Challenges to Comprehension Implied by the Logo
of Laetus in Praesens
University of Earth Alternative view of segmented documents via Kairos

1980

Liberation of Integration, Universality and Concord

through pattern, oscillation, harmony and embodiment

- / -


Originally prepared for the 5th Network Meeting (Montreal) of the United Nations University's project on Goals, Processes and Indicators of Development (GPID) as a contribution to the discussion on integration of the findings of the project. [searchable PDF version, with annexes]]

Abstract: Demonstrates that many widely advocated approaches to integration are relatively simple (if not simplistic) options in a context which is subtler and more complex (at least in a mathematical or topological sense). It is however possible that such elegance is also an indicator of vital properties of symmetry, harmony and balance. These are desirable in any domain in which integration is sought and even necessary for that integration to be both brought about and sustained (by its inherent comprehensibility). The relevance of ordered patterns of time is explored, especially in the light of the evolution of concepts of integration in music and harmony seen here as a precursor of new approaches to psycho-social organization. Attention is also drawn to the special significance this has for transforming understanding of possibilities of individual identification with processes ordered over time.


Introduction
Integrative minimalism
Approaches to integration
1. Integrative skills
2. Conventional integration (Annex 1)
3. Ecological integration
4. Meta-models
Integrating incompatibles
1. Use of incompatible modes in practice
2. Model mixes in space/time
Patterns and rhythms (Annex 2)
Oscillation
1. Conventional example
2. Psycho-social interpretation
3. "Multiphase" oscillations
4. Coupled and damped oscillators
Three-dimensional configurations
Organization of time
1. Time-binding
2. Oscillation and musical sound
3. Music
4. Significance of tone
5. Harmony
6. Consonance and dissonance (Annex 3)
7. Harmonic goal
8. Fugitive integration
Liberation of integration
1. The remaining problem
2. "Cross-harmony"
3. Comprehending the language of pattern shifting
4. Limitation of vision-based metaphors
5. Integration through four complementary languages of music
6. Challenges
Conclusions and implications
References


Introduction

This paper is an exploration of the range of ways in which 'things can be put together' or conceived as being interrelated. In undertaking this it is hoped that widely advocated approaches to integration may be shown as simple options in a context of subtler, more complex possibilities, many of which are essentially more 'elegant', if only in the mathematical or topological sense. It is possible, however, that this elegance is also an indicator of vital properties of symmetry, harmony and balance, which are desirable in any domain in which integration is sought and even necessary for that integration to be brought about.

The domains in which integration is of considerable concern may be indicated by reference to the sub-projects of the UN University's project on Goals, Processes and Indicators of Development of its Human and Social Development Programme. For example, how are the different 'models' implicit within the following sub-projects to be integrated in each case, especially when there is some degree of incompatibility between them:

In each case alternatives can be formulated in the light of different patterns of priorities. The question is how such alternatives are to be reconciled in practice in a real world.

Integrative minimalism

Although integrative skills may be successfully applied to a situation, their elusive nature can be partially defined by the ways in which such skills may fail or be used to conceal abuse. The following approaches to integration or synthesis stress the manner in which an impression of integration can be created even if little is achieved. It is partly base on material assembled in a section on 'Integrative Concepts' in the Encyclopedia of World Problems and Human Potential (2) which identifies some 620 such concepts. The bibliography therein has recently been complemented by a bibliography of 1600 items on 'the relationship between and the structure of science, philosophy, and social/political organization' (3).

(a) Reduction in variety: A simple way to ease the integrative problem is to reduce the diversity of elements present in the situation using an argument for standardization and against any 'hodge podge' mixture of elements. This of course eliminates some minority interests. In the extreme case of destructive or 'meltdown' synthesis, all variety is eliminated.

(b) Reduction in quantity: By eliminating a significant number of the elements, the problem may also be eased. The argument that can be used is that they are well-represented by the variety of elements that remain and that any 'proliferation' of elements is disorderly. In practice this results in the absorption of some elements by others, such as in the case of minority groups.

(c) Simplification: Subtleties and nuances, possibly defended by specific minority groups, may be ignored. Interconnecting webs of relations can be ignored.

(d) Tokenism: Emphasis may be placed on the image or desirability of synthesis in order to conceal inability to achieve any steps towards it.

(e) Temporary synthesis: In a dynamic situation it may be possible to achieve some measure of integration in the short-term by ignoring factors temporarily absent or only emerging over longer time cycles.

(f) Coloured synthesis: A significant degree of synthesis may be achieved, but from a particular viewpoint or in terms of a particular mode, approach or strategy. The narrowness of such a synthesis, coloured by the perspective of those who achieve it, may be difficult to communicate within the framework established by that synthesis.

(g) Enforced synthesis: In some instances, as with a dynamic set of minority interests, a form of integration may be imposed by constraining the dynamics (although without reducing the number or variety of the elements).

(h) Dogmatic synthesis: An impression of synthesis may be achieved by stating frequently and forcefully that it has been achieved and thus eroding expectation that a greater degree of synthesis is possible.

(i) Laissez faire synthesis: By reinterpreting the nature of synthesis or integration, it may be deemed to exist under any circumstances as the pattern of interaction amongst the elements. No intervention is required, although if undertaken it would merely add to the pattern of interaction.

(j) Agglomerative synthesis: Appropriate integration may be assumed to have been achieved simply by ensuring the juxtaposition of the various elements or viewpoints. This corresponds to the use of the prefix 'multi' (eg in multidisciplinary). In books reflecting such a multidisciplinary synthesis, it is the binding which provides the synthesis, given the absence of any relationship between the constituent disciplinary chapters.

(k) Comparative or cross-referential synthesis: Integration may be asumed to have been achieved by recording comparisons between the perspectives or elements. This often corresponds to the use of the prefix 'cross' (eg in cross-cultural).

(i) Cross-impact synthesis: Integration may be assumed to have been achieved by taking into account the constraints and feedback loops emerging from other disciplinary perspectives. This may correspond to use of the prefix 'inter-' (eg in interdisciplinary). Note however that it is only with the emergence of a new level of order that a synthesis breakthrough may be said to have occurred (this may correspond to the use of the prefix 'trans-' as in trans-disciplinary).

Approaches to integration

1. Integrative skills

Although during meetings there is much discussion of 'integration' and there are many attempts at producing a 'synthesis', the skills called upon seem to be poorly understood, hard to communicate, and very difficult to put into practice. It is therefore useful to note very different domains where integrative skills are practice successfully, even if it is not immediately clear what can be learnt from them for use in a meeting environment.

(a) Design and composition: This is the process through which creative intuition influences the selection of elements and the manner and proportion in which they are to be balanced -- what is to be put together and how. In each of the following the configuration of elements tends to relate to an emergent focal point:

(b) Managing dynamic situations: This is the process whereby the relationships between a complex set of given elements is kept in focus. Examples are:

(c) Analyzing complex situations: This is clearly oriented to understanding whatever can be analyzed irrespective of whether this leads to broader synthesis. Examples are:

(d) Communicating synthesis: This is the process whereby a sense of wholeness or unity among diverse parts imparted to others, even if only as a symbol or token of what may later be achieved in practice.

