Kiviuq

Kiviuq was a young inuit living far in the northern lands. The landscapes he knew were made of an everlasting tapestry of snow. Kiviuq really loved to read, but he enjoyed writing even more. He liked to write all kinds of stories: stories about the falling snow of winter, of wandering polar bears and the 24 hour day of light in the warm summer. He even wrote modern versions of the ancestral myths of his people. Everyone in the village appreciated reading his stories. Through his words he would bring tears of joy and tears of pain. Kiviuq was happy to bring stories to people and to be an important part of his community.

When the Europeans arrived, they created an alphabet to write in Inuktitut, his native language. It was a very peculiar alphabet made of hooks, probably to go fishing, of toboggans to go hunting and of triangles for which Kiviuq had no idea of the use. Even if Kiviuq was a very talented writer, he had troubles at school. He would write long and beautiful texts, but would always fail his exams because of his calligraphy. His teacher was not happy with his handwriting. His triangles were always slightly tilted on the side, as if pushed by the northern winds. The teacher wanted him to write with equilateral triangles, with three sides of exactly the same length. Sadly enough, Kiviuq was not able to fulfill his teacher’s expectations. His triangles were always isosceles or scalenes, with all kinds of different sides and sizes. Kiviuq felt the different shapes of his triangles helped express emotions in his stories. For example, a very narrow triangle would express danger, while a flattened smooth triangle would express calmness and the time that lingers on. Nevertheless, none of his readers had complained about the shape of his triangles so Kiviuq kept on with this habit. Now, because of his bad marks, his teacher told him he could never go to university and become a writer. This made Kiviuq really sad.

     inuitlanuage

So Kiviuq started working more and more on his triangles. He would practice for hours and hours drawing them and studying them. The more he would study triangles, the less he would write stories. People in the village started worrying because he would not enchant them with his words anymore. Kiviuq was adamant and decided to become a writer so he kept working hard on his triangles. With all this work, he started having good marks in mathematics and sciences. The better his calligraphy became, the worse his stories were. They became boring; the snow was static and the hunts brought no game. He did not worry about this because he had better marks in writing, even with his boring stories. Although less passionate about it, he really got amazing marks in mathematics and sciences.

So one day he finally went to university to become a scientist. Through his years of study, he learned everything that was to be known about geometry, especially about triangles. He then become known as one of world’s most eminent specialists in triangles which lead him to meet very interesting people all around the world and to see amazing marvels on all continents. He went to visit the great pyramids of Egypt, the Eiffel tour and the great Mont Everest’s peak. He worked with the artist Maurelius Escher for the art of tiling the plane with triangles, with the architect Ieoh Ming Pei for the construction of the pyramid in front of Le Louvre consisting of 666 triangles. He even met Dr. Sierpinski and helped him in his worked on his famous triangle constructed with an infinity of smaller triangles.

Vision Crystal-Alex Grey
Alex Grey -Vision Crystal :http://www.alexgrey.com/

One day Kiviuq met a man named Buckminster Füller. Mr. ller was really good with triangles as well and together they decided to construct a map of the world using triangles. After months and months of difficult labour, they finally finished the map and exposed it on a giant wall. Proudly, they spent some time admiring the result. It was a very interesting map, it placed the countries in very different places than the usual map Kiviuq knew. In particular, Kiviuq noticed the map placed the North Pole almost in the middle. Kiviuq started remembering his old land and his old friends.

