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28th June 2006 | Draft

Polarities as Pluckable Tensed Strings

Hypercomprehension through harmonics of value-based choice-making

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Annex to Hyperaction through Hypercomprehension and Hyperdrive:
necessary complement to proliferation of hypermedia in hypersociety


Pythagorean resonances: vibrating strings?
Standing waves and emergence of "objects"
"String plucking" as a metaphor for value-based choice-making
Variety of "pluckable" polarities
Integration by interlocking resonances
Storage: density and organization of self-reflexive loops?
References


Introduction

One approach to modelling "hypercomprehension" might be through the dynamics of vibrating strings fundamental to the harmonics of many musical instruments

Pythagorean resonances: vibrating strings?

The dynamics of vibrating strings fundamental to the harmonics of many musical instruments was first explored in western culture by Pythagoras (ca 500 BC). He observed that two plucked strings sounded pleasant together when their lengths were in the proportion of two small integers.The scientific basis of these observations for music theory was later established by Hermann von Helmholtz (On the Sensations of Tone as a Physiological Basis for the Theory of Music, 1863). Efforts are now made to model such phenomena through computational simulations of plucked-string instruments as an extension of physical modelling of musical instruments and model-based sound synthesis (cf M. Karjalainen, et al Plucked-string models: from Karplus-Strong algorithm to digital waveguides and beyond, 1998). The Karplus-Strong algorithm is a physical method of string synthesis that simulates the sound of a hammered or plucked string or some types of percussion.

The plucked string has been significantly used as a metaphor in different contexts (cf Drew W. Hempel, Epicenters of Justice: music theory, sound-current nondualism and radical ecology, 2000). Such a string could be understood as exemplifying a cognitive polarity. This then suggests the possibility of significance associated with particular intermediate positions between the polar extremes -- despite the logical law of excluded middle, known otherwise as the principle of tertium non datur. The suggestion also reframes discussion relating to fuzzy logic.

The behaviour of a vibrating string, excited by plucking, can be described in terms of two travelling waves traversing the string in opposite directions and then reflecting back from the terminating points. In the physics of a plucked string, the pitch (or frequency of tone) is dependent on:

Does the cognitive response to a recognized polarity indeed correspond in any way to the "plucking" of such a string undergene tension? People do effectively "play" with polarities, variously positioning themselves in relation to the extremes. Typically one "plays" with several polarities simultaneously. This is the stuff of conversation with others, of politics, and of internal dialogue with oneself. The range of "tones" evoked in this way may of course be narrow (even monotonous) or cover a wide scale -- even many octaves. The relevance of the metaphor is widely recognized in the use of the term "keynote speaker" -- who may well be experienced as "monotonous". However this metaphor has not been explored to determine the choice of "key" and the consequence for the pattern of relationship between the other "notes" at any such event.

But a plucked string does not just vibrate at a single frequency. It simultaneously vibrates at a whole series of frequencies termed the harmonics. If one string is twice the length of the other, then its lowest harmonic is at half the frequency of the other string's, and its harmonics coincide with the odd-numbered harmonics of the other string -- in a manner pleasant to the ear. In contrast, if the ratio is 1.4 to 1, then there is essentially no regular relationship between the two sets of frequencies, and many of the harmonics lie close enough in frequency to produce unpleasant beats.

Standing waves and emergence of "objects"

An overtone is a sinusoidal component of a waveform, of greater frequency than its fundamental frequency. Usually the first overtone is the second harmonic, the second overtone is the third harmonic, etc. Of particular significance is the manner in which the displacement induced by the plucked string travels from the point of plucking to both fixed ends, where it is inverted to move back in the opposite direction -- the phenomenon of the travelling wave. Interaction between such travelling waves can give rise to a standing wave (cf Joe Wolfe, Strings, standing waves and harmonics). From a mythological perspective, the striking resemblance of the sinusoidal wave to phenomena described as "snake-like" is worth consideration.