(e) Embodying synthesis: Whereas each of the above is in some way a manipulation of synthesis, however necessary, there seem to be instances where a person acts as the focal point for synthesis and is so perceived by those whose interests are reinterpreted and focused in this way. Examples are perhaps:

2. Conventional integration

The most systematic approach to interdisciplinary integration is probably that of Erich Jantsch (4), as outlined in Annex 1. There are two extremes in the conventional approach to 'integration':

The first approach tends to be unsatisfactory in an increasingly complex society in which undesirable conflict does result from incompatibilities between the models. The second tends to be unsatisfactory, either because of conflict arising from what is excluded, or because such fusion is quickly perceived as a constraint on further development if the model is successfully implemented. The more successful it is, the more its proponents will resist any further reconceptualization. Such models do not acknowledge their limitations, the need for their limitations, and the need for their eventual demise. They are conceived as a 'final solution' detached from the processes which brought them (temporarily) into being.

3. Ecological integration

As a first step in the search for subtler patterns of integration we may consider a situation exemplified in ecological systems although typical of more complex organized systems. Consider two species in an ecosystem as being analogues of two conceptual models in a psycho-social system.

The processes that characterize species X may, in the interaction with Y, be accelerated or in some way enhanced (+), or may be unaffected (0), or may be decreased (-). The same is true for species Y.

When such possibilities are cross-tabulated, it becomes evident that there are nine qualitatively different coactions. Ignoring the (0,0) coaction, the remaining eight are as follows:

       + +    symbiosis                     - -    synnecrosis
       + o    commensalism                  - o    amensalism
       + -    parasitism                    - +    predation
       o -    allopathy                     o +    allotrophy

This scheme was elaborated by E Haskell (5) who made it into the basis for a periodic coaction coordinate system. It provides a rich method for exploring species interaction.

Under certain conditions it may be desirable to view the relationship between models in this light. Note that in an ecosystem an interaction with a 'negative' component (e.g. parasitism) is not necessarily 'bad'. It may be part of a complex interspecies web or merely the corrective mechanism for a temporary imbalance between species. For the relationship between species is of course not rigidly determined. Species rise and fall numerically in relation to each other. Homeostasis is achieved by compensating patterns of oscillation around a condition of dynamic equilibrium.

4. Meta-models

It is interesting that few 'meta-models' even attempt to handle the relationship between models. Each of the latter tends of course to claim some form of exclusive validity viewing any alternatives as heretical, sub-optimal, or distortions of the most realistic. It could be said that, in comparison with the ecosystem case, models are defined at the 'species level' and that few, if any, have been successfully defined at the 'ecosystem level'.

The 'species level' provides a distinct reproducible pattern of organization, whether in the case of species or of conceptual models. The presence of other 'species' necessarily gives rise to an inter-species dynamic which is associated with a logical hiatus, since there is a distinct logical discontinuity between any such species-level perspectives. As will be discussed further below, it would appear that the ecosystem level logic cannot be adequately contained or expressed through a separate logic, it must necessarily be modelled by the a-logical pattern of inter-species dynamics.

Integrating incompatibles

1. Use of incompatible modes in practice

Before going on to a more systematic exploration of integration patterns, it is useful to note down some examples in which practical results are achieved by 'oscillation' between essentially incompatible or mutually unacceptable modes:

(a) Two or multi-party political systems: in which one of the parties takes power with the other(s) 'in opposition' to its philosophy and policies.

(b) Rotating chairmanship/secretariat/meeting locations: wherever consensus cannot be achieved on a single individual or location, perceived as emphasizing one tendency in contrast to others (which must therefore be represented in their turn).

(c) Interrogation/brain-washing procedures: in which an individual is progressively broken down by the alternating use of kindness (the traditional cigarette) and torture, often by a 2-person team of interrogators exemplifying these characteristics.

(d) Management/leadership: which is frequently required to apply appropriate doses of 'the carrot' or 'the stick' to ensure appropriate team performance.

(e) Educational challenge and encouragement: whereby the teacher alternates praise and reinforcement with exposure to new challenges through which the student achieves confidence in the skills acquired.

(f) Healing care and therapeutic challenge: through which healing is achieved by alternating periods of rest and care with periods in which the individual exercises the healing part.

(g) Formal and informal liaison: may be used as complementary alternatives essential to balanced functioning, whether in intra- or inter-organizational contact or in relations between individuals (as in the case of formal marital and informal extra-marital relationship).

(h) Strategic advance and retreat: are complementary alternatives necessary to adaptation (by a general) to changing circumstances of conflict. Inability to switch into the alternate mode is to court defeat.

Note that in each case the transition from one mode to the other is through a decision governed by qualitative judgemental factors combining art and science beyond the realm of simple models and logics.

Within some oriental cultures the ability to move smoothly between two such essentially conflicting modes is modelled by the circular process of inhalation and expiration. For this reason much emphasis is placed on eliminating defects in this process through breathing exercises for the individual. (But, strangely, never for a group).

2. Model mixes in space/time

It is important to recognize that integration can be achieved in space and/or time. For example:

Note that it is unlikely that these two limiting cases are independent of one another. It is more probable that they engender each other in space- time as do rotation and revolution in any solar system model, for example. In fact such spin and orbital motions are part of a set of six basic motions which also includes: expansion-contraction, torque, involuting- evoluting, and precession (6, point II, 400.654 and II, 986.857); These may suggest other forms of integration in space-time, as well as introducing the less-evident (spiral) evolutionary movements. As a complete set, each of these 'motions' is necessary to the existence of the others.

Further possibilities for integration are indicated by the following:

Patterns and rhythms

The question now is whether we can identify the nature of the series of patterns or rhythms in the light of the first few members of which we are more or less aware.

The simpler possibilities for N elements are clarified in Annex 2. Note that the 'relationship' indicated there by a line could, in binary terms, be considered either as one of 'compatibility' (space) or 'consonance' (time), indicated by '-', or as one of 'incompatibility' or 'dissonance', indicated by ' - '. Integration indicated by the degree of symmetry which in the time case is related to recurrence.

Annex 2 only focuses on patterns which are significant in 2 dimensions. These patterns are in fact those which have been the focus of attention in the classic communication net experiments (7). A valuable approach to more complex patterns, particularly in relation to time, are the Chladni figures visualized by vibrating powders on a metal sheet (8). Another source is the range of progressively more complex patterns from graph theory (9). Reference should also be made to Johan Galtung's suggestions regarding the use of graph theory (10).

In previous papers (11, 12) the limitations of the 2-dimensional approach have been criticized as a preliminary to considering the significance of the simple 3-dimensional patterns, especially in relation to tensegrity. The latter achieve stability as patterns of oscillation around a dynamic equilibrium condition. The concept of oscillation will now be considered in more detail.

