Karlheinz Stockhausen’s opera cycle LICHT is many things and as a great work of art it is subject to multiple, if not endless, interpretations. These interpretations are multiple because the opera is made up of living symbols. As Carl Jung taught, it is possible to distinguish between a symbol and a sign. A symbol is the best possible expression for something that is unknown, whereas a sign is something specific, such as the insignia worn by a military officer showing his specific rank.

 For this work the specific and very rich symbolism of LICHT will be set aside to look at it from a structural and systems point of view. The way Stockhausen gave his work specific limitations shaped the work in unique ways. His adept and intuitive grasp of combinatorial procedures within the limits of the system gave him a wide ranging freedom to play with the materials he had chosen, shaping the raw ingredients into an astonishing and sensual feast of sound, color, and movement.
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Opening up the lid of the opera cycle it’s possible to see how its individual components create a musical engine whose individual circuits sync together in a series allowing for a dynamic flow of energies and psychoacoustic forces. Let’s look under the hood of LICHT to see how its various pieces fit together.

Conception of LICHT: Formula & Super Formula   

            Great ideas often come as revelatory seeds into the mind of those who are prepared. By the mid-seventies Stockhausen had been composing for a quarter of a century and he had already explored a vast territory of sound implementing new ideas for the arrangement of music in time and space. He had played with intuitive music, aleatory processes, and had mastered new electronic music techniques in the studios of WDR, just for starters. The soil of his mind and spirit were fertile, waiting for the next big idea to be planted.

Another tactic basically invented by Stockhausen was formula composition and it came out of his deep engagement with serialism. It involves the projection, expansion and ausmultiplikation of either a single melody-formula, or a two- or three-voice contrapuntal construction. In serial music the structuring features remain basically abstract but in formula composition properties such as duration, pitch, tempo, timbre, and dynamics are also specified from the formula. By using concise and specific tone succession based on the single melody formula both the macro structure and micro details of the composition can be derived.
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The roots of his method of formula composition can be traced back to his once withdrawn orchestral piece Formel where the first basic pattern of notes are gradually transformed over the course of the work. The central pitch is first broadened out before the notes are removed leaving only the low and high extremes. He continued to use serial operations on his next batch of works, Kreuzspiel and Punkte, and then introduced musical pointillism into the methods as explored in Kontrapunkte and Gruppen.

​Then for a time he moved on to other musical tactics and explorations but came back to the practice with ferocity in Mantra from 1970. Written for two ring modulated pianos, the pianists are also required to play a chromatic cymbals and a wood block. One of the players also has a short-wave radio tuned to a station sending morse code, or when CW isn’t readily available live on the air, a tape recording of morse code is played. It was the first composition that he wrote where he used the term formula, and was one of many watershed moments in his musical thinking. The formula involved the expansion and contraction of counterpointed melodies. 

His next piece to use formula composition was Inori from 1974. By this time Stockhausen had already been working extensively with writing music that incorporated elements of theater. Inori took it to another level and he had the insight that he could use the formula, not just for music, but as a way to compose gestures. This was another component that would become essential in LICHT.
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Inori is a long work with performances lasting around seventy minutes. The formula for the piece is made up of fifteen notes using 5, 3, 2, 1 and 4 pitches respectively. When the formula is used on the macros scale for the work these five phrases are split into five segments Stockhausen to create a narrative sequence. Robin Maconie says it “lead[s] from pure rhythm . . . via dynamics, melody, and harmony, to polyphony: —hence, a progression from the primitive origin of music to a condition of pure intellect. The entire work is a projection of this formula onto a duration of about 70 minutes”

In 1977 Stockhausen went back to Japan to work on a commission for the National Theater of Tokyo. The idea for intermodulation of music had come to him in his first Japanese commission with Telemusik and he had played his music alongside nineteen ensemble musicians in the special spherical chamber designed for him at the World Fair in Osaka in 1970 for about five and a half hours a day, 183 days in a row. Japan had been a good country for his musical expression. The piece he came to work on when LICHT was conceived was to being written for traditional Gagaku orchestra and Noh actors. The dramatic elements for the production however came to him in a dream, just one of many dreams that gave him direct inspiration for compositions. While composing what became Der Jahreslauf, (Course of the Years), he had a revelation about a way to represent different levels of time by different instrument groups: millenniums are depicted as three harmoniums, centuries by an anvil and three piccolos, decades by a bongo and three saxophones, and years by a bass drum, harpsichord and guitar. These instrument groups became representations of vast forces and scales of time.

            This idea of composing music around the theme of various increments of time stayed with the composer for the rest of his life. While working on this commission, another idea was also transmitted into his mind, the super-formula that became the basis for LICHT. In a flash a small seed became the basis for a work of cosmic proportions. Subsequently he used Der Jahreslauf as the first act of Dienstag aus LICHT (Tuesday from Light). 

            In LICHT he realized his formula technique could be considerably expanded. The entire cycle of seven operas is based on three counterpointed melody formulas. Each of these is associated with one of the three principal characters that make up the dramatic element of the production. (Stockhausen himself said the formulas are the characters.) The melodies then define the tonal center and durations of scenes, and zooming in, give detailed melodic phrasing to more refined elements. The three characters are Eve, Lucifer, and Michael, and they are each associated with a specific instrument, bassett horn, trombone, and trumpet in turn.
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This explains formula composition, but what about a super-formula? 

In 1977 Stockhausen had been composing for just over twenty-five years.  In the super-formula he synthesized nearly all of his musical ideas into a musical tool that would occupy him for the next twenty-seven years until 2003 when the last bars for Sonntag aus LICHT were drying on the staff paper.

He had the insight to take the three formulas he had come up with for Eve, Lucifer and Michael and layer them horizontally on top of each other to make the super-formula. Now they existed as one, each with their own layer, named after the character, or force, in question. The super-formula then gets subdivided again, vertically, into seven portions, of two to four measures each. These seven vertical rows form the days of the week.

