"In the snow, flat-topped hillocks and shoulders, outlined with wavy edges, ridge below ridge, very like the grain of wood in line and in projection like relief maps. These the wind makes I think and of course drifts, which are in fact snow waves. The sharp nape of a drift is sometimes broken by slant flutes or channels. I think this must be when the wind after shaping the drift first, has changed and cast waves in the body of the wave itself. All the world is full of inscape, and chance left free to act falls into an order as well as purpose".
Gerard Manley Hopkins - lines describing a snowy landscape, from his Journal: 24th Feb. 1873.
In the early part of the last century, the influential Danish physicist Niels Bohr constructed a new atomic model, which led him to confront a difficult problem: according to James Maxwell's theory of electromagnetism, a crowning achievement of late nineteenth century science, oscillating electrons should radiate energy and quickly collapse into the nucleus of atoms. This clearly doesn't happen. Borrowing an idea pioneered by Max Planck, Bohr made a quantized model of the atom that used ratios of the harmonic series to confine electrons to discrete energy levels - a structural device providing for the atomic stability which is observed in nature. Together with Planck, Bohr was laying the foundations of quantum mechanics, a difficult to comprehend but very successful description of the physical world at the fine scale of atoms.
Musicians also inhabit a quantized world. A universe described primarily by discrete units of rhythm and pitch (and volume - though the quantized units of energy which Max Planck postulated in 1900 are far to small to be aurally perceived). Glissando and free rhythm aside, for the most part our music proceeds by quanta, little steps and jumps of pitch and duration from note to note and chord to chord.
Over many years, indeed centuries, musicians have learnt some interesting tricks about how to work such a system of discrete units, how to create structure, how to integrate detail and how the whole can be made to hang together as a coherent unit. Our rich musical heritage is the result. But it is a reward that had to be worked for, over a considerable time, with contributions from across Europe: the creation of composers, musicians and not least their patrons, the audience - by exercising choice and selection. In particular, the system of keys (tonal centers) was not invented over night but rather grew up slowly as each generation of composers explored the boundaries; until, moving forward through ever more keys, eventually brought the system back to where it had started from! This remarkable structural device, the cycle of key centers and the mechanism of modulation which allows music to step from one tonal center to another, was a collective artistic achievement, an evolutionary growth, the work of many, over an extended period of time.
The evolutionary path trod by Western Music was (is) I suspect, beyond the overall control of individuals and groups. Once musicians strode forth in harmonic steps of whole number relationships, they became unknowing players in an evolutionary game destined to find an organizational solution for music which follow fundamental principles that apply to all oscillatory systems. Essentially, rather than creating some arbitrary scheme, over the centuries I suspect, western musicians evolved or developed a rough facsimile of nature's own organizational solution.
The idea, the thread running through 'Steps Toward a Computational View of Tonal Music: Modulating Oscillatory Systems' and the other articles, is that by closely examining the structures of our musical system and the organizational principles which tonal music evolved during the period of its great flowering (roughly 1500 to 1900) - principles and techniques for the manipulation of quantized sound, regulated by the whole number relationships of modulation - some interesting, useful and general insights and perspectives might be gleaned. In these articles, set out as best I can, are what I believe to be the basic principles of tonal music's underlying scheme of organization: a physical process of oscillatory computation, driven by the law of increasing entropy and controlled by a simple, yet subtle algorithm of symmetrical exchange - modulation. It is certainly no more than a tentetive attempt to set out the ground, explore the basic model, mechanisms and techniques; and hopefully, to perhaps draw in some feedback and guidance as to the soundness of the approach. Though perhaps at times challenging, I hope you find the journey through the pages of these articles interesting, informative and... enjoyable.
1) Working on the supposition that western tonal music evolved over the heads and beyond the time scale of individual composers and musicians, to find the most effective organizational model for a relational oscillatory system; and that by a close examination of tonally organized music the nature of this underlying scheme can be divined; a model of modulating oscillatory systems is proposed as the ultimate structural device that underpins tonal music. This general relational model or little world of self-coherent relationships, is developed as the basis for a computational approach to harmonic theory and analysis in music.
2) Essentially, the model consists of relationships and a process by which the relationships can be changed. The relationships are arranged as two or more entangled harmonic (or arithmetic) series, one a fundamental nesting series, within which the other nested series reside. The mechanism whereby the constituent series interact is modulation; it is in principle the same process as that used in music to change from one key to another - though in an extended and generalized form. In computational terms the nesting and nested series form the data and the algorithm or rule which acts upon them is the process of modulation. Ultimately, the MOS model reduces a physical oscillatory structure — in principle a tonal composition — to a succession of numbers in a mutable base position value number system; a counting scheme similar to the decimal system, though richer, it that many of its numbers can be accessed by a variety of different digit sequences (Part III, Section 2).
3) It is a model that can only ever loosely describe a piece of music in its entirety, as their are many arbitrary features in such 'artistic' structures. However, when the principles of the model are applied rigorously, as if in a self-organizing system, the outcome appears essentially that of digit sequences in a counting structure. A physical form of position-value counting, founded on the spiral of fifths - the 2:3 relationships of the key centers of traditional western music. Indeed, viewed from the stand-point of this abstract model, tonal music appears to be but one of many applications of an underlying natural material process which emerges when individual oscillators are bound together, forming complex systems capable of structural evolution.
4) Probably most surprising conjecture presented in 'Elements of Music?', is that the patterns found in the Periodic Table of the Elements appear to be described with remarkable faithfulness by the model. However, I hasten to emphasise in the strongest of terms that an apparent similarity in outward form in no way establishes any real connection. The two charts below the links to articles are presented as tentative and highly speculative 'explorations' of these patterns.
The Full Cadence: Introducing Modulating Oscillatory Systems an article designed (hopefully) to set the reader thinking on the right 'wavelength' and smooth the path to the more densely written 'Steps Toward a Computational View of Tonal Music' - 160Kb PDF format.
Steps Toward a Computational View of Tonal Music: Modulating Oscillatory Systems a description of the central features, principles and mechanisms of modulating oscillatory systems and their application to music theory and analysis - 400Kb PDF format.
Steps Toward a Computational View of Tonal Music: Modulating Oscillatory Systems - Part II the application of the ideas and methods outlined in part one to the analysis of Prelude No. 1, The Well-tempered Clavier by J.S.Bach and a passage from the Piano Sonata in C major K545 by W.A. Mozart. Understanding some of the more difficult diagrams and explanations in part one will be eased by reference to the worked examples in part two. - 408Kb PDF format.
Mathematical Aspects of Modulating Oscillatory Systems a few mathematical observations concerning the nature of modulating oscillatory systems focusing on nested harmonic structure, physical numbers and a method of factorising numbers by 'listening' for resonances. - 130Kb PDF format.
Chord Types, Interpreted in Terms of Nested Harmonic Series in Modulating Oscillatory Systems. Although by no means an exhaustive catalogue of all chord types (perhaps in time others will be appended), hopefully, this collection of the basic types is enough to at least demonstrate, how the technique of nesting one harmonic series within another, can be used to elucidate the nature and attributes of chords in tonally organised music. - 51Kb PDF format.
Introduction to the book I am attempting to write: A Journey to the Heart of Music. The style and character of the book will, hopefully, put the concepts and ideas behind modulating oscillatory systems in a form accessible to the general reader, particularly those readers with some interest in, and knowledge of, classical music.
Reflection: New-from-Old Music an article on harmonic reflection - the arithmetic component in music - an area related to the dualistic theory of Arthur von Oettingen and the mirror fugues of J.S.Bach; plus example scores and MIDI files. - 280Kb PDF format
Elements of Music? - Archive Nov.03 Vol.17.2 - June.05 Vol.18.9: a speculative and exploratory set of articles and files examining structure in oscillatory systems.
The Periodic Table - Modulating Oscillatory System - a seven page wall chart to print out - 36Kb PDF format -36Kb PDF format.
The Resonances of the Planets - a chart of the Solar System, plotting the planetary orbital periods relative to the rotation of the Sun's outer layer - 35Kb PDF format.
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