Oscillation

1. Conventional example

The simplest example of oscillation occurs with 2 states (models) between which the system moves. A number of examples were given above, of which the most typical is perhaps the classic 2-party political system.

Oscillation as a phenomenon has been extensively studied in physical and electromagnetic systems where it is an aspect of vibration and wave motion. Such studies should suggest interesting questions for oscillating in psycho-social systems. For example:

(a) An oscillation results when an elastic medium (for example a spring) is displaced from its equilibrium or rest state. When the displacing force is removed, the elastic medium tends to snap back, to regain its rest state, and then to overshoot it because momentum cannot be lost instantaneously - and thus the 'simple harmonic' cycle recommences. In general, for a given mass, the greater the elasticity, the higher will be the frequency of oscillation; whereas for a given elasticity, the larger the mass, the lower will be the frequency. What is the 'elastic medium' in the psycho-social case?

(b) An oscillating system contains energy. At the extreme of the displacement, the energy is stored in the elastic material as potential energy. When passing through the equilibrium condition the material is not strained; and the energy is entirely in the form of kinetic energy of motion. Thus, oscillation involves a constant interchange between potential and kinetic energy. Can the transformation between two types of energy be recognized in the psycho-social case?

(c) Technologically, it is practically impossible to build a machine that transfers energy from one place to another without having its operation accompanied by oscillatory phenomena of some kind. Machines waste energy, and give rise to material fatigue and often noise. Are these side-effects recognized in the psycho-social case?

(d) Most physical systems deviate at least slightly from pure harmonic motion. Such deviations contribute, for example, to the quality of musical sound from instruments. It can be shown that any oscillatory phenomenon encountered can be constructed by adding together a number of component oscillations, each of which is harmonic. What are the 'deviations' in the psycho-social case?

2. Psycho-social interpretation

A useful way of describing the periodic behaviour of simple harmonic oscillation is to employ the sine curve whose complete cycle thus represents the complete oscillatory cycle. This may also be viewed as a projection of motion in a circle.

(Diagram)

a, c, and e = points of maximum kinetic energy

b and d = points of maximum potential enrgy (max. displacement)

In the psycho-social case, it is at point (a) that maximum momentum has been acquired towards the realization of Model A. At point (b) the 'maximum distortion' of the system in this direction has been achieved and a restoring force enters into play which progressively phases out Model A, such that at point (c) the maximum momentum towards the realization of Model B has been acquired. This in turn achieves maximum 'distortion' at point (d) when a reverse restoring force enters into play.

It is most important to note that within the Model A perspective, for example, there is no way in which the reversal at point (b) can be logically acknowledged or accepted (except possibly as a temporary set back). Model A necessarily strives to extrapolate along the curve (a) - (b), presumably to some static 'plateau' curve of 'A-perfection'. The increasing momentum from points (b) through (c) must necessarily be viewed by the proponents of Model A as the proliferation of unresolved problems and 'irrational' tendencies, which are seen to achieve their maximum deployment at point (d).

Obviously the proponents of Model B see this sequence in exactly the opposite light and would strive to extrapolate along the curve (c) - (d), presumably to some static 'plateau' of 'B-perfection'.

The cycle therefore involves two 'inversions' of logic (comprehension discontinuities) between the two models which are incompatible and as such mutually incomprehensible (except that what is partially 'comprehended' is used to fuel the antagonism between them). This 'irrationality', as related to the circular projection above, recalls the manner in which mathematicians have succeeded in interrelating positive and negative 'irrational' quantities by use of the Argand circular diagram.

It is this inability to handle curvature which traps the proponents of each model in the linearity of whatever portion of the curve they are associated with. It appears linear, just as the earth appears flat to a rational observer insensitive to longer range phenomena.

Nevertheless, despite the mutual incompatibility, of the models, in the real-world dynamic situation each is effectively defined in terms of change to or from the other.

3. 'Multiphase' oscillations

The argument above has focused on two extreme states (models) between which oscillation occurs. As indicated in the discussion of Annex 2, combinations of states may be envisaged between which oscillation can take place. Some of these are 'damped' as discussed in a subsequent section.

In a three phase oscillation, for example, Model A would be replaced by Model B, to be replaced by Model C, replaced in turn by Model A. Namely a triangular configuration.

Many such configurations are possible and can be represented by 2- dimensional configurations. Note however that the longer the chain of models in a circular sequence, the more difficult it is to comprehend the sequence as a whole. An interesting example of a 64-phase sequence is that of the the sequences of changes associated with the Chinese I Ching (or Book of the Changes). It is unusual in that a justification for switching from one model to the next is given, thus implying a holistic perspective 'meta' to that which is explicit.

4. Coupled and damped oscillators

Whereas the previous section focuses on a succession of models, it is also possible to envisage, for example, the simultaneous presence of 3 oscillating systems A/B, C/D, and E/F. If the oscillations of one affect another then the two are coupled. Such coupling may have amplifying or dampening effects.

A system set into oscillation by some initial displacement will not continue to remain so indefinitely unless energy is supplied from some external source. An oscillation cannot create energy. The oscillation gradually dies away and is said to be damped. The decay will be slow if the mass of the oscillator is large, and slow if the initial frequency is high. An oscillator absorbs energy from a source at maximum average rate at the resonance frequency, namely the frequency at which it prefers to oscillate. At this frequency it loads the energy source to maximum extent.

Dampening effects may be counteracted by coupling with other oscillators as note above.

Three-dimensional configurations

In Annex 2 the focus is on 2-dimensional configurations. But even in the simplest 2-phase case, the existence of that oscillating system is only possible by virtue of an appropriate contextual system within which it is embedded. (e.g. attachment of spring and gravitation governing movement of a mass). If the oscillation is described in the form of motion in a circle, the question to ask is how that circle is 'balanced' when considered as a rotating wheel. For either the wheel is unrelated to anything else (or includes everything), or it must be joined to its environment in some way, especially if energy is received to maintain the oscillation. But, just as the models associated with each phase of the oscillation are insensitive to the cycle as a whole, any cyclic perspective is equally insensitive to the forces required to maintain the cycle in stable relationship to its environment -- namely to compensate for the instabilities associated with and generated by its existence.

If the configuration of such contextual forces is rendered explicit, and integrated with the 2-dimensional cyclic configuration, one probable result would take the form of a tensegrity (12, 13). This may be considered as a spherical pattern of coupled oscillations (usually three or more interwoven cycles).

In string-and-stick models of tensegrities, for example, when a stick is displaced by application of stress, the whole system undergoes symmetrical modification to accommodate the local movement. The system's symmetry is not deformed; the system expands as a whole or contracts as a whole. Ability to respond as a system means that local stresses are being uniformly transmitted throughout the structure, and uniformly absorbed by every part of it. We have here a classic case of synergy; behaviour of whole systems, unpredicted by knowledge of the parts or of any subset of parts.