Combined the horizontal and vertical rows make up the rich matrix out of which the overall structure of LICHT is built. To expand the formula in time, every quarter note of the super-formula is equal to 16 minutes of music. This is how the maestro -or magister- used it determine the durations of the opera cycles various acts and scenes.

Stockhausen also decided to create a kind of skeleton key, bare bones version of the super formula for each of the three characters. These he called “nuclear formulas” (kernformel) and consisted of just the pitches, duration and dynamics. Boiling the bones down even further provides the broth that the music is bathed in. When the nuclear formulas are reduced to just the notes what is left is essentially a serialist tone row. These are known as the kernels, central tones, or nuclear tones. Nuclear, because they form the very atoms of the music.

With all of this in place the fun has a chance to begin. The super-formula can now be used in all manner of ways. Sometimes Stockhausen employed it in an inverted or retrograde fashion (upside down or backwards). It is very often stretched out across the time frame of scenes and whole acts. Other times it is transposed vertically. Once the listener becomes familiar with each of the formulas for the characters or forces, it is possible to pick out those forces at work in the music even though the formula is not really used as a recurring theme in the typical sense of classical music. Rather, as Ed Chang said, “In LICHT, the MICHAEL, EVE and LUCIFER formulas are used more as structural forces whose tonal characteristics exert a kind of planetary gravity over the surrounding musical ether.”

            LICHT is a complete system. The superformula, nuclear kernels, and nuclear tones form the mathematical and musical parts of the systems ecology. The content of the opera, its symbolism based around the days, and the spiritual realities of Eve, Michael, and Lucifer are another aspect of the system. All of this gave Stockhausen the raw material out of which to craft his magnum opus. The music and symbolism mix together and all are now subject to a remarkable game of combination and recombination. The system of LICHT forms the matrix of possibilities, and displayed within that matrix are an extraordinary blending and synthesis of constituent forms.

The idea of ausmultiplikation, which can be translated as "multiplying-out" bears further looking at in terms of how formula composition creates musical forms mirrored on the macro and micro scales. Stockhausen described the technique as when a long note is replaced by shorter "melodic configurations, internally animated around central tones". This bears a strong resemblance to the Renaissance musical technique of diminution or coloration, where long notes are divided into a series of shorter, frequently melodic, values. But Stockhausen also used the term to refer to when he substituted a complete or partial formula for a single long tone, often as background layer projections of the formula. Formula composition and its various components like ausmuliplikation can be seen as  Stockhausen’s way of creating a way to practice the Glass Bead Game in music.
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Robin Hartwell had the insight that when this is done at more than one level results resemble those of a fractal. If the formula compositions are fractal like, and he also used the idea of spirals throughout his work, one way of looking at LICHT is as a composed fractal music. Zooming in and out, the same structure is played in both minutely on the microscopic level, and at large on the macroscopic across the range of an entire work. Having boiled down of the musical components to microscopic levels, and having diluted them out to the macro, was one way Stockhausen prevented signal loss and maximized the transmission of his musical information. The super-formula is present and exists on every level and in every moment of LICHT.

Modular Music

Another way Licht can be seen as a musical system is by how it is structured in component modules. First of all, it should be considered that each of the operas is a work capable of being appreciated and understood unto itself, without having to hear or see the other sections. While listening to the whole cycle certainly enhances the experience of individual parts, those individual parts can also be enjoyed one at a time in and of themselves. Each opera, act, scene is self-sufficient. Even some parts of scenes can be extracted as solitary works. Certain other extra-curricular or auxiliary works have also been extrapolated out of the formulas of LICHT and its modular structure. All of these contain the essence of LICHT and give the listener one of many ways of enjoying the various elements of the cycle.  
           
This was all made possible due to the practical aspects of Stockhausen’s life as a composer. After he began LICHT, when he received a commission for a new work from this or that person or cultural institution, prescribed for this or that choir group, string quartet, or other group of instrumentation, he would incorporate the work on that commission into LICHT. It was an elegant solution that allowed him to finish the massive project.
           
​Some of the examples of modular works that can be extracted from LICHT include Klavierstucke XII and Michael’s Reise from Donnerstag; Weltraum is an assemblage of the electronic greetings and farewells of Freitag; Kathinka’s Chant for flute and electronics is an extract from Samstag; Angel Procession’s for choir comes from Sonntag; Ypssilon for flute and Xi for basset horn from Montag; the electronic layer from the second act of Dienstag becomes the piece Oktophonie; and the infamous Helicopter String Quartet is a section from Mittwoch. These are just a few of the pieces he was able to write in a modular fashion to fulfill a commission and thus complete a section of LICHT. Alternately he was able to adapt an already written section of LICHT as a module to fulfill a commission and thereby create a smaller chamber type work. 

Ars Combinatoria
            These smaller modules, extracts and auxiliary works from LICHT represent another fractal like aspect of the cycle as a system. They are separate and yet also a part of the system. The formula and super-formula interact with themselves, alongside the set symbolism of the days of the week, to produce an array of combinations perceived and permutated through Stockhausen’s intuitive imagination. Through this thoroughly disciplined act of creation and applied artistry Stockhausen has shown himself to be a “Magister Ludi” or master of the Glass Bead Game.

            He has fused mathematics and music together and along these strands and placed connecting beads from the various religious and mystical traditions of the world. He used traditional correspondences, such as in Samstag for instance, associated with Saturday, and the planet Saturn, and it’s symbolism of contraction, limitation, and death. In Samstag he wrote the section Kathinka’s Gesang as Lucifer’s Requiem. Thus the mysteries of death become a main feature of this section of the work. In this piece the flautist performs a ritual with six percussionists. The ritual consists of twenty-four exercises based on Stockhausen’s study of the Tibetan Book of the Dead. It was written as a chant protecting the soul of the recently departed (in this case Lucifer) by means of musical exercises regularly performed for 49 days after the death of the body, and lead the recently deceased into to the light of clear consciousness. For these exercises he permutated the Lucifer formula into a showstopper of extended flute techniques of deft virtuosity.
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            And the piece may really be used by the living, and played for 49 days after the departure of a loved one to help assist them in their afterlife transition. 

