Musical facilitation of integrative comprehension

Implication of Toroidal Transformation of the Crown of Thorns: Design challenge to enable integrative comprehension (Part #13)

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It is appropriate to note the extent to which music is widely appreciated and constitutes a basis for social integration and coherence. Elsewhere a case has been made for recognizing insights into musical harmony as a key to the elusive coherence required for global governance (A Singable Earth Charter, EU Constitution or Global Ethic? 2006; Polarities as Pluckable Tensed Strings: hypercomprehension through harmonics of value-based choice-making, 2006; Structuring Mnemonic Encoding of Development Plans and Ethical Charters using Musical Leitmotivs, 2001). Insight into polyphony as also been considered (All Blacks of Davos vs All Greens of Porto Alegre: reframing global strategic discord through polyphony? 2007).

As mentioned there, rich consideration has been given to the cognitive role of music by Douglas Hofstadter (Gödel, Escher, Bach: an Eternal Golden Braid, 1979).

Cognitive role of music: Especially relevant to the above argument are indications from recent research regarding the organization of music, especially in cognitive terms based on the torus. As previously discussed (Memorability: musical clues to psychosocial system sustainability, 2006), the relation of music to the functioning of the brain is a theme in the cognitive neurosciences [more].

Research by Petr Janata et al (The Cortical Topography of Tonal Structures Underlying Western Music, Science, 13 December 2002, 298. 5601, pp. 2167-2170) has indicated that knowledge about the harmonic relationships of music is maintained in the rostromedial prefrontal cortex providing a stronger foundation for the link between music, emotion and the brain. The melody used experimentally was crafted to shift in particular ways through all 24 major and minor keys. The relationships between the keys, representative of Western music, create a geometric pattern in the form of a torus (see Petr Janata, Music Mapped to the Torus, 2005, and torus dynamics movies).

The piece of music moves around on the surface of the torus offering a means of determining the pure representation of the underlying musical structure in the brain. The work clarified the mapping of melodies in the brain, as it varied from one occasion to another suggesting that the map is maintained as a changing or dynamic topography. This dynamic map may provide the key to understanding why a piece of music may elicit different behaviours at different times [more more] (see also Robert J. Zatorre and Carol L. Krumhansl, Mental Models and Musical Minds, Science, 13 December 2002: 298. 5601, pp. 2138-2139). Of particular interest was the role of any such mapping in the memorability of favourite tunes.

Toroidal harmonic space: The torus may be used as a representation of harmonic space. A piece of music moves around in this space [more]. The results of psychoacoustic experiments by C L Krumhansl and E J Kessler (Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys, Psychological Review 89(4), 1982, pp. 334-368) of the inter-key relations of all major and minor keys can be represented geometrically on a torus -- as shown by Benjamin Blankertz, Hendrik Purwins and Klaus Obermayer (Constant Q Profiles and Toroidal Models of Inter-Key Relations -- ToMIR, 1999) in the following image

Fig. 15: Geometric representation of the inter-key relations of all major and minor keys (derived from psychoacoustic experiments by Krumhansl and Kessler)

Thomas Fiore (Music and Mathematics; Beethoven and the Torus) addresses one of the central concerns of music theory, namely to find a good way to hear a piece of music and to communicate that way of hearing. He notes that music theorists make much use of mathematics in creating conceptual categories enabling them to develop taxonomies and classifications of the various sets that arise in music. In particular he shows how group theory offers a way of describing the ways that sets and pitches relate and how they can be transformed from one to another. Fiore focuses in particular on the musical relevance of the PLR group:

This is a set of functions whose inputs are major and minor chords and whose outputs are major and minor chords. These musical functions go back to the music theorist Hugo Riemann (1849-1919). As a result, the PLR group is sometimes called the neo-Riemannian group [see also Edward Gollin, Neo-Riemannian Theory, 2005].

Using the PLR group Fiore shows that the harmonic progression in the second movement of Beethoven's Ninth Symphony traces out a path on a torus. Beethoven's Ode to Joy (in the fourth and ninth movement of that symphony) has been adopted as the official anthem of Europe (Council of Europe, 1972; European Union 1985).

Given the passing recognition by the European Commission of the role of the torus in multi-agent modelling relating to sustainable development (see above), it might be asked whether a key to sustainability for Europe lies not "under the noses" of policy makers but rather "behind their ears". Of course there is indeed the possibility that odour may play an unsuspected role as suggested by Chris C. King (Fractal and Chaotic Dynamics in Nervous Systems, 1991) and may also be usefully mapped onto a torus. There may be unsuspected higher orders of significance to references in political discourse as to whether strategies "stink" or "smell right"!