A complex tensegrity is never quite still, however lightly the tendons are stretched. There will always be minute oscillations, tiny stick displacements at the order of magnitude where elasticity multiplication is truly enormous and compensating forces have enormous advantage. The equilibrium point 'is that ideal condition of rest which nothing real ever attains, and about which a tensegrity in particular dances an eternal jig of pre-Socratic derision' (14, pp. 12-19).

Organization of time

1. Time-binding

The previous sections have considered progressively more complex ways in which oscillations can be interrelated into some 'macro-pattern'. The progression has been guided by what is known about the organization of space, given the argument of Annex 2 that these indicate equivalents for organization over time. Note that this progression has in effect moved from:

The question is now whether more clarification can be obtained from the manner in which time is organized. The above argument has not focused on the possible distinctions between oscillations, and yet oscillations organize time in different ways depending on their characteristics. Some of these ways may be more significant and may thus indicate opportunities opportunities for more viable temporal configurations of models -- possibly with corresponding simplifications for configurations in space. The point of departure is the recognition that oscillation 'organizes' time by determining a characteristic complete cycle. This is composed of both the 'incompatible' half cycles of the simple example discussed earlier. It encompasses the incompatibilities typified by polarized perspectives only capable of recognizing/accepting the swing of a pendulum in one direction (or the other). It is the periodic complete cycle which characterizes the organization in time.

In developing the argument, oscillation in psycho-social contexts may be considered:

Since the micro-historical cycles are more easily perceptible and appear less abstract than the macro-historical, the remaining argument will focus on them. In fact of course, 'macro' cycles with periods of decades or centuries could also be subject to the same approach (cf the work of Pitirim Sorokim). Relationships between micro and macro cycles will be explored in a later paper.

Since the human perceptual apparatus organizes ('integrates') oscillations with considerable sophistication in the process of hearing, this argument will focus on indications suggested by musical sound. Analogous arguments could be developed on the basis of the organization of oscillations in the process of seeing colours, but since sound can be discussed with less ambiguity, and with more precise exploration of possibilities of integration, this is to be preferred. (On colour, see: Johannes Itten. Art de la Couleur. Paris, Dessain et Toira, 1974)

2. Oscillation and musical sound

The perceptual apparatus distinguishes sounds as noise or tone. Noise is usually identified not by its character but by its source. Tones are recognized as being more independent of their source, are more organized, and as such are more amenable to integration into musical compositions. The choice of sounds for music making has been severely limited in all places and periods by a diversity of physical, aesthetic, and cultural considerations. Tones are generated as oscillations.

Within this context it is now possible to consider how a series of events involving the interactions of Model A and Model B -- a characteristic defence-attack, challenge-response sequence in any diplomatic incident, for example -- might be usefully perceived as a particular 'tone'. The suggestion is that there are characteristics which enable events to be recognized as part of a familiar pattern. When such characteristics cannot be recognized, a succession of events is considered as incidental/accidental 'noise'. When the events do fall into a recognizable pattern, this 'tone' can then be used as a higher unit of analysis through which the development of the stream of events can be integrated for comprehension. The individual events generated by incompatible models responding to each other are thus encompassed by a pattern usefully characterized by a tone. The tone is independent of the particular event sequence which functions as an 'instrument' to render it perceptible.

Clearly the conventional approach in society is to recognize events generated by an opposing model and to respond with a counter-event governed by one's own model. The 'tone' is not perceptible at this level.

The question is how to create 'instruments' (meta-models?) which could generate tones, in order to move beyond the present subjection to the essentially uncontrollable reactive dynamics of event-level interaction. If society could discover when particular tones are an appropriate response to circumstances, it would no longer need to be torn by the dualities of event-level interaction. Note however that event patterns are still required, since it is through them that society functions. It is their status which is dramatically changed. In this sense 'planning' becomes 'composition', which works not by using a set of events to achieve something but by using tones (namely event/counter-event patterns).

3. Music

'Music is time made audible' (Susanne K Langer). Perception of music depends largely on the ability to associate what is happening in the present with what has happened in the past and with what one expects to happen in the future -- whatever the probability that the expectation will be fulfilled.

The previous section clarified the integration to the tonal level. It is useful to consider three dimensions of time as structured by music for comprehension (noting that recent thinking contests Einstein's one- dimension concept of time). Within this context it is again possible to consider how models in a psycho-social system might be integrated.

Four basic types of musical form are distinguished in ethnomusicology: iterative, the same phrase repeated over and over (as in some chants); reverting, with the restatement of a phrase after a contrasting one (as in sonata-allegro, aria, and rondo refrains); strophic, a larger melodic entity repeated over and to different strophes of a poetry text (as in hymns, traditional ballads, and instrumental variations); and progressive, in which new melodic material in continuously presented).

But what is the significance of tone?

4. Significance of tone

All sound is composed of a complex of oscillations of a certain frequency (which determines the wavelength). Tones are characterized by four attributes revealed in their oscillatory wave form:

Most musical tones differ from an ideal single oscillatory wave form. Any material undergoing oscillation imposes its own characteristic oscillations on the fundamental oscillation. The material would probably oscillate in parts as well as a whole. These partial wave forms bear harmonic relationships to the foundational motion that are expressible as simple integer frequency ratios of 1:2, 3:4, etc. One way of expressing this is to say that half the body (e.g. a stretched string) is oscillating at a frequency twice as great as the whole; a third of it is oscillating at a frequency three times greater, etc. Tones are in practice composites of such 'overtones' which are ignored by the untrained car. It is however the presence or absence of overtones and their relative intensities that determine the timbre of any tone.

Tone is primarily characterized by pitch, or frequency of oscillation. Man's aural perception of pitch is confined within a span of roughly 15 to 18,000 cycles per second, with 440 cycles per second having been adopted as the middle point on the keyboard. It is now useful to ask the question what is the frequency range of psycho-social event-pair patterns to which man is sensible? What are the 'tones' he can detect?

Clearly there are some event cycles whose frequency is so low that man cannot be directly sensitive to their cyclic nature. Detecting the cyclicity of such phenomena is anal;ogous to detecting the curvature of the earth's surface. The cycle of a human 'generation' is barely perceptible as such, and even a year or a month are long cycles to many. At the other extreme, one indicator is the period of seconds associated with fast conversational repartee. Even more rapid would be the mind 'experiments' of protagonists in any game or fight in which each runs through the action options open to him and the probable responses of the adversary (e.g. chess, fencing, business negotiation, etc). Cellular processes in man are however beyond the range of his sensitivity.

In this light is it possible that psycho-social functions are each associated with a characteristic event cycle frequency? And, when such functions are activated in a particular case, is this frequency accompanied by the presence of harmonic frequencies, namely associated functions? Before clarifying these possibilities, it is necessary to consider the whole question of harmony as it has been elaborated for music.

5. Harmony

Musical sound is usefully regarded as having horizontal and vertical dimensions. The horizontal aspects are those considered so far, namely those that proceed in time. The vertical aspect is the sum total of what is happening at any given moment. This includes the result of notes that sound against each other in counterpoint. In the case of melody and accompaniment, it includes the underpinning of chords that the composer gives to the principal notes of the melody.

