Algorithms of Ecstasy

22 February 2009


The scientific treatment of music began with Pythagoras.

The scientific treatment of music began with Pythagoras.

There is an old and established literature concerning the physics of music, starting with Pythagoras and Euclid in the ancient world. And ever since Euclid and his mathematical treatment of music and the medieval Quadrivium which included a systematic study of the ratios of music, there have been philosophers who see in the harmony of music a miniature of the harmony of the world, and especially the music of the spheres. It is the familiar theme of microcosm and macrocosm played out on an aesthetic level.

Before logarithms were formulated, Euclid's algorithm was used to calculate equations of musical intervals.

Before logarithms were formulated, Euclid's algorithm was used to calculate equations of musical intervals.

It is interesting to note that musical theory has greatly advanced since the days of Euclid and the Quadrivium, particularly in the twentieth century, and has gone far beyond early, simplistic notions of harmony and tonality with innovations like the twelve-tone system. But despite this remarkable advance in sophistication in the understanding of the structure and meaning of music, many people still want to see the world as “ultimately” harmonious just as they want to find the same ultimate harmony in works of art and are disturbed (and sometimes even angry) at the innovations of modernism which make no attempt to satisfy this human need for harmony and instead thrust themselves out into new territory by exploring dissonance and atonality.

One of Schoenberg's many self-portraits.

One of Schoenberg's many self-portraits.

Intuitively we do not like to think that the music of the spheres, if there was such a thing, would sound like Schoenberg — i.e., that there is a kind of parallelism between the structure of Schoenberg’s music and the structure of the world — but on reflection we ought to admit that modern physics has given us a world much less harmonious than that of Newton’s classical physics. Linnaeus also presented us with another version of the Enlightenment’s system of an orderly, rational world in his Systema Naturae, and although his method of classification survives, his world of fixed species has collapsed since Darwin’s Origin of Species offered a more chaotic but more scientifically defensible conception of the organization of processes of nature. And the shift from Linnaeus to Darwin is not from order to chaos, but from one type of order to another. The rationality of tonality gave way to the more rigorous rationality of twelve-tone composition, just as the rationality of fixed species gave way to the more rigorous rationality of evolution.

One of my favorite recordings of Pergolesi's Stabat Mater.

One of my favorite recordings of Pergolesi's Stabat Mater.

Now the important question is whether finding harmony and beauty in a certain kind of order is inherent in the order or in one who perceives the order. We may both understand the order involved in twelve tone composition, but one may find the result pleasing and another will find the result not pleasing. Perhaps these different responses are due to the different faculties brought to the task of appreciation. People who enjoy listening to twelve tone music usually find in it an intellectual fascination. My own tastes run more toward Pergolesi (or, in twentieth century music, to Webern), who could serve as a paradigm of classical tonality. But when I listen to Stabat Mater I am not merely absorbed in the sensuality of the melody (although that is a large part of the pleasure I derive from it), but I also find an intellectual satisfaction in the purity and simplicity of forms employed. It is akin to the architecture of the renaissance, a model of discipline and proportion like the Ospedale degli Innocenti in Florence.

Renaissance classicism: harmony, form, order, proportion, and symmetry at the Ospedale degli Innocenti in Florence.

Renaissance classicism: harmony, form, order, proportion, and symmetry at the Ospedale degli Innocenti in Florence.

In light of the chaotic universe we have inherited from modern science with its relativity, quantum indeterminacy, protean species, and alternative formulations of logic and mathematics, perhaps it would be after all appropriate if we came to imagine Schoenberg’s works as the Harmonia Mundi, with its dissonances and atonalities reflecting more truly the world in which we live.

The world was once a gratifyingly orderly place, and was proclaimed as such by winged cherubs. No longer.

All of the arts employ forms, but particularly in architecture and in music architectonic structure stands out, and this likens it to mathematics. Form imposes order, and we find order in all of the arts, but the structure which results from the iteration of order is a higher, more complex order, hierarchical without necessarily being hieratic (as in medieval painting in which relative importance is strictly governed by rules of size and proportion).

