The art of fugue has a thousand faces

We live in a highly specialized world. Everyone masters only a tiny fraction of the entire reality. As a consequence, we often forget the overall picture. A reminder to the intrinsic unity of reality comes (not so) unexpected from Johan Sebastian Bach and his masterpiece: the Art of Fugue.

The Art of Fugue is the musical testament of Bach. It consists of 14 fugues and 4 canons of increasing complexity. All the pieces are based on variations of a single main subject. The work is highly experimental. Bach stretched the contrapuntal techniques to their limits and came up with an impressive collection of interrelated pieces. No instrumentation is provided within the score. Indeed, this music is quite abstract and perhaps should be read rather than heard.

subject of The Art of Fugue

The main subject of The Art of Fugue.

The Art of Fugue is not only one of the best compositions ever written, but is also a pointer to other fields outside the musical domain. The work has deep relationships with mathematics, biology, linguistics and architecture.

The Art of Fugue is related to mathematics. The entire work can be regarded as a fractal structure based on recursion and self-similarity. The concept of recursion is the key to understanding both fractals and self-similarity. Recursion is the process of embedding some constructs within other constructs of the same kind. This process leads to self-similarity. Fractals display self-similar patterns. In other words, a fractal appears the same at every scale. The Art of Fugue exploits recursion, since it is based on a single musical pattern, which is restated at different levels of resolutions. Therefore, the musical structure becomes self-similar, and the work as a whole can be regarded as a musical fractal.


Fractal objects appear the same at every scale.

The Art of fugue is related to biology. Life is astonishingly diverse. Biologists estimate that nearly 2 million species live on planet Earth. There are creatures as strange as the dumbo octopus, and as regal as the tiger. Nevertheless, all life forms are based on a simple molecule: the DNA. The DNA consists of four bases (guanine, adenine, thymine and cytosine), that combined together are capable of generating every living system. Likewise, the Art of Fugue is based on a simple musical molecule: the main subject. This short string of musical information is responsible for the variety and richness of the entire work. Indeed, the subject is musical DNA, which carries all the information needed to generate the highly complex musical structures that characterize the Art of Fugue.


The Dna is the base of life.

The Art of fugue is related to linguistics. The similarities between musical and natural language have always been explored by philosophers, musicologists and linguists. Both the languages are based on syntax. In particular, the syntax of the natural language can be represented by a generative grammar. A generative grammar provides a small set of rules, that can generate all the correct grammatical sentences of one language. The Art of Fugue has its own set of rules, which considered together form a kind of musical generative grammar. The musical grammar consists of the basic rules of the counterpoint such as the inversion, diminution and augmentation, which directly shape the musical sentences. Thus, a small set of rules can generate the entire richness of the Art of Fugue.

The Art of Fugue is related to architecture. Both music and architecture are interested in structures. Music organizes sonic spaces, while architecture organizes physical spaces. Likewise, they both share the compositional process. In particular, architects have always been obsessed by the organizational process based on modules. For example, the Parthenon entirely relies on the golden ratio. The length of the temple, the height of the columns as well as the Parthenon’s façade can all be obtained from the golden ratio. Also, the architect Le Corbusier used a similar prime generator called modulor. Indeed, he exploited the modulor while working on Notre Dame du Haute and the Unite d’Habitation. The similarities between this architectural process and the compositional process used by Bach are striking. All of them are based on a single molecule of DNA that encodes the genetic instructions for the different kinds of compositions.


The Parthenon is based on the golden ratio.

The Art of Fugue in particular and the music in general are invaluable tools to understand the reality as a whole. Indeed, they can unveil the secret network that connects the different aspects of reality altogether. In this sense, listening to the Art of Fugue is not only a compelling aesthetic experience, but also a multidimensional journey that provides the listener extra-information about the structure of the world itself.

Music, the universe and Boethius

Boethius musicBoethius was a clever philosopher of the 6th century. Among other things, he investigated the relationships between music, humans and the world. The theory of music he developed is insightful and astonishingly modern. Boethius conceived three types of music: musica instrumentalis, musica humana and musica mundana. The musica instrumentalis refers to the music that can usually be heard in the concert halls. The musica humana reverberates within our bodies and spirits. The musica mundana corresponds to the music of the spheres.

Boethius thought that music was everywhere in nature. Why did he develop such a counterintuitive idea? Moreover, isn’t music just an acoustic phenomenon? Of course, it is not!  Boethius understood that music is pure order out of chaos. Composers work with raw musical materials, shaping and twisting notes and rhythms to come up with enjoyable pieces. Doing so, they act as relentless demiurges. Other arts replicate this process as well, however, none of them is as conceptual and intangible as music is. I dare say that Boethius recognized that, and was inspired by the ethereal consistence of music while working on his ideas.

Humans are order out of chaos. Our bodies are miraculous complex systems that challenge the second principle of thermodynamics. Indeed, we are nothing more than an incredible amount of organized information that breathes, perceives and feels. Likewise, mind is order out of chaos. Clusters of unintelligent neurons  organize together and entail the consciousness. So, yes… Boethius was right! The musica mundana really exists and incessantly can be heard throughout planet Earth. The humans, the animals and every living creature are the amazing musicians that play these wonderful tunes.


Musica mundana refers to the music of the universe

The universe is order out of chaos. A handful of fundamental physical laws allow the universe to unfold and develop over time. Those laws are incredible musical as well. They speak of ratios, frequencies and forces, all concepts that find an evident dual within the domain of the musica instrumentalis. Furthermore, the overture of the universe corresponded to a single powerful chord played by the tutti celestial orchestra. Unthinkably, that initial chord echoes even today as an endless ostinato that can be heard by a strange kind of audience, the radio telescopes. So, yes… Boethius was right again! Our universe is pure music. The spheres wildly dance following the musica mundana.

