Theory - The Functional Wave Fractal

Processing Opposites or The Functional Wave Fractale


Opposite“: Adjective

1. situated, placed, or lying face to face with something else or each other, or in corresponding positions with relation to an intervening line, space, or thing: opposite ends of a room.

2. contrary or radically different in some respect common to both, as in nature, qualities, direction, result, or significance; opposed:

opposite sides in a controversy; opposite directions.

3. being the other of two related or corresponding things:

friendly with many members of the opposite sex.


"opposite." Unabridged. Random House, Inc. 01 Aug. 2015. <>.


Fractal“: A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. If the replication is exactly the same at every scale, it is called a self-similar pattern.





This article with its somewhat loaded title was written in an attempt to describe an approach to musicla theory, that starting from the premise of opposites or 'dualities' one can determine extreme points, measure out points in between and by going through them step-by-step (or otherwise) within a single form to make maximum use of a variety of musical qualities – and therefore 'processing' those opposites in an interesting and engaging way. In order to do this, this article will suggest the idea of a 'Wave Fractal', which really is a system of relating various musical-technical aspects to each other after having each of them 'compressed' into a wave form (somewhat akin to a sinus wave).

The Motivation behind this was originally to find a way to make use of extremes and opposites, but later also included just to increase the manipulative ways of a musician to manipulate forms and still being able to tie them all together. This article does in no way suggest that other approaches are worse or better, more/less or invalid, or that using a 'Wave Fractal' (whatever the hell that is supposed to be) is necessary to write 'good music' (whatever the hell that is supposed to be). It is simply a new approach that was formulated to meet the musical needs of the author.



Wave Fractal

Since the possible amount of theory giveable here is quite big, I will try to formulate this section as concise and briefly as possible – at the expense that each point covered will have a lot of condensed information in very little space, therefore making it possibly necessary to ponder single points more than once, all of course depending on the reader being interested enough.


I. General Form of the Wave

First, I will establish that by defining an upper and a lower extreme, we are able to form a Continuum; Secondly, by measuring the points in between we can create a range of specific points that can be targeted within that Continuum; and thirdly, by connecting these points one after the other, we can make up a form that looks like a (Cosinus-)Wave. Why do I reference the Cosinus-/Sinuswave? It is the form we can associate with a single, very 'simple' tone, therefore establishing 'some' mental connection between form and sound and therefore making the concept more grasp- and workable (I would hope).




II. Forming specific music-related Waves

Next, I will take the freshly-formed Wave and apply it to a concrete existing musical quality – in this case, for starters, dynamics. I will for this example define ff - Fortissimo (Very Loud) and pp - Pianissimo (Quiet) as my extreme points. Next, I will fill in the middle points with forte, mezzoforte, mezzopiano, piano, and an imaginary 'exact middle point' (which for some reason doesn't quite seem to exist in musical theory). Thirdly, I will connect the points to get 'my' Sinus Wave again. What can we do with that' Well, it can be used as a 'model for dynamics', which can be played through in a composition or an improvisation – and since my aim was processing extremes and manipulatability, I have made that now happen. However, we still haven't formed that ominous 'Wave Fractal' – before we do that, I want to introduce two further processed Waves however, the first being a 'Melody Wave', compromising of a C Ionian Scale over two Octaves (and simply being arranged as ascending and descending seconds), and the second being a 'Rhytmic Density Wave', covering a possible range of density of a bar of 4/4 (limited to whole, half and quarter notes in this example). With these in the backpack, we will now go on to tie them together.




III. Forming a Wave Fractal

The observant reader might have noticed that the three aforementioned models per se don't necessarily interfere with each other – meaning, that the dynamic quality doesn't necessarily say anything about the melodic quality or the rhythmic density and vica-verse. Therefore, we are able to tie them all together into a single example of music – and in diferent ways actually. A couple of possible ways could look like this:

(Fractal A- Dynamics, Melody and Rhythmic Density as Sinus;

Fractal B – Dynamics as Cosinus, Melody and Rhythmic Density as Sinus;

Fractal C – Dynamics and Rhythmic Density as Cosinus, Melody as Sinus)




In all three instances now we used 'only' the aforementioned Waveforms – thereby making it a structure being 'more or less' compromised only out of waves, thereby making it a Wave Fractal. It goes without saying now that this example here has covered a very small portion of possible territory – not only can pretty much any abstractable musical/technical ability be 'made' into such a Wave (and be woven into a Fractal), the same quality might be 'waved' after different aspects, as for example in harmonic density (how many different notes simultaneously) versus harmoic tension (how consonant/dissonant do the notes being used sound). Or both at the same time.

What Waves/Qualities are how being used is than mainly limited and decided by the person using this kind of approach – combined with enormous amount of possible compitabilities, this makes a theoretically unlimited reservoire of ideas and possible outcomes. If one wants to use this approach of processing opposites and tying them together of course.




So, in a nutshell, what is this whole thing about now? Briefly summarized, the ideas are as follows:


  1. By defining Extremes, one is possible to form Waves (formwise akin to the Sinuswave)
  2. By processing different qualities into such Waves, a multitude of aspects become manipulatable and a 'wider range' of 'technical' expressions in a 'small room' becomes possible
  3. Tying all those together into single forms results into forms I call 'Wave Fractals', which in one swoop will contain a variety of wide ranges while maintaining a high degree of musical and theoretical explainability, while 'processing (previously defined) opposites'


And what do we need this for? Well frankly, we don't – Except of course if you are coming from the specific demands I had formulated and are interested in 'new approaches' to writing or playing music, in which case you might find this interesting.

What might be some possible criticisms of this kind of approach? First of all, the obvious enourmosly high degree of theoretical work that goes into this, and second the possible consequence of 'overmechanizing' and 'dehumanizing' (or so I would guess). I don't really have a definitive solve-it-all answer to such possible claims, except maybe the pointing out of the fact that styles such as popular and classical music are both analysable within very well defined theoretical frameworks, that (in my opinion) cover way less ground than this approach does – and there at least it doesn't seem to bother most people listening to it(because if something sounds 'dehumanized' or not will ultimately be decided by a majority vote). Also, all works within those styles are still subject to the fact that what is in the piece had to be put there by the creator of it – therefore making the content of the work more of an obligation and responsibility of the creator, not the theoretical framework used (if one was used).

And to conclude this article with some healthy shameless self-promotion, (some) practical examples of music written following (among others) this genereal approach of a Wave Fractal can be found on my Website, my Youtube Channel, various Social Media outlets and of course my recordings, available through various online outlets such as Amazon, iTunes and Google Music. For Quickstarters, try 'Elegie', 'Mirage', 'Passacaglia', 'Himmel' or 'Sonne' (all except #3 available on Youtube).

Until then, keep Waving. (Oh the Jokes...)

Find more interesting stuff on David's Music Guild, where I put up videos about everything you (n)ever wanted to know about music!