Fractal Geometry of Music

From the Rosalyn Tureck Collection,
Howard Gotlieb Archival Research Center at Boston University

[Editorial note: This letter seems to be a response to a journal article by Kenneth Hsu and Andreas Hsu entitled "Fractal geometry of music" and published in Proceedings of the National Academy of Science, Vol. 87, pp. 938-941 (February 1990). Full text available here. It is not clear whether it was ever published, or even sent, to the journal.]

Dear Sirs:

I am delighted to see the interest in the deeper levels of musical composition as expressed by Kenneth Hsu and Andreas J. Hsu. However the article sets down assumptions and conclusions about musical structure which are not only based in subjective interpretation but also contain sweeping blanket statements which cancel out any possibility of the validity of their claims. Moreover their presentation leaves a great deal to be desired. Firstly, in presenting any kind of claims having to do with musical structure, it is imperative when dealing with specific motives and compositions to include examples of those motives and/or compositions. The necessity for the inclusion of the musical example is equaled by the necessity for including the mathematical formulations. It is as necessary for the reader to have the musical material for verification as to have the mathematical formulations for consideration. Even if one knows the exact musical material the authors are referring to, their terminology, in the case of the two Bach Inventions which they refer to,is incorrect and makes impossible the analogy which they are trying to draw. They speak of the 'first movement' of the two Inventions which they have selected as a major demonstration of their theses. There is no such thing as a 'first movement' of an Invention. There is a first motive, there is an exposition, but Inventions do not contain movements, in any language. Therefore, an informed reader cannot equate their claims with any part of the Inventions except by guesswork, which is surely not the intention of such an article. Further, "the unusual deficiency of the full-tone interval (1) equals (2) is not a chance neglect, but a deliberate measure by Bach to achieve a special effect through the establishment of the small third as the most frequent note interval." The terminology continues to be either uninformed or naive. Thirds are known as major and minor thirds, not as large and small. If there is a problem of translation here it would be very helpful to choose the more universal terminology. More important the authors' certainty about Bach's "deliberate measure to achieve a special effect through the establishment of the small third as the most frequent note interval" is a totally personal assumption. Composers hardly compose in this way. It is one thing to make an analysis of the musical structure of a piece of music, it is another thing to say that the Bach worked "to achieve a special effect through the establishment of the small third as the most frequent note interval." Firstly, no one will ever know what Bach was trying to achieve in his own thinking. Secondly it is extremely presumptuous to assert that the establishment of the minor third as the most frequent note interval was a special effect that Bach was trying deliberately to achieve. This patently nonsense. That Bach created a work with a focus on the minor third is one thing, but to impose upon him his attempt to achieve special effects by deliberate measures in order to achieve the most frequent note interval is an imposition of the authors' wishes to fit their thinking to Bach's compositional processes. In the next sentence, "the excess of the sonorous fifth (i = 7), another significant deviation from the fractal relation, is also no accident." This is pure presumption and creation on the part of the authors. Bach was clearly not concerned with the fractal relations and fractal theories. The fact that the authors find a "significant deviation" simply demonstrates the fact that their theories are completely artificially created. "Deviations" are a very convenient way for excusing the inapplicability of a theory. This is not to say that the possibility of analysis of music in terms of Mandelbrot's theories is impossible. However, so far the giant strides and subjective assumptions of this paper are utterly invalid. To continue: In reference to Bach's Toccata in F# minor, BWV 910, they say of the Adagio movement here that "The F/i plot has a pattern intermediate between classical music and modern atonal music." The span between what the authors term 'classical music', and that of modern atonal music is so enormous and there have been so many styles between them and so many concepts of musical structure and so many degrees of so many different structural compositions that this phrase is absolutely meaningless. They continue by saying; "Bach tried to search for something new with his composition." How do they know that? They continue: "The Adagio movement is particularly modern in the sense that it seems to pose the question, to increase tension before the problem is resolved, and this is apparently achieved through the notable deviations from the fractal relation." Actually the structural picture that they paint here "to pose the question, to increase tension before the problem is resolved," is a situation which belongs to the kind of thinking of a short period in the 19th century. This has nothing to do with Johann Sebastian Bach's processes of thought and composition whatever, and the imposition of the authors' is astounding. They go on to say regarding this description which is totally invalid—"This is apparently achieved through the notable deviations from the fractal relation." Firstly, their assumptions about Bach's sense of structure are totally incorrect and they make a conclusion which fits just as they wish it with their theory of fractal relations. Further they claim that Bach "tried to search for something new with this composition." Actually, the Toccata in F# minor, BWV910, is one of a series of toccatas which Bach composed in the teens of the 18th century when he was still quite young. Bach was not the type to seek a new idiom. His greatest genius lay in his absorption of centuries of the development of the Western idiom structure and style and his capacity to absorb virtually everything that had happened for four hundred years. The integration which appears in his composition in the variety of the forms that he used and the variety of harmonic structures based on the developed major/ minor scales, but also harking back, at times, to earlier scales is all part of the astounding genius of Johann Sebastian. But he was not a character to search out for something new. He was also gifted in absorbing influences of his own time although on the whole he was regarded as an old fuddy-duddy, a conservative who was out of date. This was an early period in Bach's work. It was not a period in which Bach was searching for new ideas in the least. It was a period in which he was applying and amalgamating the various influences upon him. Traveling on to Mozart, here the authors do use the term again, "the first movement", this time in reference to a Sonata by Mozart. Here the terminology "first movement" is applicable but it is extremely mistaken to use such a terminology for a sonata and for an Invention in which such section does not even exist. Again the music to which they refer is required in order to make sense of their various claims about fractal relations in this work. Here again there is a "notable deviation". They speak of "The deficiency of the diminished fifth (i = 6) is common to all the compositions analyzed." Firstly, so far there are only the two short Inventions of Johann Sebastian and a first movement by Mozart, which hardly make up enough material to speak of "all of the compositions". They go on to make a vast, utterly unfounded claim, protecting themselves by saying "If so, Mozart's music could be considered pictorial, whereas Bach's is precision in mathematics." Here again is a vast statement utterly unaccounted for and not backed up by a solidity or sufficiency of evidence. In any case such claims are for the amateur. Again they are blanketing Mozart whose output and range was vast into one cubby-hole of "pictorial" and Johann Sebastian, who has suffered too long from weakminded judgments about mathematics, to say that: "Bach's is precision in mathematics." These statements are truly too superficial to be even considered with any seriousness. They go on to folk songs saying: "The most notable feature is the excess of note repetition (i = 0)." It is the authors' subjective judgment that the note repetition constitutes an "excess". Perhaps it doesn't. Their following question: "Is this a manifestation that children's music tends to be monotonous?" Again this is an extremely light-weight judgment about note repetition. It depends entirely on relationships of many factors even in a simple children's song; many factors which arise from the harmonic structure, the melodic structure, the rhythmic structure and the relationship of all these to each other. A general statement that children's music which has note repetitions tends then to be monotonous even when presented as a question is in itself utterly unbased in the reality of even what appears to be a simple form and what is judged to be a simple song. In moving on to the reference to the first movement of Mozart's Sonata in A major, K331, and asserting that this music is based on a children's song, the authors find that "The fractal relations of the two are amazingly similar."