Many people have compared music to mathematics. There is Debussy: “Music is a mysterious form of mathematics whose elements are derived from the infinite.” (Quoted in The Theories of Claude Debussy, Léon Vallas, p. 8, Dover, 1967) There is Ravel: “I make logarithms—it is for you to understand them.” (Quoted in Introduction to Contemporary Music, Joseph Machlis, Norton: New York, 1961, p. 137.) There is Herriot: “La musique est une mathématique sonore, la mathématique est une musique silencieuse.” Perhaps most famously there is Leibniz: “Musica est exercitium arithmeticae occultum nescientis se numerare animi.” The critics of Schoenberg’s technique called it “mathematical,” which reminds one of the sort of slurs that were made against the early strides of analytical philosophy (Russell noted that Bergson spoke insultingly of what is “static”, “Platonic”, and “mathematical”).

Rather too much is made of this comparison at time, but there is, I would say, the same dialectic of the intuitive and the formal in both music and mathematics. But music retains time and indeed works with time in a way in which mathematics does not. Mathematics, on the one hand, formalizes further the intuitive formalization of the natural numbers. Music, on the other hand formalizes time. It is the impulse to formalize given intuitions – the tension between intuition and formalization, between intuitive concepts and their formal surrogates –- that characterizes both.

There is in musical composition a strictly operational treatment of time by way of rhythm, tempo, meter, beat, and time signature which orders and formalizes what we cannot explain. (Recall that St. Augustine wrote in his Confessions, “What is time? If no one asks me I know, but if anyone asks me I do not know.”) And the rules of composition, like traditional axiomatizations in mathematics, enforce symmetry, repetition, and thematic development upon the pure intuition of melody — melody being a concrete form of intuition that is nevertheless closely parallel to abstract intuition represented by the number series. And what could be more intuitive, among our abstract ideas, than counting?

Saint Augustine asked 'What then is time?' and acknowledged that he could not answer the question. But, as Wittgenstein has pointed out, some things that cannot be said nevertheless can be shown.

It is as though we were given a finite set of rules to achieve the sublime state to which music can bring us: a properly constructed formalization of time, a thoroughly rational process, gives way to the rupture in rationality which is the sublime. Such practices constitute a decision procedure (Entscheidungsverfahren) for ecstatic experience. This seems counter-intuitive, as ecstasy and the suspension of the rational are thought to lie beyond all definable rules. But it is not so unusual.

Whirling Mevlevi dervishes are one of the most famous sights of Turkey, but apart from touristic interest their dance has a serious religious purpose.

Whirling Mevlevi dervishes are one of the most famous sights of Turkey, but apart from touristic interest their dance has a serious religious purpose.

The Dervishes of Turkey engage in a precisely prescribed ritual of a whirling dance accompanied by music as a method to achieve mystical experience. The Indians of the Americas take peyote cactus (Lophophora williamsii) and drum rhythmically for hours through the night during their rituals to induce an altered state of consciousness. Throughout the world prescribed rituals have been the substance of religious worship, which occasionally overflows into a fully mystical experience. It has as a result been mistakenly thought that this transcendence is uniquely religious, but it is rather a structure of consciousness which can be analyzed, defined, and even in some cases systematically reproduced, like a scientific experiment.

An illustration of a Native American peytoe ceremony.

An illustration of a Native American peyote ceremony.