The entire cycle is filled with this plentitude of subtle correspondences between music, science and various world cultures. These become the raw data for his applied musical calculus that is dancing in an elaborate play upon all these correspondences, inside a defined system, to express in multiplexed forms, that which is universal. 
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            After finishing the 29 hours of Licht, a feat some of his critics never expected him to complete, Stockhausen begin writing a series of chamber pieces called Klang, with the intent of writing one for each of the twenty-four hours of the day. Having conceived the musical forces of the days of the week, he was zooming in again to explore the musical forces behind each hour of the day. Formula composition gave him the tool he needed to explore these hours. Having written 21 of the pieces the cycle was unfinished at the time of the composer’s unexpected death in 2007 when he voyaged forth into the greater harmonies of cosmic space and time.

Read the rest of the Radiophonic Laboratory series.

References:
Other Planets: The Complete Works of Karlheinz Stockhausen 1950–2007, by Robin Maconie,Rowman & Littlefield Publishers, Maryland, 2016. 

Ed Chang's website in general has been super helpful in understanding the super-formula. It is a great journey through the Space of Stockhausen. 
http://stockhausenspace.blogspot.com/2014/08/a-brief-guide-to-licht-pt-1-drama-and.html
http://stockhausenspace.blogspot.com/2014/09/a-brief-guide-to-licht-pt-2-super.html

Threats and Promises: Lucifer, Hell, and Stockhausen's Sunday from Light" by Robin Hartwell in Perspectives of New Music 50, nos. 1 & 2 (Winter–Summer): 393–424.

Into the Middleground: Formula Syntax in Stockhausen's Licht" by Jerome Kohl in Perspectives of New Music 28, no. 2 (Summer): 262–91.

Shannon wasn’t the only one looking at the way signals were transmitted. The same year he published his breakthrough paper, another mathematician published a book that would leave a lasting impression on a number of different fields, electronic music among them. The man was Norbert Wiener and his book was Cybernetics: or control and communication in animal and machine. Wiener defined cybernetics as "the scientific study of control and communication in the animal and the machine".

Wiener was a child prodigy. Born to Polish and German Jewish immigrants, on his fathers side Nobert was related to Maimonides, the famous rabbi, philosopher and physician from Al Andalus. The predisposition to intellectual greatness was hardwired into his system. Norbert’s father Leo was a teacher of Germanic and Slavic languages and avid reader and book hound who put together an impressive personally library which his son devoured. His father also had a gift for math and gave his son additional instructions in the subject.
At age 11 Norbert graduated Ayer Highschool in Massachussettes and then began attending Tufts College where he received a BA in mathematics at the age of 14. From there he went on to study zoology at Harvard before transferring to Cornell to pursue philosophy, where he graduated at the ripe old age of 17 in 1911, when his classmates from Ayer were probably just entering college if they went at all. Then he went back to Harvard where he wrote a dissertation on mathematical logic, comparing the works of Ernst Schröder with Bertrand Russel and Albert North Whitehead. His work showed that ordered pairs could be defined according to elementary set theory. His Ph.D. was awarded before he turned twenty. Later that same year he went to Cambridge and studied under Russel, as well as at the University of Göttingen where to learn from Edmund Husserl.

After a brief period teaching philosophy at Harvard, Wiener eventually found a position at MIT that would become permanent.  In 1926, Wiener returned to Cambridge and Göttingen as a Guggenheim scholar, on a trip that would have important implications for his future work. He spent his time there investigating Brownian motion, the Fourier integral, Dirichlet's problem, harmonic analysis, and the Tauberian theorems.

​Harmonic analysis and Browninan motion in particular would go on to have a key role in the development of cybernetics.

Harmonic analysis is a branch off the great tree of math that is concerned with analyzing and describing periodic and recurrent phenomena in nature, such as the many forms of waves: musical waves, tidal waves, radio waves, alternating current, the motion and vibration of machines. And it branched off the research of French mathematician Joseph Fourier (1768-1830). Fourier was interested in the conduction of heat and other thermal effects, a trail later followed by Nyquist in his own investigations of thermal noise. 

According to the Encyclopedia Brittanica the motions of waves  “can be measured at a number of successive values of the independent variable, usually the time, and these data or a curve plotted from them will represent a function of that independent variable. Generally, the mathematical expression for the function will be unknown. However, with the periodic functions found in nature, the function can be expressed as the sum of a number of sine and cosine terms.” The sum of these is known as a Fourier series. The determination of the coefficients of these terms is became known as harmonic analysis.

Brownian motion or movement relates to a variety of physical phenomena where some quantity of substance undergoes small and constant but random fluctuations. When those particles that are subject to Brownian motion are moving inside a given medium, and there is no preferred direction for these random oscillations to go, the particles will over time, spread out evenly in the substance.

Both Browninan motion and harmonic analysis can be considered stochastic processes. A stochastic process is, at its core, a process that involves the operation of chance. It is a process where values change in a random way over time. Markov chains are another important form of stochastic process that has been applied to music. Stochastic process can also be used to study noise, and Wiener was a student of this mathematical noise.

Amidst the conflicts of WWII Norbert was called upon to use his prodigious brain for solving technical problems associated with warfare. He attacked the problem of automatic aiming and firing of anti-aircraft guns. This required the development and further branching of even more specialized math. It also introduced statistical methods into the recondite area of control and communications engineering, which in turn led to his formulation of the cybernetics concept.

His concept of cybernetics was eerily close to Claude Shannon’s information theory. What they both had in common was knowledge of the influence of noise and the desire to communicate or find signals in, above, and around the noise. One of the ways Wiener figured out how to do this was through filtering. Enter the Wiener filter. It works by computing statistical estimates of unknown signals using a related signal as an input and filtering that to produce an estimated output. Say a signal has been obscured by the addition of noise. The Wiener filter removes the added noise from the signal to give an estimate of the original signal.