3e2845d9db8c64f81b98f2d10559be78--buckminster-fuller-world-maps
Richard Buckminster Fuller’s projection

Kiviuq decided to go home after all these years of work and traveling. Coming home, he felt like a stranger. He realised people were really interested in his traveling stories, but none would bring tears of joy nor tears of pain. Kiviuq missed these days when he felt he could reach people’s hearts, so he started to write stories again. In his stories, the snow was not shining like crystal under the sun, the fish tasted like nothing and the polar bears were gone. He had forgot the feeling of his own culture and lost himself. Kiviuq realised he was again really sad. One day, wandering in the village he saw some kids playing in the snow. After some hesitation, he decided to go play with them. He had forgotten the simple pleasure of playing in the snow and the pleasure of building igloos. The more time he spent in his town, the more he was rediscovering the deepness of his culture. One day, when he was ready, he decided to write a story, his own story. He was able again to make the wind dance and the northern lights shine. Moreover, he was now able to place his own culture in a world mixed with myths and sciences. The wind had a soul but as well a precise velocity. It came from and went to places Kiviuq could now describe. Finally he could describe precisely the extremely complex geometry of every single dancing snowflake. Now again, people were reading his story and snow made of tears of joy and tears of pain were to be found in his northern lands.

fractal___snow_flake_by_p1_2004gsb

Félix Lambert – copyright story- 2018

metonym.information@gmail.com

http://metonym.io/?lang=en

Advertisements

Exhibition In Copenhagen

I have the chance to have three paintings on the walls of the Lighthouse cultural center in Copenhagen for few weeks. To visit, look at the calendar if the space is open, the paitings are available only when the space is open for activities.

18740583_129557070933318_8175427971919959530_n

The paintings relate to my researches in narratology that can be found in favious articles of the blog. The main useful articles are:

Narrative Sculptures: Graph Theory, Topology and New Perspectives in Narratology.

and for french readers:

Narration et mathématiques: L’utilisation des graphes au cinéma et dans la bande dessinée (1, 2, 3, 4)

The four pieces presented there are the following:

Infinite Walls

Infinite Walls
Infinite Walls by Felix Lambert. Two Circles on a torus. Copyright: Felix Lambert, 2017

Handcuffs

Handcuffs
Handcuffs by Felix Lambert. Two circles on the plane. Copyright: Felix Lambert, 2017.

Lost in Days and Nights

Lost in Days and Nights
Lost in Days and Nights by Felix Lambert. Two circles on the torus. Copyright: Felix Lambert, 2017.

To complete stories represented by the paintings are presented by the side of the paintings at The Lighthouse. Visit the place, Pasteursvenj 8, Copenhagen, Denmark, to read the stories.

To visit the rest of my work, please visit my website.

http://metonym.io/

Felix Lambert

The Spanish tinge: a hidden treasure of blues, jazz and dance History

”The blues were played in New Orleans in the early days very, very slowly, and not like today, but in a Spanish rhythm.”     Baby Dodds, musician.

When I started to dance and DJ, I quickly became interested in the history of American blues music to find out what it could add to the scene. Being very passionate by nature, I started out with this simple motivation but ended up with the desire to write a book on blues history. In the process, after going through quite an unhealthy amount of documentation, I had the chance to give history classes to blues dancers in Montreal and Toronto. I was then surprised to learn that even long-time dancers or teachers seemed unaware of one of the most important Latin influences on the blues: the Spanish tinge. It reminded me that there’s always a difference between how a dance really looked back in the good old days and what we can see in modern competitions and dancefloors. In between lies a big gap in knowledge and a long list of innovations, adaptations and all natural elements contributing to the evolution of any dance. This article reaches back in history to try to find some hints about blues’ birth and the meaning it could have for dancing.

Back at the beginning of the last century

Blues music probably appeared somewhere in the 1890’s in Afro-American communities and slowly spread around. While the dance itself remained obscure, the music started to appear in partitions during the next 20 years or so. The first white craze for blues music exploded around Handy’s composition ”The Memphis Blues” (recorded in 1914) that Vernon and Irene Castle were using to promote the new dance in vogue: the foxtrot. Although the following video doesn’t show us this particular dance, it does present the general movement of the famous dancers.