In a cognitive world characterized by polarities of every kind, it is worth considering whether "objects" of any permanence, to which cognition attends, are the manifestation of "cognitive standing waves" associated with standing waves in the human brain (cf discussion in Paul L. Nunez, Neocortical Dynamics and EEG, 1994; Matt C. Keener, Resonance Phenomena and Quantum Resonance Theory, 2000) . Their "reality", and existence, is then a consequence of "where, and how, the string was plucked" between relevant polarities. This argument focuses specifically on the "objects" to which cognition is attending in the moment -- not on those which it ignores. Again the emergence of objects of some permanence from "snake-like" dynamics is worth consideration given the mythological association between the snake and knowledge -- and the constraining archetypal polarity of "good" and "evil" (to say nothing of the subsequent challenge of the Ouroboros).

The phenomenon of standing waves has also been the subject of significant attention on 2-dimensional vibrated surfaces as Chladni patterns -- especially from the perspective of the acoustics of musical instruments [more]. Rather than being "plucked" as in the case of tensed strings, the surfaces are "bowed" using a violin bow (cf Paul Bourke, Chladni Plate Mathematics, 2003). Again definable "objects" emerge from disorder -- raising questions as to how explicate order emerges to comprehension, perhaps in terms of some form of "cognitive vibration" (cf David Bohm, Wholeness and the Implicate Order, 1980). The approach has been extended to 3-dimensional Chladni plate interference surfaces. These are defined as positions where N harmonics cancel -- giving rise to a rich set of surfaces from 3 orthogonal harmonics (cf Paul Bourke, Chladni plate interference surfaces, 2001). Chladni figures might be seen as having a form reminiscent of yantras. They became a central metaphor for Danish physicist Hans Christian Ørsted in his reflections on the relationship between intellect and understanding [more].

This series raises the question of cognition within a dynamic configuration of multiple polarities together engendering more complex standing wave patterns. How might comprehension within this context take a form to be understood as "hypercomprehension"?

Further reflections in support of such understanding are to be found in concern with the emergence of macro-structures in systems biology. As noted by Stuart A. Newman (The Fall and Rise of Systems Biology), such preoccupations originated in the early work of William Bateson (Materials for the Study of Variation, 1894) who was concerned with the repetitive organization of certain animal parts, such as the segments of earthworms, the backbones of vertebrates, and the digits of the hand. Bateson proposed a physical metaphor for the generation of such repetitions in terms of Chladni figures. The knowledge at that time did not permit making a mechanistic connection between global organizing principles and a privileged set of small-scale processes, namely interactions of genes and their products.

In the 1950s, however, the mathematician Alan Türing showed that reacting and diffusing molecules in living tissues could spontaneously arrange into Chladni-like concentration distributions, thus providing the conceptual link that Bateson lacked. Newman concludes that, as of 1995, no satisfying picture had appeared of how processes capable of generating organized forms, structures, and behaviors had emerged over the course of evolution -- for reasons he sees as primarily ideological. He sees a new approach to systems biology -- 'integrative biology' and 'biocomplexity" -- following the "failed promises of genetic reductionism". The argument is further developed by Denis Noble (The Music of Life: beyond the Genome, 2006) who rejects the computer metaphor of life being "programmed" by genes in favour of life as polyphonic music.

"String plucking" as a metaphor for value-based choice-making

If, as suggested above, the cognitive response to a recognized polarity does indeed correspond to the way in which a tensed string is "plucked", then this process is effectively a rich model of choice-making or decision-making. The choice is "where to pluck" between the polar extremes. This decision is governed by the kind of tone it is designed to generate. Making music is a process of making decisions, especially when there is any degree of improvisation -- rather than reproducing the decisions of the composer of a score. Such decisions, as in other arenas, are effectively made in relation to a number of distinct "tensed strings" -- the polarities which challenge and enable decision-making. Value-based choice-making may therefore be explored as "playing on a stringed instrument" -- a guitar, for example.