Cybernetics is also related to systems theory, and studied in particular the idea of feedback, or a closed signaling loop. Wiener originally referred to the way information or signals effect relationships in system as “circular causal”. Feedback occurs when some action within the system triggers a change in the environment. The environment in turn effects another change in the system when it feeds back the now transformed signal into the originating source.  Wiener, through his study of zoology was applicable to biological and social systems, as well as the mechanical ones his research had originally grown out of. Cognitive systems could also be understood in terms of these circular causal chains of action and reaction feeding back in on itself.

Cybernetic’s essential idea of feedback was also directly applicable to the new electronic musical systems defined by the advent of the microphone, amplifier, and speaker. When these devices are connected together in a circuit audio feedback is one possible result stemming from holding the mic close to the speaker. Everyone has experienced the unintentional noise when a PA is being tested. Musicians quickly adapted the idea of using intentional feedback, and distortion (noise on a signal) to give their recordings and live performances a new sound.

Cybernetics is not limited to mapping the flow of information, distorted or otherwise, in and out of systems. It also includes concepts of learning and adaption, social control, connectivity and communication, efficiency, efficacy, and emergence.
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The related fields of information theory, cybernetics and systems theory would have huge impacts on music and the arts, as the theories trickled down from places like Bell Labs, the Macy Conferences with their focus on communication across scientific disciplines, and the success of Wiener’s book outside of strictly scientific circles.

The word cybernetics sounds kind of cold and inhuman. It conjures up the chrome clad computerized villains made famous by Doctor Who, the cybermen who speak only in monotone and whose overriding program is to delete organic life. Yet the word cybernetics itself comes from the Greek kybernḗtēs, or "steersman, governor, pilot, or rudder.” Human systems require a guide, someone to steer them. Wiener had picked up the word from the French mathematician and physicist André-Marie Ampère who coined the word "cybernetique" in an 1834 essay on science and civil government. Governments and other systems of human invention require steersman and guides with a firm hand on the rudder to give direction and control the effects of feedback.
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The creation of systems is a human trait, and their guidance, via our input, doesn’t have to be cold. It can be done with intuition, insight, and artistic flair. Writing on systems in the world of art for the 1968 Cybernetic Serendipity art and music show at the ICA gallery in London, Jasia Reichardt wrote, "The very notion of having a system in relation to making paintings is often anathema to those who value the mysterious and the intuitive, the free and the expressionistic, in art. Systems, nevertheless, dispense neither with intuition nor mystery. Intuition is instrumental in the design of the system and mystery always remains in the final result."

The Discreet Music of Brian Eno
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Designing musical systems can result in extraordinary beauty. In the mid-1960s while attending Ipswich Art School Brian Eno had his first encounter with cybernetics. It would go on to have a lasting influence. Under the mentorship of Roy Ascott who had developed the controversial “Groundcourse” curriculum adopted by a number of other art colleges Eno absorbed Ascott’s philosophy of systems learning, making mind maps, and playing mental games.

Eno started thinking of the music studio and groups of musicians in terms of cybernetic systems. Making great musical compositions started with designing the parameters, limits, inputs and outputs that would give a composition its ultimate form. Creating these systems and letting them run was how many of his first, and the first, ambient music records were made. 

The liner notes for Eno’s 1975 album Discreet Music contain a block diagram of the system he created for the music. He had been given an album of 18th century harp music to listen to while laying in the bed in the hospital, where he was recovering from a car accident injury. A friend who had been visiting put the record on for him before she left but the volume was turned down too low. Outside it was raining and he listened to “these odd notes of the harp that were just loud enough to be heard above the rain.” The experience “presented what was for me a new way of hearing music—as part of the ambience of the environment just as the color of the light and the sound of the rain were parts of that ambience.”

Eno connected this experience to Erik Satie’s idea of “furniture music” that was intended to blend into the ambient atmosphere of the room, and not be something focused on directly. Furniture music could mix and combine with the sounds of forks, knives, tinkling glasses and conversation at a dinner.

After Eno’s listening experience in the hospital he set out to make his own ambient music, setting off a musical cascade and defining and kick-starting a genre that at the time of this writing is now forty-five years old.

In the liner notes to Discreet Music, Eno wrote these now famous lines, “Since I have always preferred making plans to executing them, I have gravitated towards situations and systems that, once set into operation, could create music with little or no intervention on my part. That is to say, I tend towards the roles of the planner and programmer, and then become an audience to the results.”

The liner notes also contain a block diagram of the system he set up. Eno had wanted to create a background drone for guitarist Robert Fripp to play along with. He was working with an EMS Synthi AKS with built-in memory and a tape delay system. He kept being interrupted in his musical work by knocks on the door and phone calls. He says, “I was answering the phone and adjusting all this stuff as it ran. I almost made that without listening to it. It was really automatic music.”

Discreet music started with two melodic phrases of differing lengths played back from the digital recall of the synth. That signal was then ran through a graphic equalizer to change its timbre. After the EQ the audio went into an echo unit and the output of that was recorded to a tape machine. That tape runs to the take-up reel of  a second tape machine, whose output is fed back into the first machine which records the overlapping signals and sounds. When Fripp came by the next day to have a listen Eno accidentally played the recording back at half-speed. Eno says of the result “it was probably one of the best things I’d ever done and I didn’t even realize I was doing it at the time.”

Autonomous Dynamical Systems

Another example of musical systems in practice comes from the work of David Dunn. David is a composer, sound artist, bioacoustics researcher and an expert at making audio recordings of wildlife. A deep interest in acoustic ecology informs his work. Ecological thinking and systems thinking go hand in hand and this sensibility is present in many of David’s works.