In music, harmony can be broadly defined as the sound of two or more notes heard simultaneously. In practice this can include notes sounded one after the other in cases when the ear creates its own simultaneity and perceives the harmony that would have resulted had the notes been sounded together. It is also the succession of harmonies that gives a piece of music its distinctive personality.

It should be understood that harmony is an optional additional form of organization or integration. Rhythm and melody can exist without harmony and in fact most of the world's music is nonharmonic, using unharmonized melodic lines often with a sophisticated rhythmic organization.

The concept of harmony and harmonic relationships is not an arbitrary creation. It is based on certain relationship among musical tones that the ear accepts almost reflexively, and that are also expressible through elementary scientific investigation. A stretched string divided by simple arithmetical ratios (1/2, 2/3..) and plucked can demonstrate that the intervals (or distances between tones) sounded before and after the division are the most fundamental that the ear perceives. Occurring in the music of nearly all cultures, whether in melody or harmony, these intervals are the octave, the fifth, and the fourth.

Returning to the considerations of the previous section, the question now becomes one of whether there are natural or fundamental harmonies between psycho-social functions? In other words when considering the expression of a particular function, should advantage be taken of the possibility of accompanying it by a function 'harmonic' to create a 'chord' which imbues the lifestyle with greater quality -- or, at least, increases the quality of the moment? The quality of the moment rta has remained a prime concern in the development of sanskrit- based culture.

6. Consonance and dissonance

The interplay of consonance and dissonance is the very foundation of harmonic music. Consonance can be defined as the normal range of tone combinations accepted as implying 'repose' by theorists and composers of a musical culture during a given period. Dissonance therefore refers to any sounds outside this range. Many attempts have been made to link consonant with pleasant, smooth, stable, beautiful, and dissonant with unpleasant, grating, unstable, and ugly. Whilst such attributions may be meaningful in a giving musical context, difficulty arises in generalizing to all such contexts. Such 'objective' classifications are now held to be value judgements.

Dissonance has always been recognized as the prime element creating movement in harmony. When the ear recognizes a certain harmony as unstable within the given musical context, it 'demands' that this instability or tension be rectified by resolution to a stable harmony. Without dissonance music would be hopelessly static. The historical development of music can however be seen as one of exploring different approaches to the treatment of dissonance so that the musical flow is an ordered alternation of tension and relaxation.

This suggests a useful way of perceiving the relationship between psycho- social models in society. How compatible or 'consonant' should they be? Is some degree of incompatibility necessary for the dynamic of society? What methods are available for managing the transition between compatibility and incompatibility?

In music, the understanding of which specific chords and intervals constitute consonance has altered dramatically from the origin of (Western) harmony as indicated in Annex 3.

Musical composition is in a stage of intense experiment. Although concepts of classical harmony have lost their importance it is not a question of the dissolution of harmony but rather of the uses to which such harmonies are put, and the changing relationship of harmony to musical structure. There are organizational systems emerging that point to a clear control and regulation of musical elements that may in future be analyzed in terms of a new, fundamentally different harmony. This may offer a means of relating the above, essentially Western concepts of composer-oriented music, to the body of unharmonic, non-Western music which is often performer-oriented. Significantly, however, music of every epoch and harmonic organization is appreciated by growing audiences.

The above stages are not only helpful as an indication concerning the nature and concept of integration, it has been suggested by many authors that the musical form or code conditions social organization in many subtle ways. Furthermore, the new developments in music are a response to the condition of society and the emerging codes indicate the basis of the organization of society in the future. These points have been most recently and forcefully argued by Jacques Attali (15). This perspective will be further discussed below.

7. Harmonic goal

In the previous section the descriptive emphasis is on the vertical harmonic relationships, if any, between tones and the ordering which governed them. Here the focus will be on the horizontal ordering 'where the music is going to'. This is of course helpful in understanding how an integrative goal, or the goal of an integration process, can be understood. It is also helpful in showing how the status of such goals has been brought into question. The evolution of the harmonic goal is described in Annex 3.

Contemporary music may be said to be 'goal free', or to call upon the listener to be responsible for any goal he chooses to derive from the music. The emphasis is very much on: the response of the individual listener, the context to which performers respond (including audience response), and increasingly the process of improvisation. The separation between traditional musical roles is breaking down (e.g. between listener/audience, performer, conductor, composer). The goal lies in the appreciation of the moment whatever the range of sounds which define it.

8. Fugitive integration

The fugue is often considered to be the most complex and highly developed type of composition in Western music. The argument above suggests that much could be learnt about new approaches to socio-political integration from a study of integration in music.

Many aspects of have already been explored. Here the focus is on the significance of the relationship between distinct 'voices' and themes, which is of course basic to polyphonic and symphonic integration. In the socio-political sphere it is usual for advocates of a proposal, a model, a cosmology or an ideology to propagate it as though it alone should achieve dominance (ad aeternam), effectively excluding alternative approaches. Within the musical framework this can lead to pieces which are either immediately monotonous and boring, or whose interesting characteristics quickly become an intolerable imposition unless balanced by other pieces in the musical diet (cf. the life cycle of a 'hit' record). There is of course no musical continuity between the succession of such separate pieces of music. This lack of integration is analogous to the equivalent situation in society where the advocates of the alternatives, evoked by overstress on a particular model, compete in parallel or in succession for constituencies and resources for their own approaches. In the fugue however the relationship between these competing voices is explored within a musical continuum. This represents a new level of integration. In effect the concept (model, etc) is explored, inverted, countered, distorted, etc within an overriding set of rules which permit a new level of freedom. The rules ensure a more exciting balance of tension and harmony.

Could it be that one dimension of the challenge of socio-political integration is illustrated by the problem of interrelating seemingly hostile or incompatible 'voices'. And that counterproposals and counterarguments need to be set in a larger context to which we are as yet insensitive? And does the time dimension over which arguments and counterarguments are developed need to be better understood in terms of an integrated socio-political process? Such a process brings out both the essential inadequacy of any particular proposal and the manner whereby it can be counterbalanced and enriched by complementary proposals which together as a process bring about a new level of integration. The implications of the fact that music has 'progressed beyond' the fugue are discussed in the main paper. But it would appear that the lessons it offers for socio-political integration have not yet been considered.

Liberation of integration

1. The remaining problem

Such an outcome may well be satisfactory to those who favour the appearance of anarchy, however positively it is evaluated. The reality of the situation, as mentioned above, it that music of every type continues to have its adherents. In effect the musical goal of contemporary music has been 'detached' from the music and reappears in the search for increasing degrees of liberty which composers/performer/listener can be brought to share. This marks the achievement of the progressive liberation from the various forms of domination built into the harmonic goals noted above.

But, by now emphasizing the experimental styles of contemporary music, such a goal effectively aims at domination, by a particular 'liberating' style, of the other musical styles, each preferred by significant 'constituencies'. Any such stance of 'contemporary is best' inserts such music into a linear historical stream in which that dominating statement has been made at every stage. It does not face up to the atemporal reality of musical appreciation in society - and the corresponding range of preferences for forms and degrees of order and integration.

