Brain organization, cognition, comprehension -- and music

Framing Cognitive Space for Higher Order Coherence (Part #6)

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Tuning systems and music: Whilst the complexity indicated above is variously beyond ordinary comprehension, as formally presented or illustrated, it is surprising to note the capacity of many to engage with higher orders of complexity through music.

Given the emphasis above on the apparent inexplicability of different patterns of N-foldness in the distinction of sets of concepts and strategies, there is a strong case for exploring the extent to which the sense of coherence ("feeling right") may derive in part from unrecognized preference for particular musical tuning systems. This possibility merits exploration in the light of Cognitive constraints in comprehending strategic coherence (2019) and chunking required for memorability, dating from the famed paper of George Miller (The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing information, Psychological Review. 63, 1956, 2).

The validity of this widely accepted argument is now challenged (Nelson Cowan, The Magical Number 4 in Short-term Memory: a reconsideration of mental storage capacity. Behavioral and Brain Sciences. 24, 2001, 1; Nelson Cowan, et al., The Legend of the Magical Number Seven, 2007; Timothy Brady, et al., A Review of Visual Memory Capacity: beyond individual items and toward structured representations, Journal of Vision, 11, 2011, 4).

A tuning system is the system used to define which tones, or pitches, to use when playing music, namely the choice of number and spacing of frequency values used. The creation of a tuning system is complicated because musicians want to make music with more than just a few differing tones -- arguably, as with the creation of sets of concepts or strategies. As the number of tones is increased, conflicts arise in how each tone combines with every other -- again, as in the case of strategies. Finding a successful combination of tunings has been the cause of debate, and has led to the creation of many different tuning systems across the world. Each tuning system has its own characteristics, strengths and weaknesses. The distinctive sets of global strategies could then be explored in this light.

The unexplained preference for 12-fold sets was noted above (Checklist of 12-fold Principles, Plans, Symbols and Concepts: web resources, 2011), Any quest for clarity could then be explored in terms of the 12-note chromatic scale, each note being unique, and the various compromises in tuning it. It is not known why there are 12, as stressed by Daniel White (Potential Mathematical Models for the Western Musical Scale A Historical and Empirical Comparison, 2007; summary). This presents a number of explanatory theories.

A musical scale is any set of musical notes ordered by fundamental frequency or pitch. Scales may be described according to the number of different pitch classes they contain. The notes of a scale form harmonic intervals with each of the other notes of the chord in combination.

Given the confusion with regard to such chunking, the provocative question could be raised -- in the light of the self-referential perspective -- as to whether attention could then be usefully accorded to "14 plus-or-minus-2" rather than "7 plus-or-minus-2". This would neatly encompass the range of strategies from 12 to 16 at least.

Hexany
Of particular interest to the argument with regard to musical tuning systems is the hexany invented by Erv Wilson. This can be thought of as analogous to the octahedron (geometric dual of the cube). The notes are arranged so that each point represents a pitch and every edge and interval with each face represents a triad. It thus has eight just intonation triads where each triad has two notes in common with three of the other chords. Each triad occurs just once with its inversion represented by the opposing 3 tones. The edges of the octahedron show musical intervals between the vertices, usually chosen to be consonant intervals from the harmonic series. The points represent musical notes, and the three notes that make each of the triangular faces represent musical triads. Wilson also pointed out and explored the idea of melodic hexanies. (Robert Walker, Hexany)
	Modified by Robert Walker from Tilman Piesk's Hypercubestar on Wikipedia

Polyhedra and music: This association dates back to the reflections of the pythagoreans and the Harmony of the Spheres. A recent exercise includes that of W. Douglas Maurer (A Musical Suite Based on the Platonic Solids, Bridges: Mathematical Connections in Art, Music, and Science, 2002). Other references include (Polyhedra: frozen music for the eyes; Caspar Schwabe, The Zonohedra Music Chart; Iulia Millesima, The Pythagorean Acoustic: Geometry and Music of Sirens, Labyrinth Designers; Peter Pesic. Music and the Making of Modern Science, MIT Press, 2014; Bruce Stephenson, The Music of the Heavens: Kepler's harmonic astronomy, Princeton University Press, 2014).

Consideration has also been given to organization of musical scales in terms of a hypercube by R. W. Peck (A Hypercube-Graph Model for n-Tone Rows and Relations. In: J. Yust, et al, (eds), Mathematics and Computation in Music, MCM 2013. Lecture Notes in Computer Science, 7937, 2013).

Music and I Ching: One extensive discussion of this relationship is provided by Richard O. Burdick (I Ching as a Structural Foundation for Music). The author offers a relationship of musical scales to the Shao Yong circle of hexagrams (Richard O. Burdick, I Ching Music -- Shau Yung's Circle).

With the 8-eements of the BaGua associated with the musical octave, the images above can be further modified to suggest a mapping of the hexagram houses associated with each trigram, as shown below (Organization of I Ching hexagrams in terms of traditional "houses", 1995).

Indication of BaGua "sub-cubes" positioned outward along diagonals
Screen shot of virtual reality image (as above with trigram coding)	Trigrams as "houses" from image on left (with associated hexagrams)	Sub-cube movement outward on diagonals (animation)

Strategic credibility: Such arguments usefully frame the challenge of comprehending the nature of "belief" in any N-fold strategic framework or set of "goal" -- and its credibility when communicated widely. Since equivalent sets have long been a feature of the theology of different religions, there is a case for exploring such belief otherwise (Mathematical Theology -- Future Science of Confidence in Belief, 2011; Gregory Benford, Applied Mathematical Theology, Nature, 440, 2006; James Bradley, Theology and Mathematics, Theology and Science, 9, 2011, 1)

The chunking issue can be usefully related to the central preoccupation of this argument as framed by the cube:

In arguing the point through a musical metaphor, there is an irony to any criticism of the study by Jacques Attali ( Demain, qui gouvernera le monde? 2011) in that Attali himself (in a previous study) specifically indicated that cultures articulated their social organization through the musical structure favoured in the immediate past (Noise: the political economy of music, 1977). Thus he specifically relates the currently favoured pattern of organization to that of classical Western music. As discussed separately, with respect to "tomorrow", should then at least take account of the pattern of music currently favoured by the voters of the future (Tomorrow, Who Will Govern the World? 2011).

Neuroscience: The arguments above merit comparison with the results of recent neuroscience research indicating the remarkable possibility of cognitive processes of up to 11-dimensional form in the light of emergent neuronal connectivity in the human brain. As summarized:

Using mathematics in a novel way in neuroscience, the Blue Brain Project shows that the brain operates on many dimensions, not just the three dimensions that we are accustomed to. For most people, it is a stretch of the imagination to understand the world in four dimensions but a new study has discovered structures in the brain with up to eleven dimensions - ground-breaking work that is beginning to reveal the brain's deepest architectural secrets..... these structures arise when a group of neurons forms a clique: each neuron connects to every other neuron in the group in a very specific way that generates a precise geometric object. The more neurons there are in a clique, the higher the dimension of the geometric object. ...

The appearance of high-dimensional cavities when the brain is processing information means that the neurons in the network react to stimuli in an extremely organized manner. It is as if the brain reacts to a stimulus by building then razing a tower of multi-dimensional blocks, starting with rods (1D), then planks (2D), then cubes (3D), and then more complex geometries with 4D, 5D, etc. The progression of activity through the brain resembles a multi-dimensional sandcastle that materializes out of the sand and then disintegrates. (Blue Brain Team Discovers a Multi-Dimensional Universe in Brain Networks, Frontiers Communications in Neuroscience 12 June 2017)

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