His 2007 album Autonomous Dynamical Systems touches on ecology, fractals and chaos theory, graphic imagery to sound conversions, and feedback loops. The album consists of four compositions. Lorenz from 2005 is a collaboration with chaos scientist James Crutchfield. James has a long history of work in the areas of  nonlinear dynamics, solid-state physics, astrophysics, fluid mechanics, critical phenomena and phase transitions, chaos, and pattern formation, having published over 100 papers in his field of mathematics and physics.

The Lorenz attractor was first studied by meteorologist Edward Lorenz in 1963. He derived the math from a simplified model of convection in the earth's atmosphere and is most frequently expressed as a set of three coupled non-linear differential equations. In popular culture the idea of the “butterfly effect” comes from the physical implications of the Lorenz attractor. In any deterministic nonlinear system one small change, even the small disturbances in air made by the flight of a butterfly, can result in huge differences to the system at a later time. This shows that systems can be deterministic and unpredictable at the same time. When the Lorenz attractor is plotted out graphically it has two large interconnected oval shapes resembling a butterfly or a pair of wings.

For the piece Lorenz, David Dunn used a piece of software written by Crutchfield called MODE (Multiple Ordinary Differential Equations) plugged into the interface program OSC (Open Sound Control), a networking protocol that allows synthesizers, computers, and other multimedia devices. OSC is then in turn fed into sound synthesis program. The sound synthesis program is then fed back into OSC and again into MODE. The entire piece is a feedback loop originating from chaos controlled sound. As such its structure embodies the very principles it seeks to express as music. Another piece on the album, Nine Strange Attractors from 2006 steps up the game even further in its creative use of mathematics to explore feedback loops.
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Another piece uses feedback loops in a different way. Autonomous Systems: Red Rocks from 2003 used environmental field recordings fed into computer systems. Saved in the memory a chaos generator program chooses from among the sounds in a non-linear fashion and plays them back, sometimes electronically transformed, other times not. The composition is done, not by performing live, but by setting up and programming the system, then stepping away, sitting back, and listening to the results.   
John Cage said, “My compositions arise by asking questions.”  The music of systems proceeds from this same curious spirit. When designing new electronic works the composer must begin by asking questions of herself. Then systems can be designed to ask that question in different ways and to find out different answers. 

Wobbly and his Smart Phone System

Wobbly, aka Jon Leidecker, a solo artist, member of Negativland, and now host of radio show Over the Edge after the death of Don Joyce has also made a very interesting album by working with systems.

Between 2015 and 2018 Wobbly worked on an album called Monitress, released in 2019. He created an innovative system leveraging musical pitch tracking apps and synthesizers on a group of mobile phones and other mobile devices. Each of the devices was sent an audio signal. This was picked up by the pitch tracking app and coverted to MIDI data used to drive the synth. The resulting sound is then fed into an analog mixer. Once the signal is going into the mixer it can be routed and fed back into another mobile device also running a pitch tracking app and synth. The resulting effect is a cascade of sound between the devices.

​As Jon writes in the liner notes for the album, “ Feedback loops similar to acoustic or electrical feedback occur when you close the circle. The pitch-tracking apps are prone to errors, especially when presented with complex multiphonics or polyphonies;  they get quite a few notes fascinatingly wrong.  But more striking is the audible reality of their listening to each other.  Unison lines are an elemental sign of musical intelligence; we are entrained to emotional reactions when hearing multiple voices attempting the same melody.  These machines may not meet our current criterion for consciousness, but every audience I’ve played this piece in front of quickly realizes they're not listening to a solo…

The technology used to create these sounds existed before the mobiles, but this music would not have been made on earlier equipment -- it's a result of the relationship developed with a machine that is always present, and always listening. This was the project I dug into as we woke up to the true owners of these tools, a frame to make the relationship between ourselves and our machines audible while we think about the necessary steps to take next.”

The textures on this album are sublime, the kind of things that could only be heard through this a cascade of forces, each triggered by the preceding and affecting the whole in tandem. Wobbly did do post production editing of this work, but the initial results he captured once the process was set in motion is where the real magic lies. This is the kind of music that can’t be predicted. It couldn’t be written by a composer note for note. Rather the job of the composer is to design systems capable of eliciting beauty.
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The three examples of systems music explored here are only a few of many. Musical systems is a large category within the new common practice generally. Other ways of thinking about it is in terms of modular set ups, various configurations of test equipment, systems of feedback in the way guitar pedals are arranged, and more. I don’t know if Norbert Wiener ever thought of music as one of the places where cybernetics would take flight. To hear the music made with its principles is an artistic way of exploring the rich ecology of sound.  

Read the rest of the Radiophonic Laboratory  series. 

References: 

The Information: a history, a theory, a flood by James Gleick, Pantheon, 2011

A Mind at Play: How Claude Shannon Invented the Information Age by Jimmy Soni and Rob Goodman, Simon & Schuster, 2018

Encyclopedia Britannica: https://www.britannica.com/science/harmonic-analysis

Brian Eno, Discreet Music, Obscure Records, 1975

David Dunn,  Autonomous and Dynamical Systems, New World Records 2007

Wobbly, Monitress: https://hausumountain.bandcamp.com/album/monitress

Stockhausen picked up his interest in information theory by way of Werner Meyer-Eppler during his time as a student at the Bonn. Meyer-Eppler himself was something of a scientific polymath, having studied mathematics, chemistry and physics at the University of Cologne in the late 1930s before going to the Bonn where he became a scientific assistant in the physics department, and then a lecturer in experimental physics. After WWII ended his attention turned with laser beam focus to the subject of phonetics and speech synthesis. In 1947 Paul Menzerath brought him into faculty of the Phonetic Institute of the University of Bonn. It was in this time period when Meyer-Eppler started publishing essays on the Voder, Vocoder, and Visible Speech Machine. One of his contributions to the field that is still in use today was his work on the development of the electrolarynx.

            Information theory made many contributions to many fields. Linguistics was one of those fields where it was influential in studying how frequently words were used, word length, and the speed words could be read. Shannon had tested the information theory principle of redundancy, or the amount of wasted space used to transmit a message, by having his wife predict the number of repeating letters in a random crime novel he pulled off his bookshelf. Sometimes redundant is better, when it comes to getting a message across. Redundancy is added while communicating over noisy channels as a method of error correction.