(end side I)

Side Two

(cf. fig. I, E, F). It is not difficult to recognize that a number of relations would be similar and this would not be an "amazing" occurrence if the movement or even a section of the movement is based on a children's song. Obviously the children's song is the basis of whatever is composed through or around it and therefore the similarity is inevitable. In fact, it is not possible to envision a lack of similarity between a composer's composition and the work upon which it is based. When the authors say: "Not surprising is the total absence of the 'devil's note' the diminished fifth." Without spending as much time as the subject of the diminished fifth deserves, it is not surprising at all that the diminished fifth is generally avoided by composers for as many as four centuries or so because (1) it was, in fact, one of the intervals that was considered forbidden as was the diminished fourth. Therefore the avoidance of the diminished fifth is nothing very special in the very consonant music of Mozart's idiom, by that time. By the mid to late 18th century such intervals had virtually lost their existence so there is no justification in pinning poor Mozart down to omitting the diminished fifth within the context of the authors' theory of fractal relations in music. It is a pity that they continue to characterize via the term 'pictorial' a characteristic of Mozart, despite the fact that this appears in this article in the form of a question: "Is that a coincidence or is being pictorial a characteristic of Mozart?" What by the way do the authors mean by the "pictorial"? They never define the term. There are many ways of characterizing pictorial and to speak of a characteristic of Mozart" as being "pictorial" is such a sweeping statement that it becomes meaningless. They are in difficulties with their fractal theory admittedly, when they say: "The fractal relation of Mozart's Sonata, imperfect as it is, ..." Alas, here again Mozart's Sonata does not fit very neatly into their theory. To go on by saying "to illustrate the obvious difference between classical and modern music," is to mistake the entire history of the development of structure, style, form, compositional processes, psychological rapprochement with sound, sonority. etc. etc. All this occurs in the ever moving changes from era to era. One cannot speak of "the obvious difference between classical and modern music". This is ludicrous. Firstly when they say "classical" music are they referring to the classical era which is exemplified by Haydn and Mozart, or are they using the term as amateurs usually do as referring to non-popular music. Or it appears as if they may he lumping the entire development from Johann Sebastian Bach's time until the time of Stockhausen—a mere matter of two hundred years of extremely subtle as well as powerful changes in sense of form, sonority and structure. There is no such thing as "the" obvious difference. The density of the kaleidoscopic movement and development in music can never identified under any "the". It is astonishing to read the sweeping statement: "The notable deficiency of the great third [by which obviously they mean the major third] (i = 4) and the extreme excess of the diminished fifth (i = 6) are what make modern music atonal". If life and art were as simple as the identification of the deficiency of one particular interval and the excess of another particular interval, clearly art wouldn't be worth very much, and virtually anybody could do it. As a result of this very slight study which so far has not produced anything that might be seriously considered because of its repeated assumptions.

The frequency of incidents of note intervals of music is by no means a methodology for determining the relationships that constitute an artistic composition. If one is looking for a fractal geometry of music, which I applaud wholeheartedly, then the steps taken must be based on something immeasurably more solid and accurate than what appears in this article.