This of course epitomizes the dramatic problem in society today. Different models are preferred simultaneously (or as alternatives) whilst often being mutually incompatible. This was the point of departure of this paper. Are there any indications of the nature of the next level of integration which would respond to this paradox without introducing new forms of dominance?

2. 'Cross-harmony'

As a first indication of a direction for exploration, the notion of a harmonic analogue to the crossword puzzle can be considered. In such a puzzle rows and columns would be all based on different harmonic principles - covering the complete range. The art would lie in making the points of intersection meaningful. But, as a flat bounded matrix, this is not satisfactory. As argued elsewhere (11), a more interesting development would be to wrap the flat surface around a sphere (or a polyhedral approximation to a sphere). In this way the surface is unbounded and without any privileged central points. Symmetry relationships then emerge as factors in a new level of integration. The tensegrity approach is particularly relevant to relating incompatibles brought together in this way (16).

In such a 'cross-harmony' model each row and column would need to correspond to a different (non)harmonic organization. How can different styles of music be interrelated, preserving their identity, but bringing out a transcendent harmonic order? If it can be done with music, then may be it could be done with psycho-social models.

3. Comprehending the language of pattern shifting

The suggestion in the previous section can be viewed as a crude approximation to a highly sophisticated approach based on the 4,000 year- old chanted hymns of the Rg Veda of the Indian tradition. A very powerful exploration of this work by a philosopher, Antonio de Nicolas (17), using the non-Boolean logic of quantum mechanics (18), opens up valuable approaches to integration. The following themes are explored in the de Nicolas study:

The unique feature of the approach is that it is grounded in tone and the shifting relationships between tone; It is through the pattern of musical tones that the significance of the Rg Veda is to be found.

'Therefore, from a linguistic and cultural perspective, we have to be aware that we are dealing with a language where tonal and arithmetical relations establish the epistemological invariances... Language grounded in music is grounded thereby on context dependency; any tone can have any possible relation to other tones, and the shift from one tone to another, which alone makes melody possible, is a shift in perspective which the singer himself embodies. Any perspective (tone) must be 'sacrificed' for a new one to come into being; the song is a radical activity which requires innovation while maintaining continuity, and the 'world' is the creation of the singer, who shares its dimensions with the song.' (17, p. 57)

4. Limitation of vision-based metaphors

De Nicolas contrasts this perspective with that of languages governed by vision:

'Thus, in a language ruled by the criteria of sight, vision may mean the sum of perspectives from which a fixed object can be seen, plus the theoretical perspective of the relationships holding amongst different perspectives of the object, plus the mental acts by which those perspectives, relationships and visions are performed. In any event, the invariant object is the condition for the variations in the meaning of vision. The object is the condition for the variations in the meaning of vision. The invariant object is, therefore, not a reality, but a theoretical precondition (phenomenal or noumenal) for a whole system or method for establishing facts. Therefore, it is no wonder that when people speak of transcendence, within this framework, they are mostly forced to speak in mystical terms of things unseen or unseeable, either in terms of religious experiences, or in terms of modern physics. In a literal sense, in the latter two cases, speech is aboutno things by the same criteria of the speech used to designate things.

Whereas in a language governed by sound:

'In a language ruled by the criteria of sound, perspectives, the change of perspectives and vision, stand for what musicologists call 'modulation'. Modulation in music is the ability to change keys within a composition. To focus within this language, and by its criteria, is primarily the activity of being able to run the scale backwards and forwards, up and down, with these sudden shifts in perspectives. Through this ability, the singer, the body, the song and the perspective become an inseparable whole. In this language, transcendence is precisely the ability to perform the song without any theoretical construct impeding its movement a priori, or determining the result of following such movement a priori. Nor can any theoretical compromise substitute for the discovery of the movement of 'modulation' itself in history. The human body would then be asked to lose the memory of its origins; a task the human body refuses to do by its constant return to crisis.

It is up to the philosophers to discover the language ruled by the criteria of sound, rather than presuppose a priori that the only language universally human is the one ruled by the criteria of sight.' (17, p. 192)

5. Integration through four complementary languages of music

Given the importance of sound and music as a major integrative factor across cultural boundaries, and given the size of the audience which music now has through radio and cassettes, the possibilities of this route merit further exploration. Integration modelled on sound may be inherently more comprehensible to more people than integration modelled on sight.

It is interesting to note a tantalizing relationships between this study and that of Jacques Attali who structures his study of the political economy of music in terms of 'quatre formes possibles de diffusion de la musique, retrouvant les quatre structures fondamentales que peut avoir un graphe' (15, p. 63). These are 'networks' associated with: ritual murder or sacrifice, presentation, composition, and reproduction.

De Nicolas on the other hand structures his study in terms of four 'languages' distinguished by their intentionality: images and sacrifice, existence, embodied vision, and non-existence. The first three seem to be related to those of Attali, and it is the non-relationship of the last which is significant in both perspectives. Attali's concept of the sacrificial aspect (15, pp. 43-91) as an attribute essential for the renewal of social structures is intimately related to the Rg Vedic concept (17, pp. 139-154). Such efforts to show the functional significance of sacrifice in relation to social integration need attention in a period when 'nobody is willing to sacrifice' advantages acquired under the present systems in crisis. The implications of the relation between the two studies could also be further explored in the light of number-governed sets (19).

De Nicolas study has already inspired an exploration of the tonal underpinnings of the Rg Veda by a musicologist, Ernest McClain (21), which is interesting in its own right. This helps to understand the interrelatedness of perspectives and the mnemonic value of their expression through vivid symbols (gods, dragons, etc). McClain clarifies the musical significance of the four languages and, in this context, their relevance to integration:

'The four Rgvedic 'languages' de Nicolas defines have their counterparts in the foundation of all theories of music. His 'language of Non-Existence' (Asat) is exemplified by the pitch continuum within each musical interval as well as by the whole undifferentiated gamut -- chaos - - from low to high. His 'language of Existence' (Sat) is exemplified by every tone, by every distinction of pitch, thus ultimately by every number which defines an interval, a scale, a tuning system, or the associated metric schemes of the poets, which are quite elaborate in the Rg Veda.

The 'language of Images and Sacrifice' (Yajna) is exemplified by the multitude of alternate tone-sets and the conflict of alternate values which always results in some accuracy being 'sacrificed' to keep the system within manageable limits. The 'language of Embodied Vision' is required to protect the validity of alternate tuning systems and alternate metric schemes by refusing to grant dominion to any one of them'. (21, p. 3). 'The embodiment of Rg Vedic man was understood... as an effort at integrating the languages of Asat, Sat and Yajna to reach the dhih, the effective viewpoint, which would make these worlds continue in their efficient embodiment' (17, p. 136).

The whole notion of dialogue between perspectives which is the basis of the Rg Vedic approach needs, however, to be related to current investigations of conversation theory as summarized by Gordon Pask (20).

