Introducing all-interval fractal sets

Introducing all-interval fractal sets

My interest in writing music that is technically systematic, possessing a strong inner logic, dates back over several decades. I began many years ago to build what I called “micro-macro” structures. When I eventually encountered the work of the Polish-born mathematician Benoit Mandelbrot, I realised that these structures were examples of fractal forms. In 1979 Mandelbrot revealed his discovery of what came to be known as the Mandelbrot Set. He used it to create breathtakingly complex computerised images. He argued that the multiplicity of organic forms in the natural world is the result of the multiplication of simple self-generating patterns capable of replicating themselves at every structural level from the microscopic to the macroscopic. The structure of a fern leaf, the florets of a cauliflower, the pattern of streams and tributaries of a river valley were all examples of what Mandelbrot called Fractal Geometry.

As far back as 1975 I began using 7-note all-interval sets as a way of governing all the structural levels of a composition, from pivotal notes marking the succession of movements, through sections within movements, down to the smallest detail of melodic contour and ornamentation. I’m certainly not the only composer to interest themselves in processes of this sort. Per Nørgård’s “Infinity Series” is a classic example.  One of the characteristics of Nørgård’s series is that we keep on finding the original series at different scales, extended in time, at different pitches, inverted and non-inverted. These features are characteristic of fractal geometry. Nørgård’s series exhibits two of the fundamental principles of fractal geometry: self similarity and structural invariance.

Of course there is nothing new about all-interval sets. The earliest example I know of is in the first movement of Alban Berg’s Lyric Suite for string quartet (1926):

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Its symmetry emerges if we convert all the secondary intervals (those larger than a tritone) to primary intervals (smaller than a tritone). Berg has constructed a circular 12-note row consisting of two interlocking hexachords, the second a retrograde transposition of the first:

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In 2009, I embarked on a somewhat gruelling process of trial and error to see if there were more sets of this kind. At first glance the task seemed daunting. The total number of permutations of six interval numbers is factorial 6 (6!) = 720. Taking account of the fact that each digit may be preceded by a plus or minus sign, the number of possibilities expands 42 times, bringing the total to 30,240. However symmetries began to emerge enabling me to speed up the process very considerably and to double-check my results. I discovered that Berg’s set and the ones I had been using for several years were defective in various ways. They were indeed all-interval sets, but they were not strictly fractal. In order to be classed as fractal, a set must meet three conditions:

  1. It must contain 7 different pitch classes.
  2. It must contain all six primary intervals as melody intervals.
  3. It must contain all six primary intervals as resultant intervals (the interval between the first pitch and each successive pitch)

The following example of a truly fractal all-interval set makes this clear:

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In the first line all the melody intervals are shown as primary intervals. (Interval 6, the tritone, is plus or minus.) In the second line the numbers indicate resultant intervals relative to the first pitch. Some of the pitches have been transposed up an octave so that the resultant intervals are all primary.

The following is an example of the kind of pattern which fractal sets can produce. In the first line a succession of fragments of different lengths using the above set produces a higher-order structure in which the first note of each bar is part of another all-interval fractal set. In the second line, this new set is multiplied upon itself to produce another pattern which is rhythmically complimentary to the first. That is to say that when a bar in one voice contains six pitches, the other voice has one; when one voice contains five pitches, the other contains two; when one voice contains four pitches, the other contains three. Thus every bar contains six different pitches but differently distributed between the two voices. The letters A and C refer to various modal classifications. It so happens that Mode A is the well-known octatonic scale of alternate tones and semitones much used by Messiean.

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In fact very few of the vast number of possible permutations meet these three conditions. The actual number of permutations of six interval numbers that produce true fractal sets is only 53. Of these, 47 each exist in two forms differentiated by varying patterns of partial inversion. The remaining 6 permutations have no viable partial inversion. Among the resulting total of 100 patterns there is a subset of just 12 unique forms that exist in relationships of complimentary rhythmic pairing. The pairing shown above is one of them.

As a composer what use can I make of musical forms that are so uncompromisingly mathematical in their derivation? What can I possibly communicate with them? Paradoxically the sets are extraordinarily lyrical and song-like in quality and not without a hint of Second-Viennese angst. I have found them immensely fruitful. They have been productive of both melody and harmony, as well as rhythm and musical form, first in a set of 12 preludes for piano, then in a string quartet, followed by two pieces for the CoMA Bristol Ensemble and three song cycles. The most recent, “Dark Seas”, is a setting of five poems of Philip Larkin for coloratura soprano, clarinet and piano written for Sarah Leonard with the help of a grant from Arts Council England.

I do sometimes privately lament the apparent lack of technical rigour in some of the contemporary compositions I encounter, but I wouldn’t presume to criticise other composers for such a lack. Neither would I presume to suggest that my fractal hexachords are the ultimate solution like some sort of successor to the 12-note method, but they work for me. If other composers want to use them, or something like them, it’s entirely up to them.

Nevertheless, when I discovered these forms, I felt sure I was on to something and wanted to share what I had discovered with other composers and musicians. It seems to me that they embody some kind of musical truth or, to be more precise, a mathematical truth expressed in music. I know that there must be a mathematical principle behind them but I do not know what it is. I have shown the patterns to several mathematicians, but they have been unable to understand the musical concepts with sufficient clarity to discern what the mathematical basis might be. On the other hand fellow composers have appreciated their musical elegance but have been no better able to discern the mathematical basis than I have. I invite readers of the Institute of Composing Journal who feel they can throw any light on this issue to read my extended paper on all-interval fractal sets and get back to me with any useful feedback.

The paper can be downloaded as a PDF from my website at jolyonlaycock.uk/theoretical-writings

You can also download a copy of the powerpoint presentation which accompanied my talk “Of Fern Leaves and Cauliflower Florets – fractal patterns in music” given at Bristol University in March 2014 at: severnsidecomposersalliance.co.uk/newsdetails.asp?news_id=156

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