Born to be musical… or maybe not!

Music and brainOne feature that defines us all as humans is our impressive musical ability. No other species shows such a rich musical activity as humans do. Although some birds have a quite original musical language, however, their overall musical activity has a low-level of complexity and is dictated by pure instinct. On the other hand, humans compose very articulated music, perform it, and of course enjoy it. What’s more, humans are moved by music and are able to map simple acoustic signals into emotions, words and memories.

All these processes are amazing, but really complex as well. Therefore, the question of whether or not music is a kind of built-in human feature arises. Indeed, some people believe that our music ability is directly embedded in our genes. Others think that music is the result of cultural pressure. Both points of view have strengths and weaknesses. The intellectual fight between the two perspectives has a long history and is just another example of the never-ending nature versus nurture debate.

Let’s delve into some implications of the hypothesis of music as a natural trait first.

  • Music has universals shared by all humans. Indeed, musical scales, the octave, and  a small amount of tones within a scale (typically between 5 and 7) are all examples of musical universals.
  • Music is a product of evolution. Music can be seen as an evolutionary advantage that improves the chance of survival of individuals who developed musical traits. How? Probably, increasing the chance of mating.
  • The human brain processes music in specific musical modules. This idea directly opposes to the claim that music is processed by the same brain modules of natural language. Fortunately, we already have experimental evidences that support this hypothesis.

Now, let’s discover the main implications of the hypothesis of music as a cultural trait.

  • Music entirely depends on society. Music is but one of a number of activities which rely on society such as literature, art, political organization. Particularly, music is based on the exchange of information between the members of a society. The society develops a musical tradition, which undergoes a continuous process of evolution and refinement.
  • Different musical environments produce radically different musical outputs. Since there aren’t music universals, music develops different paths in different cultures. As a consequence, all the rules of music as well as the music compositions only relies on the here and now.
  • Music is an artificial human construction. Music is an artefact comparable to other artefacts such as hammers, cars or tables. The comparison might sound a bit unorthodox at first, however, all these items share a common characteristic. They are all produced by humans and are not directly coded inside us.

Which of the two approaches is correct? Of course, there’s no absolute answer to this question. However, I suggest that we consider the two approaches complementary rather than contradictory. Since in medio stat virtus, we can say that a mixed approach would be best. There are strong evidences of music as a natural trait, but a simple innatist approach to music would ignore all the cultural pressures which every human experiences during her life.

As a consequence, we can think of music as a double layered phenomenon. The first layer is the natural level, which is shared by all humans. The second layer is the cultural level, which is built upon the natural level. While the natural layer is  culture-independent, the cultural layer is related to the society we live in and the stimuli we are exposed to. Hopefully, this mixed approach could help solving the eternal nature versus nurture musical debate shading some light on the understanding of music.

Can we define music?

No matter how hard we try, we can’t define music. Music is a fleeting idea that all of us grasp, but no one can formalize. A myriad of musicians, scholars as well as ordinary men attempted to find a comprehensive definition of music, but no one succeeded. Every definition focuses on some aspects of music, ignoring other relevant characteristics. The results are crippled definitions that tell just a part of the entire story. So, what are the main problems with the definition of music? First, music is a very broad idea. Secondly, music works on different levels of reality. Thirdly, music is too complex to be defined in a reductionist fashion.

Music spans several disciplines and fields. Although music is a form of art, however, it’s just impossible to describe music only from the artistic point of view, without appearing at least naive. Indeed, music is strictly related to a bunch of other disciplines such as philosophy, biology, cognitive science, sociology, mathematics, physics and linguistics to name a few. Therefore, an ontological definition of music must consider all these different angles to be credible. However, the effort to harmonize all these different perspectives is simply overwhelming.

Music is a phenomenon that appears at the physical, psychological, social and cultural levels. Yet again, a thorough definition of music should be capable of harmonizing all of these different levels of reality. However, definitions of music usually focus on one of the different levels. This process leads to a problem of perspective. Rather than describing music as a multi-layered phenomenon, the definitions of music usually flatten the idea of music. On the other hand, music is a high dimensional concept that embraces several domains. This richness is what makes music a unique phenomenon within the human realm. However, this power of expression entails an overall ambiguity and fuzziness which surround the concept of music, and make it impossible to formally define.

Music is a highly complex phenomenon that can’t be reduced in a sentence or two. This issue is directly grounded within the Western mind-set. The idea of isolating deterministic behaviours which guide natural as well as cultural phenomenon is a well-established Western tradition. Although this approach paid off endless times in science, however, it can’t work for music. Indeed, music is too complex to be synthesized in a simple deterministic formula. As a consequence, all the attempts made to find an ontological definition of music are doomed to failure. The impossibility of defining music is but one of the numerous barriers nature put in front of us when we try to discover things. This is a knowledge boundary similar to the uncertainty principle by Heisenberg in physics, or the theorem of incompleteness by Godel in logic.

So, should we desist from further attempts to define music? Well…I’d say yes, but I’d also say that this is the wrong question to ask. I believe that we should focus on finding a functional description of music, rather than spending time in an effortless exercise to capture the ontological nature of music. This is a more pragmatic approach, that aims at finding a number of important features that music shows. Following this path, I propose a systemic approach to analyze music, which leads to a functional description of music. Indeed, I describe music as a recursive, complex, adaptive, living system. Since every single word in this sentence deserves a rich explanation, I’ve planned to publish some posts to cover these topics later.

How do you define music?