6. Challenges

But, despite the richness of the Rg Vedic model, the de Nicolas study raises questions which he does not address:

These are however points to be fed into any further exploration of the possibilities so successfully opened up by de Nicolas. It is interesting, in the light of this paper's opening comments on the UN University's GPID project, that many of the governing preoccupations of that project are interrelated in the Rg Vedic approach.

Conclusions and implications

1. Range of approaches: This exercise has brought into focus an intriguing range of approaches to integration. As intended this exploration reached as far as seemed feasible. Clearly in doing so it moved beyond familiar ground and the approaches to integration which have been attempted by the social sciences. So whilst in the early stages the approaches appear as 'models' of integration, the subsequent stages take on the aspect of 'analogies', and the final stages could simply be considered as 'metaphors'. This is not to deny the value of the final possibilities, rather it highlights the fact that it may be our attitude or problems of comprehension which reduce possible models to 'superficial' metaphors. Conversely, the sterile inadequacy associated with models may result from their function as metaphors -- 'taken too far'.

2. Locus of incompatibility: A key factor in considering the value of these various possibilities appears to be where one chooses to 'locate' the 'incompatibility' or hiatus between alternative modes (or models). Locating it at the level of the single oscillatory cycle must necessarily 'push' any more sophisticated approaches to integration into the realm of analogy or metaphor. For conceptually we have not yet digested such a cycle involving incompatibility. If however the incompatibility is modelled by dissonance between tones (based on oscillations), the apparent drift is only encountered further towards the end of what could usefully be perceived as a continuum. This raises the interesting question of how incompatibility or discontinuity is 'contained' (or nested) at a lower level. (For example the oscillatory left-right motion of walking is a complex problem of balance which a child must learn -- once learnt the jerky discontinuity of the broken cycle disappears and 'walking' may be comprehended at the 'tonal' level, later still walking merely becomes an aspect of dancing or other patterns governed by 'harmony' of some kind).

3. Beyond the broken cycle: It is certainly appropriate to consider that in our attempts to reconcile societal alternatives we are at the 'broken cycle' level (just 'limping' along). The attempts to move beyond this in envisaged policy proposals seem 'monotonous' (lacking any sense of 'rhythm' and 'melody'). Perhaps the non-Western cultures endowed with a greater sense of rhythm and rhythmic organization will provide the understanding to move us all into a new mode in which the Western understanding of harmony can be used. On the other hand, it is important to recognize that it is often those rhythmically-oriented segments of Western culture, for which current social science thinking is sterile and meaningless, but which need to be integrated into the process whereby social alternatives are envisioned. (This aspect of integration also needs to be considered in relation to the social scientist, who in other roles expresses a well-developed sense of rhythm and harmony, but whose professional thinking may in this context resemble that of a learner on a musical instrument).

4. Resonance: The possible significance of the concept of resonance between an oscillatory cycle and its bounding environment merits further reflection. It is significant in relation to the use of available energy, to affecting the environment, to enhancing environment cycles, to destroying 'crystallized' structures, etc. Entrainment of one oscillatory cycle by another should also be considered. In a world seen in terms of movement these two phenomena are significant in terms of the prenuptial dance and the process of conception.

5. Harmonic goals: The concept of harmony and a harmonic goal may also be used in evaluating such phenomena as a conference or an academic paper as exercises in integration. In the case of a conference there is some merit in seeing it at the broken cycle level trying to get together a coherent tone. But it may also be seen in terms of polyphony and the challenge of harmonizing different voices. But in the light of the historical evolution of harmony, the value of moving the conference through a series of consonant and dissonant 'chords' to a tonic goal may be viewed as somewhat simplistic (The closing phases of most conferences these days reveal the superficiality of that goal). In a separate paper suggestions have been made concerning non-linear agendas, weaving consonance and dissonance at a 'tensegrity conference', which could correspond to another level of harmonic integration (22). It is tempting to see the use of non-sequiturs, inconsistency and anecdotes in a conference or a paper as corresponding to chromaticism in music. Edward de Bono's call for the use of 'po' to break sequence is also relevant (23). For a more extreme example consistent with the historical evolution of harmony, Christopher Jones random number generated conference paper should be noted (23). It aimed to break down the whole approach to 'conference think'.

6. Concepts of progress: The Rg Vedic approach however casts a valuable light on many aspects of integration. Firstly the whole motion of a historical sequence progressing towards 'greater' or 'better' integration is seem to be only partially associated with linear thinking - partially because it would be a Boolean-logic approach to stress only the non-linear aspect. This relates to the whole question of whether progress is an illusion (25). In one sense there is no convergence to better integration, only a more integrated understanding of the present moment - which is however - where we live.

7. Choice and integration: It is however this ahistorical context which frees the individual's power of choice. The individual can choose to be moved by a 'less sophisticated' form of harmony - rejecting the greater freedom of contemporary music, for example. The ultimate freedom is the freedom to be unfree, bound by particular constraints.

8. Dialogue and integration: Given, the UN University/GPID stress on dialogues and integration, the Rg Vedic focus on dialogue between contrasting perspectives and its relationship to integration is valuable.

9. Harmony of harmonies: Perhaps the subtlest aspect of the Rg Vedic model is that which confronts the problem of the integration of alternative approaches to integration. This is the distinction between languages which can each only partially express the insights of a Language which can only be expressed through them. This is the problem of the harmony of harmonies, in which the alternatives appear and disappear as patterns of ripples on the surface of a sphere.

10. Theory vs. Praxis: Extremely valuable is the Rg Vedic oscillation between theory and praxis as inherently incompatible but necessary to each other. It is refreshing to see those related as yin- yang complementaries - with the hope that some androgynous perspective may emerge (26).

11. Integration of what?: The whole notion of a range of tones suggests a fundamental question: what are the different things that need to be integrated to bring about a quantum change? And what is the necessary harmony of that integration in order for the change to occur? Why cannot we focus more clearly on the relationship between the factors which generate and control the various 'waves' (27) to which our societies are subject? (*) What are the forms of organization that are required?

12. Individual self-image: But aside from its relevance to the various aspects of human and social development mentioned at the beginning of this paper, the whole notion of harmony is of special relevance to the concept of the individual and to the individual's concept of herself or himself. Given the variety of roles/persona through which the individual expresses himself, the concept of 'harmonizing' such voices in one way or another is very intriguing (e.g. the notion of polyphony). Also the notion of organizing one's time and awareness in terms of a 'melody' of such voices, perhaps with some overall harmonic goal. At this level a certain convergence can be seen with the affective and dramatic content which composers have conventionally associated with certain forms of melody and harmonic organization - most crudely in the musical accompaniment to films. A challenge for the individual is then to experiment with subtler harmonies whilst maintaining the option to move between them. This is an organic approach to maintaining continuity of awareness, rather than being subjected to a 'broken cycle' style of life. As such, concepts of harmony are relevant to the question of social indicators.

