is capable of being in uncertainties, mysteries, doubts, without any irritable reaching after fact and reason... [rather than] being incapable of remaining content with half-knowledge (1899)

Nonlinearity and multidimensionality: Negative capability is otherwise comparable with the Zen Buddhist understanding of shoshin (Christian Jarrett, How to foster "shoshin", Aeon-Psyche). This contrasts with the assumption of perfection, or the righteous "unidirectional" quest for a form of unity, characteristic of many belief systems having problematic dynamics with whatever calls them into question. Paradoxically that process of designing out any sense of imperfection, from what they assume they embody, engenders a deniable process of enantiodromia whereby they eventually embody imperfection of a kind. This is especially evident in religions (most obviously Catholicism at the present time), political ideologies, and science itself (Knowledge Processes Neglected by Science: insights from the crisis of science and belief, 2012).

The absence of any poetic aspiration is especially evident in the essentially linear articulations of conventional strategies and principles -- dependent on forgettable "reason" in the absence of memorable "rhyme". In their assumption of righteousness, they could indeed be understood as characterized by "negative incapability" -- otherwise evident in avoidance of any form of negative feedback or acknowledgement of ignorance. This is currently a documented characteristic of world leaders.

In contrast with the asystemic approach of strategic linearity, this can be understood as an essentially nonlinear narrative, whether poetic or note (Difference Between Linear and Nonlinear Text, Difference Between, 18 June 2018). It is in this sense that any linear understanding of degrees of agreeability, unity, perfection or memorability can be fruitfully called into question. As with polyhedral forms, strategies may be differently memorable and variously exemplifying unity.