A slightly sarcastic but maybe accurate description would be: white people dancing to Afro-American music. Which is, indeed, not a problem by itself. It simply demonstrates the complications of trying to find out what is the ”real” way of dancing a dance. The star couple had a very strict idea of what dancing was all about: no shimmies, no hip movement, no hopping, no dips, no twisting the body and so on since such movements were obviously sinful and disgraceful. (Basically, no fun.) After Vernon and Irene, things could have easily found another path in history and their dance could have ended up being called blues dancing.

What could have changed as well, in similar fashion, are the labels for the music. ”Memphis Blues” can actually be considered a rag, no musician of the time seems to reject fast versions of ”The St. Louis Blues” as being non-blues, and even some songs long after were still labeled foxtrots, like many of Washboard Sam’s recordings. In a similar fashion, Louis Armstrong would explain to Bing Crosby: ”Ah, swing, well, we used to call it ragtime, then blues -then jazz. Now, it’s swing. Ha! Ha! White folks yo’all sho is a mess. Ha! Ha! Swing!”.

Indeed, the multiple variations on this same topic are almost infinite and we end up with the conclusion that it’s very hard to define what blues music is. This forces us to keep two possibilities in mind: either the blues was a very narrow and specific musical trend that has been stretched out for commercial value by recording companies, or it should be considered as a wider variety of music. We’ll probably never get a fully satisfactory answer, but the fact is that the fun part of the debate is hunting down some specific historical components of the music itself.

The question of Latin influences on the blues finds a hint of resolution in various anecdotal contexts. It can be Skip James’ tuning, which he got from a man named Stuckey, who learned it in Europe from Bahamian soldiers during WWI; or, slightly more convincingly, in some songs bearing Caribbean influences, like ”Coal Mine Blues” by Georgia’s songster Peg Leg Howell, which sounds like Mighty Sparrow’s old calypso recordings; or even more explicitly in Clara Smith’s ”West Indies Blues”.

It still remains rather obscure in official American music, and Latin influences might predate the blues and jazz era. In his 1897 publication, Rag Time Instructor, Ben Harney points to some Spanish origins of ragtime music based on the habanera rhythm. Sadly, not much more evidence can be found, either in his own text or in other ragtime publications of the time.

Probably the most famous Latin section of any blues comes in the introduction of the main blues anthem; ”The Saint-Louis Blues” published by Handy in 1914. In his biography he mentions that while playing at the Dixie Park in Memphis, his band went through the habanera section of Will H. Tyer’s ”Maori”. He was impressed by what he described as the natural gracefulness of the black dancers during that section and he suspected something inherently ”black” about this rhythm. He tested his hypothesis by playing ”La Paloma”, the famous Spanish song with its clear habanera section. As predicted, dancers seemed particularly at ease with it so he decided to keep this rhythm in mind for later use. After hiding it in ”The Memphis Blues”, he added it to the introduction part of his well-known blues composition. These few measures are often described as a tango introduction since the habanera rhythm is also very present in Argentina’s national music. It is as well present in various African music. Conclusion; a main Latin element in blues is not strictly Latin.

Where did the rhythm come from?

We know a large quantity of slaves were brought first to the Caribbean for ”seasoning”, basically to prepare them to become good slaves and have more value on the market. Some would stay just a little while, some would stay there forever. The conditions in the Caribbean allowed them to continue to perform music and dances rooted in Africa but evolving in the specific context of the various islands. These dances, like the Bamboula, the Chicta, Calinda and many others involved various hips and shoulders movements, often described as lascivious by white observers.

With the independence of Saint-Domingue in 1804, a lot of white and Creoles fled Haiti and transited to Cuba where their music and dances mixed with the local traditions like the contradanza habanera and others. Then many ended up in New Orleans, some 10 000 already by 1815, where Latin, French and multiple African influences finally all mixed in the pot of the Crescent City and surroundings. Lower class blacks with their very specific dances would often face racial discrimination from whiter, therefore more privileged creoles. Nevertheless, both bore an Afro-Caribbean heritage.