​            Shannon had the insight that this was a baked-in purpose behind the repetition of letters in Englsih. He had also showed that he could use stochastic processes to build something that resembled the English language from scratch.

Werner had been following these developments of information theory with special attention to their applications in linguistics and speech. Later in the 50’s Meyer-Eppler became concerned with how statistics and probability, core tools of information theory, might be applied to creating electronic music as explored in his book Statistic and Psychologic Problems of Sound. In this work Meyer-Eppler introduced the word “aleatoric” into the musical lexicon. According to his definition “a process is said to be aleatoric ... if its course is determined in general but depends on chance in detail”.

Aleatoric music is made when some element of the composition is left to chance or when a significant portion of how the composition is realized is left up to the performer or performers. Aleatoric composition has a precedent in the dice games of the 18th century. The word itself comes from the Latin alea, meaning dice.

There are many methods for applying aleatoric processes to music. One of the ways Stockhausen tackled it was by using a polyvalent structure, or writing a piece that was open to a number of different interpretations. Klavierstucke XI is an example of such a piece that he wrote for piano.

The piece is made up of 19 fragments printed on a very large piece of paper. There is no turning of the sheet music. The pianist may start with any fragment they wish and from there continue on to any other fragment they wish to play. It is polyvalent because each performance could begin and end in new places. There is no set musical narrative; it is more like reading a choose-your-own adventure, or wondering through a maze, or labyrinth which the pianist enters, circumnavigates, and then returns. Each time the pianist may enter the labyrinth from a new entrance and likewise, reemerge in a different place.

The pianist shares responsibility with the composer for the eventual shape of any given performances. The possible permutations are vast, yet even in different interpretations it may be heard as the same piece of music, its essential characteristics remaining the same no matter the order they are played.
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Commenting on the piece the composer said, "Piano Piece XI is nothing but a sound in which certain partials, components, are behaving statistically...  If I make a whole piece similar to the ways in which (a complex noise) is organized, then naturally the individual components of this piece could also be exchanged, permutated, without changing its basic quality."

Considered as a whole Piece XI will sound the same even though every time it is played it will sound different. It is a system unto itself, and as a system, even when the component parts are rearranged in the order they are played it is still the same system, and will still sound like itself. Listened to statistically the musical values remain the same.

Stockhausen would go on to use polyvalent form again and again. In his percussion piece Zyklus (Cycle) from 1959 the score is printed as a spiral and the performer may start anywhere within the spiral he or she chooses. Furthermore, they may play the piece from left to right or right to left. The piece is finished when the player reaches the original starting point. In the performance space the cycle is shown again visually with the percussion pieces laid out in a circle with the performer moving around them in the manner determined by a chosen starting point.
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Zyklus also shows the amazing diversity of possible interpretations demonstrated before in Piece XI. It is however the interpretation of the scored is a bit more closed. On one side of the score the music becomes increasingly aleatoric, giving more freedom to the player in how it is interpreted. On the other side of the spiral the composition is exactly fixed and predetermined. Played on way it moves from fixed to open, and in the other direction from open to closed. 

Stockhausen was obsessed with cycles. Specifically cycles of time. His mid-seventies composition Tierkreis (Zodiac) consisted of twelve melodies for each of the twelve zodiac signs. Originally written for custom made music boxes, Tierkreis can be played on any melody instrument and peformed in a number of different ways. For the purpose here a complete performance begins with the melody for the corresponding zodiac sign for the day when the performance is being held. For instance, if the performance was held on August 22 the performers would begin with the Leo melody and proceed through Virgo, Libra, and the rest until they return to the starting melody of Leo. Each melody is played at least three times and may be improvised upon. This gives considerable variation to individual performances. Further variations are specified by the composer.
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  In his chamber opera Sirius written a few years later the Tierkreis melodies are employed again in a section of the piece called The Wheel. Here the music may be heard in four different ways, depending on the season it is performed. If played in the Winter the section starts with the melody for Capricorn, if in the Spring with Aries, Summer starts with Cancer, and Autumn with Libra. 

In all of these cyclical works an echo of the tape loop may be heard. Stockhausen had worked with tape loops extensively in his piece Kontakte, using them to show relationships between pitch, timbre and the way musical events can be perceived in time and space through the process of slowing things up or down. I wonder, if besides the strong grounding Stockhausen had in religion, philosophy, and science if the eternal return and recurrence of the tape loop at all framed his cosmic conception of the vast cycles of time. 

The cycles continued in his magnum opus LICHT (Light): Die sieben Tage der Woche (The Seven Days of the Week). Written between 1977 and 2003 it is a cycle of seven operas, one for each of the seven days of the week. Stockhausen described the work as an “eternal spiral” considering there to be “neither end nor beginning to the week.” Clocking in at a total duration of 29 hours, deft intricacies exist within the piece on a micro and macro scale and many volumes have already been and will continue to be written about it. Within the broad palette afforded by an opera cycle longer than Wagner’s the Ring, Stockhausen was able to play the role of a Magister Ludi, or master of the Glass Bead Game. LICHT is a system, and within that system Stockhausen playfully and masterfully displayed with pyrotechnic virtuosity a comprehensive knowledge of combinatorial and permutative arts as applied to music.

These arts of combination were a central component of the Glass Bead Game as played in the novel.

To show how all of these interlocking parts fit together the basic structure of the opera must be examined. And to understand LICHT as a system a slight change of lanes onto the parallel track of Norbert Wiener and his theory of cybernetics is in order.

Read the rest of the Radiophonic Laboratory Series. 

REFERENCES: 

Other Planets: The Complete Works of Karlheinz Stockhausen 1950–2007, by Robin Maconie,Rowman & Littlefield Publishers, Maryland, 2016. 