The choice of "where to pluck" any string is a judicious exercise in engendering a particular (tonal) value, appropriately balanced against other such values. The implications of this choice could indeed be further explored in terms of:

In musical terms, the choice of "where to pluck" is guided by a prior choice of tuning system -- effectively a paradigm. A tuning system defines which tones, or pitches, to use when playing music. It is therefore the choice of the number and spacing of the frequency values which are used. Because of the psychoacoustic properties of tones (as studied by Helmholtz), various pitch combinations will sound more or less "natural" when used in combination. For musicians the creation of a tuning system is complicated because of their need to make music with a wider variety of distinct tones. As the number of these tones is increased, conflicts arise in how each tone combines with every other.

As with the elaboration of sets of non-musical values:

The range of all twelve pitches of the Western tempered scale is termed the chromatic scale. Since it it is impossible to tune the twelve-note chromatic scale so that all intervals are "perfect", many different methods have been explored [more]. Each represents a particular set of compromises. The following examples can be usefully reviewed in the light of the implications for value-based choices -- between polar extremes -- in other fields:

Other tuning scale systems include both traditional forms and recent proposals and experiments. In Indonesia the tunings of the 5 notes of the gamelan music are intentionally different for each orchestra -- each has its own harmonic personality. Scales of 22 steps are used in India. Arab melodies use tones half-way between western notes, giving rise to 24 notes. Australian aborigines chant to a 2 note scale. Most Chinese music is based on the five-tone, or pentatonic, scale, although the seven-tone scale is also used. (cf Martin Braun, Bell Tuning in Ancient China: a six-tone scale in a 12-tone system based on fifths and thirds, 2003).

The "compromises" in each tuning system give a sense of the challenge in musical terms to comprehension of "appropriateness" -- as judged by the ear. This may well be an excellent model of the "design" problem of "goodness of fit" and balancing the dimensions of appropriateness in relation to other values (cf Comprehension of Appropriateness, 1986; I. Narsky, Goodness of Fit: what do we really want to know? 2003). This helps to highlight the nature of the challenge of defining universal values, notions and any sense of balance. In fact it is not a matter of definition, unless closure on exploration is sought through commitment to a particular tuning system precluding other explorations. The challenge for governance of any kind is rather one of continual creativity in exploring possible definitions and their particular lilitations (cf Poetry making and Policy making: arranging a marriage between Beauty and the Beast, 1993).

Variety of "pluckable" polarities

It is useful to consider how different kind of polarities are reframed in terms of the tonal (possibly musical) metaphor of cognition.

"Spatial": Interesting possibilities include:

"Temporal": Different approaches to a polarized understanding of time could be considered:

"Subject-Object": Polarities of this nature typically give rise to extreme forms of polarization in practice with relatively little conscious recognition of intermediate situations. Such situations are however regularly experienced, notably in human relationships.

"Reality-Hyperreality": As noted earlier, hyperreality is a situation in which nothing and everything is "real"; it is a situation in which people have lost the ability to distinguish reality and fiction [more]. However this also suggests a wide spectrum of conditions between the unquestionably "real" and the purely "fictional" -- and many complex combinations with which many people are familiar in daily life.

Polarized ethical preoccupations: These polarities underlie and condition many debates. The polarization precludes exploration of the intermediary zones in which are obliged to live their daily lives and which typically reflect the compromises of governance.

Polarized preoccupations of governance: The following polarities, as conventionally understood, tend to preclude exploration of the complex patterns of intermediate conditions. It is in relation to these that people live, whether in terms of serendipitous opportunities or unrecognized problems. The identification of these different conditions could be more fruitfully explored using a musical metaphor that highlights harmonious and unharmonious patterns.