Of the lower class, we know for instance voodoo priest Marie Laveau would hold secret events at Lake Pontchartrain, where dances were different than what you would see publicly at Congo Square. As for the creoles, some famous musicians came out of that social group, like Alphonse Picou, Emmanuel Perez, Sidney Bechet and many others. The 1890’s Jim Crow laws forced some more interactions between black and creole musicians since before Jim Crow creoles lived more as a separate upper class from blacks, while after Jim Crow they were more considered all in the same boat by whites. Of the various traces of such collaborations and influences, one can listen to the Latin-sounding trumpet solo on King Oliver’s Creole Jazz Band version of ”St-James Infirmary”, or simply in the various biguine-like songs of Sidney Bechet and the Haitian Serenaders or the Baby Dodds Trio, sung in creole dialect. (Even if these recordings date from 1938 and 1946, biguine music’s popularity goes back to the beginning of the century, and like the tango, it found a large audience in France).

As segments of the The Spirit Moves by Mura Dehn clearly evoke, both the brothel’s social function of blues and the ceremonial Afro-Caribbean dancing probably influenced the basis of blues dancing in New Orleans. The dancers here were professional stage artists, but they seem aware of the underground background of the dance. Sandra Gibson’s very sensual motions might come from lascivious movements from the red-light district while Al Minns’ performance near the end of the video looks like a slow Afro-Caribbean dance.

In the small jazz revival of the late thirties, when collectors suddenly started to realize or remember that jazz came from people of color, some aficionados decided to go back to its roots and talk to old-time musicians. This period is also the beginning of a long series of records of Latin jazz et Creole jazz by various artists. One of these musicians was no other than one of the most important jazz composer of the century.

When Jelly Roll Morton crossed the door at the Library of Congress in 1938, he was already going downhill in his career. After benefiting from the instant money and gratification of his popularity in the music world, he had to see publishers, record companies and white musicians make a fortune from his music. The 20-hour long interview he was going to give to Alan Lomax in the following months ended up becoming one of the greatest jazz interviews ever made, even considering the various inaccuracies.

Of course, we need to challenge Morton’s main claim of having invented jazz music in 1902, but the interview still provides a lot of interesting information. One of the key moments of the oral history he left us is the precise description he gives of the influence of the certain rhythm and in creation of blues and jazz, starting from an example we already know:

‘’Now take La Paloma, which I transformed in New Orleans style. You leave the left hand just the same. The difference comes in the right hand –in the syncopation, which gives it an entirely different color that really changes the color from red to blue. Now in one of my earliest tunes, New Orleans Blues, you can notice the Spanish tinge. In fact, if you can’t manage put tinges of Spanish in your tunes, you will never be able to get the right seasoning, I call it, for jazz.’’

This “Spanish tinge” is the habanera rhythm. It is believed he has learnt it from a Spanish guitar teacher he had in his teens. It seems plausible since there was also a strong Latin presence in New Orleans and the rhythm had already spread around (as we can hear later in singer Lydya Mendoza and others). Its direct influence on blacks and creoles seems to have stuck to piano since guitarists from New Orleans don’t share this musical figure, nor can it be found in Harry Oster’s various recordings made in Angola prison years later. Morton’s use of seasoning relates most likely to food, but since it shares the meaning of adding value it makes it all slightly disturbing.

What is so fascinating in this interview segment is that you can clearly hear the drift from a Spanish song to the blues, all of this simply by changing the complementary syncopation of the left and right hands, which gives the music a particularly appealing kick that makes you want to move. It is also interesting to notice in the interview that he starts from the Spanish tinge and slowly modifies it into another closely related rhythm known as the Charleston rhythm, which is the one we can hear on his recording of ”New Orleans Blues”.