Klavierstucke XI essay by Ed Chang: 
http://stockhausenspace.blogspot.com/2015/06/klavierstuck-xi.html

The Glass Bead Game by Herman Hesse, translated by Clara and Richard Winston, Holt, Rinehart and Winston, 1990

From the ice cold farms and fields of Michigan to the halls of MIT and then onwards to Bell Labs at Murray Hill, Claude Shannon was a mathematical maverick and inveterate tinkerer. In the 1920s, in those places where the phone company had not deigned to bring their network, around three million farmers built their own by connecting telegraph keys to the barbed wire fences that stretched between properties. As a young boy Shannon rigged up one of these “farm networks so he and one his friend who lived half a mile away could talk to each other at night in Morse code. He was also the local kid people in the town would bring their radios to when they needed  repair and he got them to work. He had the knack.
 
He also had an aptitude for the more abstract side of a math and his mind could handle complex equations with ease. At the age of seventeen he was already in college at the University of Michigan and had published his first work in an academic journal, a solution to a math problem presented in the pages of American Mathematical Monthly. He did a double major in school and graduated with degrees in electrical engineering and mathematics then headed off to MIT for his masters.
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While there he got under the wing of Vannevar Bush. Vannevar had followed in the footsteps of Lord Kelvin, who had created one of the world’s first analog computers, the harmonic analyzer, used to measure the ebb and flow of the tides. Vannevar’s differential analyzer was a huge electromechanical computer that was the size of a room. It solved differential equations by integration, using a wheel-and-disc mechanisms to perform the integration.
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At school he was also introduced to the work of mathematician George Boole, whose 1854 book on algebraic logic The Laws of Thought laid down some of the essential foundations for the creation of computers. George Boole had in turn taken up the system of logic developed by Gottfried Wilhelm Leibniz. Might Boole have also been familiar with Leibniz’s book De Arte Combinatoria? In this book Leibniz proposed an alphabet of human thought, and was himself inspired by the Ars Magna of Ramon Lull. Leibniz wanted to take the Ars Magna, or “ultimate general art” developed by Lull as a debating tool that helped speakers combine ideas through a compilation of lists, and bring it closer to mathematics and turn it into a kind of calculus. Shannon became the inheritor of these strands of thought, through their development in the mathematics and formal logic that became Boolean algebra.  

​Between working with Bush’s differential analyzer and his study of Boolean algebra, Shannon was able to design switching circuits. This became the subject of his 1937 master thesis, A Symbolic Analysis of Relay and Switching Circuits. 

 Shannon was able to prove his switching circuit could be used simplify the complex and baroque system of electromechanical relays used in AT&T’s routing switches. Then he expanded his concept and showed that his circuits could solve any Boolean algebra problem. He finalized the work with a series of circuit diagrams.

In writing his paper Shannon took George Boole’s algebraic insights and made them practical. Electrical switches could now implement logic. It was a watershed moment that established the integral concept behind all electronic digital computers. Digital circuit design was born.

Next he had to get his PhD. It took him three more years, and his subject matter showed the first signs of multidisciplinary inclination that would later become a dominant feature of information theory. Vannevar Bush compelled him to go to Cold Spring Harbor Laboratory to work on his dissertation in the field of genetics. For Vannevar the logic was that if Shannon’s algebra could work on electrical relays it might also prove to be of value in the study of Mendelian heredity. His research in this area resulted in his work An Algebra for Theoretical Genetics, for which he received his PhD in 1940.

The work proved to be too abstract to be useful and during his time at Cold Spring Harbor he was often distracted. In a letter to his mentor Vannevar he wrote, “I’ve been working on three different ideas simultaneously, and strangely enough it seems a more productive method that sticking to one problem… Off and on I have been working on an analysis of some of the fundamental properties of general systems for the transmission of intelligence, including telephony, radio, television, telegraphy, etc…”

With a doctorate under his belt Shannon went on to the Institute of Advanced Study in Princeton, New Jersey where his mind was able to wonder across disciplines and where he rubbed elbows with other great minds, including on occasion, Albert Einstein and Kurt Gödel. He discussed science, math and engineering with Hermann Weyl and John Von Neumann. All of these encounters fed his mind.
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It wasn’t long before Shannon went elsewhere in New Jersey, to Bell Labs. There he got to rub elbows with other great minds such as Thornton Fry and Alan Turing. His prodigious talents were also being put to work for the war effort. 

 It started with a study of noise. During WWII Shannon had worked on the SIGSALY system that was used for encrypting voice conversations between Franklin D. Roosevelt and Winston Churchill. It worked by sampling the voice signal fifty times a second, digitizing it, and then masking it with a random key that sounded like the circuit noise so familiar to electrical engineers.

Shannon hadn’t designed the system, but he had been tasked with trying to break it, like a hacker, to see what its weak spots were, to find out if it was an impenetrable fortress that could withstand the attempts of an enemy assault.
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Alan Turing was also working at Bell Labs on SIGSALY. The British had sent him over to also make sure the system was secure. If Churchill was to be communicating on it, it needed to be uncrackable. During the war effort Turing got to know Claude. The two weren’t allowed to talk about their top secret projects, cryptography, or anything related to their efforts against the Axis powers but they had plenty of other stuff to talk about, and they explored their shared passions, namely, math and the idea that machines might one day be able to learn and think.

Are all numbers computable? This was a question Turing asked in his famous 1937 paper On Computable Numbers. He had shown the paper to Shannon. In it Turing defined calculation as a mechanical procedure or algorithm.

This paper got the pistons in Shannon’s mind firing. Alan had said, “It is always possible to use sequences of symbols in the place of single symbols.” Shannon was already thinking of the way information gets transmitted from one place to the next. Turing used statistical analysis as part of his arsenal when breaking the Enigma ciphers. Information theory in turn ended up being based on statistics and probability theory.

The meeting of these two preeminent minds was just one catalyst for the creation of the large field and sandbox of information theory. Important legwork had already been done by other investigators who had made brief excursions into the territory later mapped out by Shannon.

