Self-similarity

The coherencies which exist in music have to agree with each other in any scale and dimension in which they are being perceived. The auditory experience which comes from a performance in the way that the sound of the instrument (a single instrument or an orchestra) matches the music, gives complete freedom to the listener to choose the scale of audition. However, it enslaves him or her by providing the same message at every level. The listener is free to tune in at any scale of perception. However, the composition has a single feeling to it. In fact this feeling may change in every performance. However those different feelings are in turn related to each other by the integrity of the piece. A score of a composition is the coding of a musical idea in some accepted dimensions as parameters for the sake of communication. Schoenberg says[40, page 220]:

THE TWO-OR-MORE-DIMENSIONAL SPACE IN WHICH MUSICAL IDEAS ARE PRESENTED IS A UNIT.^2.17Though the elements of these ideas appear separate and independent to the eye and the ear, they reveal their true meaning only through their co-operation, even as no single word alone can express a thought without relation to other words. All that happens at any point of this musical space has more than a local effect. It functions not only in its own plane, but also in all other directions and planes, and is not without influence even at remote points. For instance, the effect of progressive rhythmical subdivision, through what I call `the tendency of the shortest notes' to multiply themselves, can be observed in every classic composition.

Such a definition perhaps takes a dimensionless concept such as the musical idea and projects it onto a plexus of dimensions for communication. The fact that every part is part of a whole and abides by global law while at the same time, as Schoenberg says, ``all that happens at any point of this musical space has more than a local effect'', can be modeled with self-similar structures and self-referentiality. It is true that such a model tries to capture a sense of aesthetics and romantic feeling about music; however, there is no need to fear since we will never reach a true self-similar shape since they only exist in infinity. The idea of self-similarity can also capture the uniformity of time and perception. Stockhausen has noticed this fact as well, and one of his acoustical piece, Mantra (1970), may be called a fractal piece. About it he writes[47, page 57]:

I can give an example of a more recent concept of sequential form, my composition MANTRA for two pianos and electronic modulation. In this work I use a 13-note formula, and nothing but this formula throughout the whole duration of the composition. The formula is expanded and compressed in its pitch and time intervals, but it is always the same formula. Each note of the original statement of the formula has certain characteristics: a periodic repetition, an accent at the end of the note, an ornament, and so on, these characteristics are seeds of later development. The structure of the whole composition is an enlargement in time of that one small formula to more than 60 minutes, and the sections of the composition correspond to the notes of the original formula, and their characteristics. The form is sequential, but with an overall development.