Charleston_rhythm_(with_notes)

As Morton points out, it is for him also an essential part of jazz composition: he uses it for instance in ”The Craze”, ”Creepy Feelings” and others. The Spanish tinge probably traveled with some early piano players as far as New York. Willie the Lion Smith and James P. Johnson remember Jack the Bear playing the salty song ”The Dream”, again with the same rhythm.

It is difficult to know why the Spanish tinge doesn’t appear on more records from the 20’s. It might be that for recording companies, its Latin flavor made it less suitable for the race record market. In any case, we know it spread at least a little at the time, as the recording of ”Tia Juana Man” by Ada Brown testifies. Not surprisingly, the various musicians on the song are also New Orleans old timers like Albert Nicholas, Barney Bigard, Johnny St-Cyr, George Mitchell and Luis Russell.

The Spanish tinge almost disappeared when New Orleans pianists moved up the Mississippi to reach Chicago. Luckily though, some pianists like Doug Suggs and Little David Alexander were using it in the Windy City. It appears on many of Jimmy Yancey’s recordings, from ”At the Window” to his interpretation of Leroy Carr’s classic ”How Long Blues”.

From there, the multiple exchanges and cutting contests between musicians might have worked to spread and modify its structure. Its influence can still be heard on some random recordings like the piano line of Jazz Gillium’s ”Gonna Take My Nap”, some Champion Jack Dupree (from Louisiana) or Sunnyland Slim’s songs and multiple other recordings. It also appears in closely-related patterns on Memphis Slim’s left hand, and he gave us to understand that such a form was also practical, since it was possible to play it while holding a cigarette!

It also reached the West Coast as it can be heard on multiple recordings by Lloyd Glenn: ”Southbound Special”, his interpretation of Yancey’s composition ”Yancey Special”, on ”Savage Boy”, and so on. Its use on ”Old Time Shuffle” might indicate that the rhythm has been used for a long time. (The shuffle was danced on plantations, but other song labeled as shuffles, like ”T-Bone Shuffle” or ”Ballroom Shuffle” don’t use the Spanish tinge.)

Indeed, once you started spotting the Spanish tinge or its offshoots, you see it everywhere, especially in the R&B era and Jump Blues/Jive where variations of it can be found at different speeds. Ruth Brown, Clarence Garlow, Professor Longhair, Percy Mayfield, Howlin’ Wolf, John Lee Hooker, Chuck Berry and others were influenced by it, at least for some of their songs.

What does this mean for dancers:

Since this asymmetrical rhythm seems to have been present for a long time in the blues and related music, it is natural to think a specific step could have followed it. In a Latin dance like salsa, the asymmetry is reflected in the steps. From the lead’s perspective it goes: Left foot front, right foot under, left foot under followed by its anti-mirror right foot back, left foot under, right foot under. The equivalent for follows is right foot back, left foot under, right foot under followed by its anti-mirror left foot front, right foot under, left foot under. It is very plausible that similar steps were used in New Orleans more than a century ago. Of course, a possibility would be the mirror version of salsa steps starting with the left foot back followed by right foot front. (for the lead)

9600022a3d6b86c6d16ea0956836ea0c

The same holds for half of that series of steps, which is also a structure present in current dances, and that can be simplified as step, step rock step. The rock step here holds the same function of following the asymmetrical structure of the music (it can be seen as half the mirror version of salsa steps while for follows as half the normal salsa steps). The rhythm here can be understood as the superposition of two patterns, binary and ternary. We’ll see how it could be thought in terms of dancing.

In that video, from the lead’s perspective, the beat at 1 could be for the rock step behind, on the left foot and the step under when the red spot hits 2. As a result, the beats at a and & are used by the right foot to balance back the motion of the left foot.

In follow’s perspective, the beat at 1 could be for the right step in front and when the red spot hits 2 the right foot steps under. The left foot then serves to balance that motion at a and &.

The asymetry of the rhythm can appear in simpler forms in the dance. From the Memphis Slim’s song previously presented, the slow dance could very well include the asymetric basic steps: left-right-left, pause, right-left-right, pause.