Telecommunications in general already contained within it many ideas that would later become part of the theories core. Starting with telegraphy and Morse code in the 1830s common letters expressed with the least amount of variation, as in E, one dot. Letters not used as often have a longer expression, such as B, a dash and three dots. The whole idea of lossless data compression is embedded as a seed pattern within this system of encoding information. 

In 1924 Harry Nyquist published the exciting Certain Factors Affecting Telegraph Speed in the Bell System Technical Journal. Nyquist’s research was focused on increasing the speed of a telegraph circuit. One of the first things an engineer runs into when working on this problem is how to transmit the maximum amount of intelligence on a given range of frequencies without causing interference in the circuit or others that it might be connected to. In other words how do you increase speed and amount of intelligence without adding distortion, noise or create spurious signals?

In 1928, Ralph Hartley, also at Bell Labs, wrote his paper the Transmission of Information. He made it explicit that information was a measurable quantity. Information could only reflect the ability of the receiver to distinguish that one sequence of symbols had been intended by the sender rather than any other, that the letter A means A and not E.

Jump forward another decade to the invention of the vocoder. It was designed to use less bandwidth, compressing the voice of the speaker into less space. Now that same technology is used in cellphones as codecs to compress the voice and so more lines of communication can be used on the phone companies allocated frequencies.

WWII had a way of producing scientific side effects, discoveries that would break on through to affect civilian life after the war. While Shannon worked on SIGSALY and other cryptic work he continued to tinker on other projects. Shannon’s paper was one of the things he tinkered and had profound side effects. Twenty years after Hartley addressed the way information is transmitted, Shannon stated it this way, "The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point."

In addition to the ideas of clear communication across a channel Information theory also brought the following ideas into play:

-The Bit, or binary digit. One bit is the information entropy of a binary random variable that is 0 or 1 with equal probability, or the information that is gained when the value of such a variable becomes known.

-The Shannon Limit: A formula for channel capacity. This is the speed limit for a given communication channel.
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-Within that limit there must always be techniques for error correction that can overcome the noise level on a given channel. A transmitter may have to send more bits to a receiver at a slower rate but eventually the message will get there.

His theory was a strange attractor in a chaotic system of noisy information. Noise itself tends to bring diverse disciplinary approaches together, interfering in their constitution and their dynamics. Information theory, in transmitting its own intelligence, has in its own way, interfered with other circuits of knowledge it has come in contact with.

A few years later psychologist and computer scientist J.C. R. Licklider said, “It is probably dangerous to use this theory of information in fields for which it was not designed, but I think the danger will not keep people from using it.”

Information theory encompasses every other field it can get its hands on. It’s like a black hole, and everything in its gravitational path gets sucked in. Formed at the spoked crossroads of cryptography, mathematics, statistics, computer science, thermal physics, neurobiology, information engineering, and electrical engineering it has been applied to even more fields of study and practice: statistical inference, natural language processing, the evolution and function of molecular codes (bioinformatics), model selection in statistics, quantum computing, linguistics, plagiarism detection. It is the source code behind pattern recognition and anomaly detection, two human skills in great demand in the 21st century.
 
I wonder if Shannon knew when he wrote ‘A Mathematical Theory of Communication’ for the 1948 issue of the Bell Systems Technical Journal that his theory would go on to unify, fragment, and spin off into multiple disciplines and fields of human endeavor, music just one among a plethora.

Yet music is a form of information. It is always in formation.  And information can be sonified and used to make music. Raw data becomes audio dada. Music is communication and one way of listening to it is as a transmission of information. The principles Shannon elucidated are form of noise in the systems of world knowledge, and highlight one way of connecting different fields of study together. As information theory exploded it was quickly picked up as  a tool among the more adventurous music composers.

Information theory could be at the heart of making the fictional Glass Bead Game of Herman Hesse a reality. Herman Hesse also dropped several hints and clues in his work that connected it with the same thinkers whose work served as a link to Boolean algebra, namely Athanasius Kircher, Lull and Leibniz who were all practitioners and advocates of the mnemonic and combinatorial arts. Like its predecessors, Information Theory is well suited to connecting the spaces between different fields. In Hesse’s masterpiece the game was created by a musician as a way of “represent[ing] with beads musical quotations or invented themes, could alter, transpose, and develop them, change them and set them in counterpoint to one another.” After some time passed the game was taken up by mathematicians. “…the Game was so far developed it was capable of expressing mathematical processes by special symbols and abbreviations. The players, mutually elaborating these processes, threw these abstract formulas at one another, displaying the sequences and possibilities of their science.”

Hesse goes on to explain, “At various times the Game was taken up and imitated by nearly all the scientific and scholarly disciplines, that is, adapted to the special fields. There is documented evidence for its application to the fields of classical philology and logic. The analytical study had led to the reduction of musical events to physical and mathematical formulas. Soon after philology borrowed this method and began to measure linguistic configurations as physics measured processes in nature. The visual arts soon followed suit, architecture having already led the way in establishing the links between visual art and mathematics. Thereafter more and more new relations, analogies, and correspondences were discovered among the abstract formulas obtained this way.”

In the next sections I will explore the way information theory was used and applied in the music of Karlheinz Stockhausen.

Read the rest of the Radiophonic Laboratory series.

REFERENCES:
A Mind at Play: How Claude Shannon Invented the Information Age by Jimmy Soni and Rob Goodman, Simon & Schuster, 2018

The Information: a history, a theory, a flood by James Gleick, Pantheon, 2011

The Glass Bead Game by Herman Hesse, translated by Clara and Richard Winston, Holt, Rinehart and Winston, 1990

Information Theory and Music by Joel Cohen, Behavioral Science, 7:2
(1962:Apr.) 

​Information Theory and the Digital Age by Aftab, Cheung, Kim, Thakkar, Yeddanapudi

​Logic and the art of memory: the quest for a universal language, by Paolo Rossi, The Athlone Press, University of Chicago, 2000.