13. Individual as keystone for integration: This paper started by suggesting that more elegant forms of integration might not only be desirable but also necessary for effective integration to be achieved. But whilst our values now stress the importance of centering social development on the human being, the Rg Vedic model would appear to go beyond an indication that this is desirable. It would appear that, to bring about the kind of integration which is required by the conditions of our society, it is essential that integration be embodied, transmuted and expressed through the individual in movement. The individual is in this sense the dynamic 'keystone' to an integration relevant to human social development.


'With the Classical Physics viewpoint of the world, we were primarily concerned with discovering notes, classifying them and aiming, through their different classifications, to compose a symphony that would eventually sing the world so conceived. Suddenly, with the discoveries of Modern Physics we have realized that the symphony was already there; that the songs were already being sung, and that the true sound of any note was embedded. One could say that one single note is the whole song; that in order to play one single note, one has to know the whole song; or that while playing one single note one is playing the whole symphony. Silence underlies both as a condition of possibility. Perhaps this musical metaphor helps clarify what has preceded it.' (De Nicolas, 17)

'We can't put it together; it is together' (The Updated Last Whole Earth Catalog, 1974)

 


Annex 3: EVOLUTION OF HARMONY

For Western music, harmony has evolved over the centuries

A. Vertical harmonic organization (the moment)

(a) Ancient Greece: Harmony based on the succession of tones within an octave. Scales were used as a basis for singing in unison. Melody was synonymous with harmony.

(b) 6th to 9th century: Use of any 12 such modes (scale patterns of tones and semi-tones) in which the notes also had characteristic functions.

(c) 9th century: Only the simplest 'perfect' harmonic ratios were accepted: fourth, fifth, octave. This allowed the addition of one or two voices which exactly paralleled the original melody. Later these voices acquired melodic independence, possibly moving contrary to the original melody.

(d) 12th to 15th century: Inclusion of other intervals, thirds and sixths, and in some cases, seconds and sevenths. This was associated with the development beyond 3-part scoring to 4-, 5-, and 6-part scores, thus further enriching the harmony of voices.

(e) 15th century: Introduction of additional notes outside the mode, thus breaking down the distinction between the 12 classical modes and foreshadowing the major/minor mode system.

(f) 16th century: The tonic, or keynote, triad then became the point of departure and of arrival in a composition and in its component phrases.

(g) 17th century: Greater emphasis was then placed on expressive melodic line harmonically underpinned by a base line as the generating force upon which harmonics were built (often by improvisation) -- contrasting markedly with the interweaving of parts of equal importance.

(h) 19th century: Deliberate use was then made of unresolved harmonies (unstable chords used as self-sufficient entities) and of ambiguous chords. Although rooted in tonality, every possible device is used to complicate or obscure the tonal sense.

(i) 20th century: Use of chords seemingly conforming to classical practice but which are resolved in unexpected directions. Tonality exists in the sense that there are extended stable areas that give the impression of being in some definable key, but the intense use of notes outside the scale of the basic key (chromatism) makes it nearly impossible to group the unity of a work in terms of its adherence to a clear tonal plan.

(j) 20th century: Use of atonality, abandoning the traditional duality of consonance and dissonance (eliminating the concept of a single predominant key as tonic). Break away from traditional scales in recognition of the power of context and the sense of a continuum between consonance and dissonance.

(k) 20th century: Emphasis on performer improvisation/interpretation catalyzed by indeterminacy procedures making any concept of overall harmonic direction irrelevant.

Musical composition is currently in a phase of intense experiment. Although concepts of classical harmony have lost their importance, it is not a question of the dissolution of harmony but rather of the uses to which such harmonies are put, and the changing relationship of harmony to musical structure -- and the emergence of a new, fundamentally different harmony.

B. Horizontal harmonic organization (over time)

(a) Up to 15th century: In the use of 7 to 12 harmonic modes, the harmonic goal was governed by the given scale pattern. Although chants were sung unharmonized and in a rhythmically free manner, there were constraints and there was a proper final note for a modal melody.

(b) With the development of melodic independence between voices (polyphony) and the use of dissonances within the composition, emerged an emphasis on the resolution of such tensions through consonances at the end of compositions as the point of arrival. This reinforces the idea of the cadence, or the finality of the keynote of a mode on which pieces normally ended.

(c) From the 16th century: Devices such as the suspension were used as a way of enhancing, through dissonance, the resolution to consonance and the sense of completion of the final chord. In a suspension one note of a chord is sustained while the other voices change to a new chord. In the new chord the suspended note is dissonant. One or two beats later the suspended note changes pitch so that it resolves into, or becomes consonant with, the chord of the remaining voices. This reinforced awareness that harmony moves through individual chords towards a goal.

(d) 17th century: The concept of a key was developed as a group of related notes (belonging to either a major or minor scale), plus the chords formed from those notes, and the hierarchy of relationships among those chords. The keynote, and the chord built on it is a focal chord towards which all chords and notes in the key gravitate. Given chords assumed specific functions in moving toward or away from harmonic goals, the main goal being the tonic key or keynote - of which there were a total of 24 possibilities. These derive from the 12 major key scales and 12 minor key scales (each of 7 tones). The most common movement from chord to chord is through 'strong' intervals (fourths, fifths, seconds) which have the' fewest notes in common.

(e) 18th century: Modulation, or change of key, became an important factor because it allowed the composer to exploit the listener's ability to sense the relation between the keys. Modulation was usually to a 'dominant' key which was a 'strong' interval' apart. After the modulation there is a process of return to the initial key. During this process the harmonic movement tends to pass rapidly through many chords and often with momentary diversions into many new keys thus dramatized as unstable -- and in this way lending greater impact to the eventual return to the stability of the original key. This modulatory scheme from tonic to dominant key and back to tonic key formed the basis of large-scale musical forms, although often with additional refinements (such as secondary dominants) to strengthen the sense of completion of the tonal journey.

(f) 19th century: There was increasing disavowal of modulation, in terms of any tonal goal. By deliberately failing to resolve dissonances, or by creating ambiguity so that it was unclear whether resolution had been achieved, the status of the harmonic goal was redefined. The listener was called into an active role to respond to the 'questions' raised by the unresolved elements and to define the unity to be supplied. This blurring was also counter-balanced by an emphasis on continuous, goal-less melody. Two simultaneous tonalities (polytonality), neither dominating the other as a tonal goal, were also used.

(g) 20th century: With the advent of serialism, no single note could any longer serve as a harmonic goal. Whereas melody, from being synonymous with harmony (Ancient Greece), became the surface of underlying harmonies (16th century), and then bore its own harmonies (into the 19th century), serialism provided a melodic sequence out of which harmonies were generated. Such harmony effectively became the surface, or final result, of melody.

Contemporary music may be said to be 'goal-free', or to call upon the listener to be responsible for any goal he chooses to derive from the music. The emphasis is very much on: the response of the individual listener, the context to which performers respond (including audience response), and increasingly the process of improvisation. The goal lies in appreciation of the moment whatever the range of sounds which define it.


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