It would not be terribly difficult to formulate some well-crafted experiments to test this hypothesis further, and with some experimentation and refinement we might converge, in time, on a precise algorithm of ecstasy. Of course, we would have to account for cultural relativity, but if the ecstatic response is a human universal, we should be able to strip away the accretions of historical accident and get to the core of ecstasy itself. With this to work with, we could test particular individuals for their ecstatic response.

altered_states

There is something of this in the film Altered States, in which a sensory deprivation chamber is used to induce altered states of mind. The film is reportedly loosely based on the life of Dr. John C. Lilly (as related in the novel by Paddy Chayevsky), who criticized the film’s failure to acknowledge the use of hallucinogenic drugs in combination with the isolation tank. In an interview with OMNI in January 1983 Lilly said, “The tank scenes were fine — except that in reality there are no vertical tanks, only horizontal ones — and the film implied that use of the tank itself would cause those out-of-the-body trips, which it doesn’t.” Also, the film goes off the Hollywood deep end, taking leave of both science and naturalism by having the protagonist experience evolutionary retrogression when dreaming of himself as primitive man. Worse yet, in the final climax of the film all the stops are pulled out and the mystical experience is manifested by events in the real world. (I also feel compelled to point out how dated the film now looks; it did not age particularly gracefully.)

Romain Rolland thought that Freud's The Future of an Illusion had missed the point, and that religion has its origins in an "experience of eternity" that he called the "oceanic experience".

Romain Rolland thought that Freud's The Future of an Illusion had missed the point, and that religion has its origins in an "experience of eternity" that he called the "oceanic experience".

Freud opens his wonderful essay Civilization and its Discontents with a discussion of his correspondence with a friend (later revealed to be Romain Rolland) about Freud’s The Future of an Illusion. Rolland agreed with Freud about religion, but felt that Freud had neglected the ultimate source of religious sentiments. When Rolland described the experience he took to be the “true source” of religion, Freud wrote in perfect deadpan: “I cannot discover this ‘oceanic’ feeling in myself. It is not easy to deal scientifically with feelings.” I have always loved this quote from Freud, not least because I am in the same boat; but while it is not easy to deal scientifically with feelings, with effort and proper diligence it may be possible to scientifically capture even the heights of religious ecstasy. An algorithm for induced ecstasy tested on Freud or myself would register zero on the scale; with others (like Dr. John C. Lilly), it might be necessary to allow the use of chemicals for them to register on the ecstasy scale; still others (genuine mystics, for example) might top out the scale with no assistance from mind-altering substances.

freud_civilization

In The Life of the Spirit in the Age of the Machine, I discussed the changes manifested in the religious life of man as a result of the industrial revolution, and how this change took different forms in Europe and North America. It now occurs to me that the Age of the Machine has also produced naturalistic science, and this science may, when we are ready for it, revolutionize the spiritual life of man as it has already revolutionized the material conditions of life in the industrialized world. The last of Freud’s New Introductory Lectures of Psychoanalysis (Lecture XXXV) suggests that, with psychoanalysis, science finally has an instrument with which to approach the ideals and aspirations of man, which science had previously neglected. I have always found this to be one of Freud’s more pregnant observations, and deserving of fuller treatment. And while the psychoanalytic treatment of our ideals may not be flattering to the same, it is one scientific approach. I have above suggested a different approach, but in so far as both presuppose a methodological naturalism, they are of the same spirit and of the same aspiration.

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2 Responses to “Algorithms of Ecstasy”

  1. “Music, … formalizes time. It is the impulse to formalize given intuitions – the tension between intuition and formalization, between intuitive concepts and their formal surrogates – that characterizes both” – makes sense for me, as a mathematician.

    “Music formalizes time” – I love it. And I love the “tension between intuition and formalization”. Really, this is the most difficult and fascinating thing about mathematics. For me, it took years to get used to it.

    • geopolicraticus said

      Dear Marina,

      I’m very pleased that you liked my formulation. At one time I worked on this extensively, so there is a lot of thought behind some of the more seemingly elliptical statements. I won’t attempt any systematic exposition of my views here, but let me suggest a few questions in this vein: Given that music is the formalization of time, a theory of musical abstraction follows from this. Exactly what kind of abstraction is involved in the formalization of time? What is distinctive about musical abstraction? Why is time amenable to formalization in this way? Is not music among the most concrete abstractions conceivable?

      Best wishes,

      Nick

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