The Octave As Fractal Numeric Object

 

53-step Musical System

 

The Basic Symmetry of I Ching

 

 

 

2005 B.Svarog

waruna@yandex.ru

 

This whole system impresses with a perfect regularity showing for any unprejudiced glance a certificate of absolute mathematical strictness of the quint system. It does not matter, that in the musical theory an appearance of commatic changed intervals, has served only for a greater proof of "impracticability" of the quint build more than three centuries.

Music, as it is not difficult to make sure, by its nature is close "to mothers" - subconscious, laws of number and time and nor to a small extent is a result of agreements (convention) or collection of arbitrary rules. It is another thing, that the internal need, developed by the European musical culture, was going contrary with objective existing progressions of "real harmony" or, better to say, only the "relations of I order" were significant for its musical practice - namely those assigned by the first 12 steps.

A comma was perceived as undesirable ("13-th" in the quint circle) and stimulated searches for a closed musical system (there the end in accuracy may be coincided with the beginning). "To curve" was in the nature of an Octave, the rationalism of the XVII century decided to unbend it. An utterance of lister Crawly is very much to the point here:

 

For the blindness to beauties and wonders

of the Universe the mankind is obliged to their illusion of rectilinearity".

 

The dizzy success of temperation - of whole European classics beginning from J.-S. Bach - has led to a "partial mnesia" about the nature of numeric roots of music. "Having completely leveled in rights" twelve intervals - simply by giving them a value - a new musical theory, however could not help saving an asymmetry (harmonic pattern) explained only in Pythagorean theory - just, the relation between white and black keys: 7 main + 5 changed = 12 . However, we must admit that European classics, developed on abstract harmonic principles, in its most perfect samples could approach closely to "objective art" ideals.

 

 



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