A key principle of this document is that Nature is full of interconnecting scale structures. This will be elaborated in later sections, but first the terminology needs to be defined.

The term scale structure is applied here in the broadest sense of a musical scale: relative dimensions and relations among parts that together form logical wholes.


…ordering divisions, combinations, ratios, or operations, applying to fields of elements, or events. Scale structures may apply horizontally across a range of parameters of any generic quality or quantity; and/or may apply vertically spanning hierarchies or dimensions; or may apply complexly and variously as syntactical organization (frequently reflecting the emergent, self-organizing behavior of complex systems). They will often exhibit the phenomenon of ‘resonance’ and recursion. Many are fractals, or have fractal-like properties. They organize, and are organized. They determine, and are determined.
W. Roberts2, 1996

To qualify as a scale structure in this document, the structure or principle (which may be abstract or ‘ideal’ as in geometry) must be applicable in more than one situation: it must tend to the generic rather than the specific; it must concern relative values rather than so-called ‘absolute values’. In this there is no room for absolute pitch, absolute measurement, etc. Every entity can only be defined in terms of some other entity or entities. This follows from the profound interconnectedness of the universe.

The meaning, understanding, and relevance of scale structures will become apparent as we look at the precedents for scale structures and their ubiquitous occurrence in nature, mathematics, music, science, history, language, and finally, their potentially enormous application in the visual arts and ‘thinking’.

Visual art which syntactically links its apparently haphazard external forms to the rhyme and measure of various covert scale structures will result in visual works of enormous diversity which combine logic and intuition in concord with the physiology of the mind. Progress in this direction is likely to occur hand-in-hand with the development of new paradigms for thinking across the disciplines.

This haphazard arrangement of forms may be the future of artistic harmony. Their fundamental relationship will finally be able to be expressed in mathematical form, but in terms irregular rather than regular.
W. Kandinsky3, 1911

  • early methods of recording numbers were cumbersome and impractical
  • efficiency in representing numbers improved when it was realized a single symbol could be used to represent intermediate numbers. These were then variously combined to represent larger numbers (eg groups of tally marks)
  • the invention of new symbols (eg the Roman numerals) for a range of intermediate values allowed for the economical recording of significantly larger numbers.
  • further developments saw more syntactical ways of combining numerals such as the rules which governed the value of groups of numerals in Roman notation. The advantage of this was that additive and subtractive rules (depending on whether a numeral came before or after another numeral) allowed for large numbers to be represented with even fewer numerals
  • a major development came with the introduction of fully positional systems. In these a numeral acquired a second meaning which derived from its position (or column) within a number, and which acted as a multiplier of the numeral’s face-value. For example in our base ten number system , the numeral “3” in the number “300” represented 100 multiplied by 3.

The discovery of an efficient method for writing or representing numbers was a critical development in the history of mathematics. Even the simplest of patterns among numbers had until then remained largely hidden, obscured by an often cumbersome or inconsistent system of numerical notation. For example, although the Roman system of representing numbers possessed a syntax which determined the reading of numerals within a sequence, this resulted in numbers whose component numerals bore no consistent positional meaning from one number to the next but instead obtained their meaning locally – according to the adjacent numerals within a given number. This meant that although numbers could be organised into rows and columns of numerals, there was no consistent positional meaning of a numeral in a given column from number to number. A special device (abacus) was invented in order to achieve speedy operations (such as addition and multiplication) between numbers, and it is interesting that the abacus which itself possessed a columnar logic did not quickly ‘straighten out’ the columnar inconsistencies of the Roman notation it served.

With many of these early number systems, operations which should have been simple (such as addition and multiplication) were unnecessarily difficult. The difficulties stemmed from the fact that they did not have a place-value system; the notation and its organisation had no intrinsic logic or resonant scale structure. Nor did many of them have a symbol for a null quantity, the zero.

The combined innovations of, firstly, unique symbols for the integral divisions of a base (for example, in base 10, nine numerals were necessary) , secondly, a consistent place-value or positional system in which each column represented incremental powers of the base (eg units, tens, hundreds, etc), and thirdly, the adoption of a symbol for zero, were first harmoniously combined within a system of numerical notation in subcontinental India. This system supplanted virtually every other competitor it met and is more or less the system of notation used to this day.

These combined innovations resounded like harmonics on a vibrating string. The number notation came of age and opened a door to a new and fascinating abstract world of number we now know as mathematics. It was a closing of the circle. This new way of writing numbers contained an inner logic which enigmatically reflected relationships on earth and in the heavens. Ultimately this logic would enable humankind to predict the motions of planets, to measure time, and navigate space. John Barrow sums up the eventual breakthrough in the history of math this way,

. . . something strange happened. These symbols were found to have a life of their own that dictated how they should be manipulated. And the ways in which they were used enabled new facts about the world to be predicted.2

He goes on to note that the move to abstraction in mathematics (the concept of ‘number’ detached from ‘numbers of things’), occurred when the method for recording numbers (the notation) reached a certain threshold of logical organisation itself.

. . . something strange happened. These symbols were found to have a life of their own that dictated how they should be manipulated. And the ways in which they were used enabled new facts about the world to be predicted.
J. Barrow2

to Linda,

Natasha , Matt, and Jackson

to my parents & family on both sides
and to friends

doctors, medical workers & researchers.
In the hope of the discovery of
new cures in medicine.

 To musicians and artists of the future,
new harmonies are in your hands

To the memory of
Sir Alexander Fleming (1881–1955)
Nobel Prize-winner, Medicine, 1945

This book & web site would not exist were it not for his discovery—my mother’s life-threatening childhood septicaemia was cured by some of the first ampoules of Penicillin available in Sydney, Australia, many years before I was born.

To all
who bring, or aspire to bring,
healing to the Earth.

and to (the late) Dr Darryl Reanney,
whose books inspired this one