Axes of bias: A particularly insightful approach to a set of value polarities is that of W T Jones (The Romantic Syndrome; toward a new methodology in cultural anthropology and the history of ideas, 1961). He demonstrates that the discontinuities in communication can be described in terms of the different positions of the participants (or schools of thought) on seven pre-rational axes of bias. These differences are reflected in aesthetical, theoretical, value, life-style, policy, and action preferences, as well as in the preferred style of discussion. Any difference between people in position "along" an axis gives rise to discontinuity which it is difficult to handle within a rational frame of reference. The axes identified by Jones are:

In the musical metaphor, these axes could then be viewed as strings, "plucked" at particular positions in the course of a given person's discourse in a debate -- effectively characterizing the contribution of that person to the debate.

"Value polarities": The Human Values Project of the Encyclopedia of World Problems and Human Potential distinguished some 220 "value polarities" ("constructive" vs "destructive") as a means of ordering 1,100 value antonyms and synonyms. These polarities were themselves clustered into 45 "types". The project addressed the question of whether the set of values so configured could be seen as self-constraining or self-organizing.

This implied a shift from value sets as linear or tabular arrays to exploration of the possibility of mappings onto various approximations to a sphere as implying an integrated whole. The challenge was to discover how the spherical array got "tensed" and "tuned" by an appropriate disposition of polarities. Given the evident failure of value systems based on value checklists and value matrices, there is a case for exploring spherically mapped values. These offer new, and intuitively appealing, ways of exemplifying: a sense of holism and whole; a sense of integration and interlocking; mutual visibility; a possibility of complexification and decomplexification; a sense of checks and balances that uses differences rather than being undermined by them. The many attempts over recent years to reduce the set of values to 5 to 10 should be seen as a decomplexification which conceals the variety required for practical policy making.

Integration by interlocking resonances

The variants above suggest two distinct approaches to hypercomprehension:

The experiential environment of an individual may therefore be understood as centered at the confluence of a set of polarities -- of which the limbs might be considered a physical exemplification.

Storage: density and organization of self-reflexive loops?

In the configurative approach to integration (above), there is a risk of descriptive objectification/reification that is specifically challenged in the generic (existential) approach. The challenge lies at the intersection of "storage of identity", "mnemo-memetic storage" and "self-reflexivity" -- with the possibility that:

Clues to the modes of storage might then include:

Of particular interest is the manner in which "tones" associated with playing on polarities are fundamental to the storage of meaning within a culture. Pointers include:

Also of interest are the associated psychosocial "energy" processes:


References

Joachim-Ernst Berendt. The Third Ear: on listening to the world. Dorset, Element Books, 1988

Alain Daniélou. Music and the Power of Sound: the influence of tuning and interval on consciousness. Rochester, Inner Traditions International, 1943 (1995)

D C De Roure, D G Cruickshank, D. T. Michaelides, K. R Page and M. J. Weal. On Hyperstructure and Musical Structure. In Proceedings of The Thirteenth ACM Conference on Hypertext and Hypermedia (Hypertext 2002), pp. 95-104 [text]

Joscelyn Godwin. Cosmic Music: three musical keys to the interpretation of reality. West Sotckbridge, Lindisfarne Press, 1987

Drew W. Hempel. Epicenters of Justice: music theory, sound-current nondualism and radical ecology. 2000 [text]

Tom Irvine. An Introduction to Music Theory, 2000 [text]

Anthony Judge:

M. Karjalainen, V. Välimäki, and T. Tolonen. Plucked-string models: from Karplus-Strong algorithm to digital waveguides and beyond. Computer Music Journal, vol. 22, 1998, no. 3, pp. 17-32 [text]

Ernest G. McClain. The Myth of Invariance: the origins of the Gods, Mathematics and Music from the Rg Veda to Plato. Shambhala, 1978

Guy Murchie. The Music of the Spheres. Dover, 1961

Denis Noble. The Music of Life: beyond the Genome. Oxford, 2006

Rudolph Steiner. The Inner Nature of Music and the Experience of Tone. Anthrosophic Press, 1983

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