Of course, we’ll never know exactly how much such asymmetrical steps were commonly used since it was easier to make money from people’s recordings than from filming their dance events. We know from some songs that many steps were used in country blues-related music, like the grizzly bear, the chicken scratch, the turkey trot and many others. A step behind in a kind of a salsa-ish or rock-step-ish fashion seems also very probable.

We have some footage of jazz dancers in big cities like New York and Chicago, which includes some rock steps, but there’s pretty much none for blues dancing, especially in areas like New Orleans and the Piedmont. It is very plausible that Piedmont dancers were either tap dancing or doing a mixture of bouncing steps, like some stationary Charleston. Maybe some bouncy rock steps were there as well. Blind Blake did mention the Charleston in ‘’Dry Bone Shuffle’’ and Blind Willie McTell’s wife remembers dancing the Charleston in a club on her husbands music. It is difficult to know how much this was similar to the Geechie and Gullahs dances that inspired James P. Johnson to compose his ‘’Charleston’’ that made the dance popular through the show Runnin’ Wild in 1923. To go full circle, we can notice that the Spanish tinge can be obtained from the Charleston rhythm by substituting the second dotted crochet (or quarter) by three eighth notes, the first of the three remaining silent to obtain the rhythm. It is exactly the reverse process of what Morton does in the interview for Lomax at the Library of Congress. How these rhythms influenced the dancing remains an interesting question since Piedmont blues represents a fair deal of blues recordings and records travel fast. For instance, it could have reached the Mississippi delta since a guitarist like Robert Johnson knew how to play covers of Blind Blake, Blind Boy Fuller or Blind Willie McTell. Such a dance would also be well suited for other upbeat music of the time like the Mississippi Sheiks and others.

As much as the Spanish tinge seems to have sometimes only few occurrences throughout a song, at various speeds, some rock step equivalent might very well have been part of the various steps that created the blues or all related dances of black people at the time. Even more interesting, is that we can use the Spanish tinge to link both some parts of blues and jazz music and even, maybe, dancing. Nevertheless, it remains difficult to estimate which proportion of recordings really contain direct influences of the Spanish tinge. If it was a steady left-hand pattern of some early 1900’s compositions, it evolved into various forms or was absorbed in other songs as only an occasional punctuation. In all cases, it’s hidden there in many songs.

Finally, it reminds us that there is a lot we don’t know about blues dance history. Whatever we learn about how to dance will always remain useful indications, hints and advice, not an absolute. Dances grew as combinations of various steps, often invented by long-forgotten folks in juke joints, rent parties or other dancing venues. The same holds for present-day dancing, where everyone’s personal touch can add to how we dance. In any case, we should keep in mind learning to dance well with the people in our own community. The center of the scene are the dancers with their valid preferences, points of view and skills. The only absolute criteria we should always keep in mind are to be safe and respectful, with a touch of bluesiness to wrap it all up.

©Félix Lambert, Montreal

Contact: metonym.information@gmail.com

Thanks to Andrea Rosenberg, Debbie Carman, Evelyne Batoula and Dominique Perras Saint-Jean for editing and suggestions. Thanks to Felix-Antoine Hamel for sharing musical passion and knowledge. Thanks to all the dancers I enjoyed sharing a moment with on the dancefloor.

Selected Bibliography:

Borneman, Ernest. Jazz and the Creole Tradtition. Presented to Penrose Library, University of Denver

Broven, John. Rhythm & Blues In New Orleans. Pelican Publishing Company

Brown, Scott. James P. Johnson; A Case of Mistaken Identity. Scarecrow Press

Emery, Lynne Fauley. Black Dance from 1619 to Today. Dance Horizon Book

Fernández, Raúl. Latin Jazz: The Perfect Combination/La Combinación Perfecta. Chronicle Book, 2002.