Torus and tonnetz: In a helpful overview Justin London (Some Non-Isomorphisms Between Pitch and Time, Music Theory Midwest, April 2001) points to the recognition of the role of the torus by Brian Hyer (Re-Imagining Riemann, Journal of Music Theory 39(1), 1995, pp. 101-38) and Richard Cohn (Neo-Riemannian Operators, Parsimonious Trichords, and their Tonnetz Representations. Journal of Music Theory, 41(1), 1997, pp. 1-66). The tonnetz on which he focuses is a tonal lattice invented by Hugo Riemann as a model for just intonation. He notes that:

Music theorists are not alone in recognizing the torroidal shape of tonal space. Researchers in music perception and cognition have empirically measured the goodness-of-fit for notes and chords in a tonally-primed context, and they too have mapped tonal space onto the surface of a torus.

London concludes:

... metric space is planar, tonal space is non-planar; therefore the two spaces are non-isomorphic. And if the two spaces are non-isomorphic, then there are fundamental problems in trying to map elements or relationships (i.e., functions which employ those elements) from one space to another....What is gained in this exercise is that by trying to follow the same "rules" in constructing graphic representations of tonal and metric relationships, we are forced to confront the differences between them. We also are reminded of how the topologies of ... the metric tree and the tonal tonnetz -- arise from the combination of formal relationships among their component elements as well as the way human beings hear and understand those relationships. As in all of our musical representations, what we can hear and what we can imagine are intertwined and interdependent.

Justin Hoffman (Listening with Two Ears: conflicting perceptions of space in tonal music, 2011): ***

The Tonnetz is a spatial model of tonal pitch, constructed by placing fifths along the horizontal axis of a coordinate plane and thirds along the vertical axis.... Though the Tonnetz can be constructed in several different ways, all of these representations have in common that they appear in some type of space (whether a coordinate plane or a non-Euclidian torus) with fifth-related pitches along the horizontal axis and third-related pitches along the vertical axis.... The second space, the pitch-class Tonnetz, is a torus, a finite space representing relationships among the twelve pitch classes that result from equal temperament

Music and sustainability: Given the remarkable sustainability of religious orders, monasteries and ashrams, it is worth exploring the role that regular use of bells -- audible throughout the community -- have played in ensuring coherence. Such possibilities are linked to understandings of the cognitive organization associated with any understanding of "attunement" in relation to individual or collective meditation (as well as to the torus-like halo associated with holy people, and even a particular form of clerical tonsure).

Is it possible that particular patterns -- mappable onto a torus -- are more appropriate to ensuring community sustainability? What is to be learnt from bell tuning in ancient China in this respect -- in sustaining scattered communities within the imperial domain? Especially noteworthy was the function of the Yellow Bell in ancient China -- a carefully tuned instrument upon which all other pitches and measurements were based, which was carefully reworked by every new emperor [more | more]. Might this be a key to the challenges of development in Africa (cf Knowledge Gardening through Music: patterns of coherence for future African management as an alternative to Project Logic, 2000)

Orbifolds: Especially relevant to this argument is the work of Dmitri Tymoczko (The Geometry of Musical Chords, Science, 313. 5783, 7 July 2007, pp. 72 - 74):

A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by utilizing short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant chords possess different near-symmetries, and suggest different musical uses.

Of interest to a potential ambiguity in the significance attached to the Knight's move as mapped onto a torus, in a subsequent work Dmitri Tymoczko (The Generalized Tonnetz, 2011) relates two categories of music-theoretical graphs, those in which points represent chords and those in which points represent notes.

Music theorists typically represent voice-leading possibilities using two different types of graph. In note-based graphs, notes correspond to points and chords are represented by extended shapes of some kind; the prototypical example is the Tonnetz, where major and minor triads are triangles, and efficient voice leadings are reflections ("flips") preserving a triangle's edge. In chord-based graphs, by contrast, each point represents an entire chord and efficient voice leading corresponds to minimal motion in the space, typically along an edge in a discrete graph.

Harmonic relationship between Knight's moves: Such considerations highlight the possibility of insights into ways of ordering the Knight's moves in terms of their harmonic significance -- potentially through their appropriate sonification. Is appropriate integration of relevance to governance then to be understood through patterns of resonance as previously argued? (Liberation of Integration through pattern, oscillation, harmony and embodiment, 1980 with a Fugitive Integration: a musical addendum, 1980). Is it through such insights that a sense of how a "crown" may "work" -- as with a "Round Table"? More provocatively, with what sense of shared harmony do "keynote speakers" now address an "audience".

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