|Wayne Roberts © 2003-2015
Scale structures & asymmetric symmetries
Scale structures result from the outworkings of Universal interconnectedness. There exist resonant points and divisions of wholes and these generally reflect an integrative process in which every part contributes to the emergence of 'resonant forms'.
Sometimes a new perspective or paradigm shift is necessary in order to recognise an often hitherto-covert natural rationalism or scale structure (as will be demonstrated later in this document). Such forms may entail asymmetric but no-less 'rational divisions'. The resonances are like harmonics on a vibrating string—the harmonic positions (scale structures) are not independent of the length of the string but in fact dependent upon it. Scale structures as discussed in this text are thus generally dependent (in an integrative sense) and may be seen as the emergent outworkings and forms within systems (be they linear or nonlinear).
Introduction to scale structures (from a musical perspective)
Scale structures often simultaneously entail symmetric and asymmetric qualities. As I am invoking the term "scale" in the musical sense, music is a logical place to look for a good example of this. Take, for example, the ubiquitous major diatonic scale of western music and JS Bach's division of an octave into twelve equally-spaced semitones. [On the modern-day keyboard these semitone (half-tone) intervals are played sequentially and ordinally by pressing the next most adjacent key in the upward or downward direction and irrespective of whether it be a black or white key.]
Curiously and incredibly, the 'major scale' of classical and popular music simultaneously combines both symmetric and asymmetric attributes: