Saturday, 9 March 2019

Implication-Realisation and the Entropic Structure of Everything

The basic structure of any sound is that it starts from nothing, becomes something, and then fades to nothing again. In terms of the flow of time, this is a process of an increase in entropy as the features of the note appear, a process of subtle variation around a stable point (the sustain of a note, vibrato, dynamics, etc) where entropy will decrease (because there is less variation than when the note first appeared), and finally an increase in entropy again when the note is released.

A single note is rarely enough. It must be followed or accompanied by others. There is something in the process of the growth of a piece of music which entails an increase in the "alphabet" in the music. So we start with a single sound, and add new sounds, which add richness to the music. What determines the need for an increase in the alphabet of the sound?

In the Implication-Realisation theory of music of Eugene Narmour, there is a basic idea that if there is an A, there must be an A* which negates and compliments it. What it doesn't say is that if the A* does not exactly match the A, then there is a need to create new dimensions. So we have A, B, A*, B*, AB and AB*. That is no longer as simple as a single note - for the completion of this alphabet, we not only require the increase and decrease of entropy in a single variable, but in another variable too, alongside an increase and decrease in entropy of the composite relations of AB and AB*. The graph below shows the entropy of intervals in Bach's 3-part invention no. 9:

What happens when that alphabet is near-complete, but potentially not fully complete? We need a new dimension, C. So then we require A, A*, B, B*, AB, AB*, C, C*, AC, AC*, BC, BC*, ABC, ABC*. That requires a more complex set of increases and decreases of entropy to satisfy.

The relational values AB, AB*, AC, AC*, ABC, ABC* are particularly interesting because one way in which the entropy can increase for all of these at once is for the music to fall to silence. At that moment, all variables change at the same time. So music breathes in order to fulfil the logic of an increasing alphabet. In the end, everything falls into silence.

The actual empirical values for A, B and C might be very simple (rhythm, melody, harmony) etc. But equally, the most important feature of music is that new ideas emerge as composite features of basic variables - melodies, motivic patterns, and so on. So while at an early stage of the alphabet's emergence we might discern the entropy of notes, or intervals or rhythms, at a later stage, we might look for the repetition of patterns of intervals or rhythms.

It is fairly easy to first look for the entropy of a single interval, and then to look for the entropy of a pair of intervals, and so on. This is very similar to text analysis techniques which look for digrams and trigrams in a text (sequences of contiguous words).

However, music's harmonic dimension presents something different. One of the interesting features of its harmony is that the frequency spectrum itself has an entropy, and that across the flow of time, while there may be much melodic activity, the overtones may display more coherence across the piece. So, once again, there is another variable...

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