Goffin, Robert. La Nouvelle-Orléans, capitale de jazz. Éditions de la Maison Francaise

Gray, Michael. Hand Me My Travelin’ Shoes: In Search of Blind Willie McTell . Chicago Review Press

Guralnick, Peter. À la recherche de Robert Jonhson. Traduit par Nicholas Guichard. Le Castor Astral

Handy, W.C. Father of the Blues. MacMillian

Herzhaft, Gérard, Americanaè Histoire des musiques de l’Amérique du Nord. Librairie Arthème Fayard

Kubik, Gerhard. Africa and the Blues. University Press of Mississippi

Lomax, Alan. Mr. Jelly Roll: the Fortunes of Mr Jelly Roll Morton, New Orlans Creole and ”Inventor of Jazz”. University of California Press

Lornell, Kip. Introducing American Folk Music; Ethnic and Grassroot Traditions in the United States. McGraHill

Reich, Howard and William Gaines. Jelly’s Blues: The Life, Music, and Redemption of Jelly Roll Morton. Da Capo Press

Rosemain, Jacqueline. Jazz et Biguine: Les musiques noires du Nouveau Monde. L’Harmattan

Schuller, Gunther. Early Jazz: It’s Roots and Musical Development. Oxford University Press

Sharing Paradoxes: Impossible Spaces, Impossible Times and Impossible Facts. The Function of Self-Contradictory Structures in Arts, Sciences and Philosophy. (Part 4)

A first step in dealing with paradoxes is then to accept their double existence as being true and false. Outside academic studies, it is a natural process implied in the appreciation of any narrative art. As it has been described about cinema, there is a point where we accept the false to be true, as if trying to find a proof reducto ad absurdum. This process is called the suspension of disbelief (Walton, 7). Proposed first by the poet Samuel Taylor Coleridge, it has extended in the study of literature, cinema and videogames to name a few. This refers to any action where the false is taken as possibly true in order to appreciate a narration and facilitate the immersion process. Youssef Ishaghpour describes the duality implied as the reality of the image and the image of reality (Ishaghpour, 8-11). The suspension of disbelief is therefore a way to conciliate this duality in order to appreciate the fiction.

The difference when working in a fictional environment rather than in a scientific one is that every time a contradiction or unearthly events appears, it is usually taken as an extension of the perceived diegetic world: when the staircase optical illusion appears in Inception, it is accepted as part of the fictional world. Instead of defying physical rules, it is simply accepted as a new information contained within the fiction. Again, as in Kierkegaard or Bohr’s vision, it is an extension of the paradigm. The same happens for multiple worlds’ diegetic construction such as previously mentioned in movies like ExistenZ or Avalon. The multiple ontological world, very similar in their nature to the Russell’s paradox are accepted as such. The suspension of disbelief catalyses the conceptual acceptation of such constructions and even changes them into interesting and pleasing artistic visions.

Cat'.s Paradox

Figure 7: The cat’s Paradox. Source: Wikipedia

The use of paradoxical constructions taken from science does not end here. Another case comes from quantic physic. Erwin Schrödinger described the nature of some quantic events by the metaphor of a cat in a box. Let say there is a cat and some poison inside a box. There is as well a 50% chances for the poison to be relieved and therefore for the cat to die. The way quantic physics works is that as long as the information about the cat has not been extract from the situation, the cat is in both states: alive and dead. Both states excludes each other and therefore it leads to a paradox that of ‘’ the living and the dead cat mixed or smeared out in equal parts.’’ (Schrödinger) What modern physics proposes as a solution the acceptation of both state for a certain period of time. This process is known as quantic bifurcation. Even if this is very difficult to accept as being true for neophytes, when transferred to fiction it leads to acceptable and interesting narrative constructions. An example of a movie using this type of multi-linear time frame is Source Code by Duncan Jones. In this movie, a soldier is sent multiple times in the past to prevent a terrorist attack. After failing multiple times, he achieves his goal and life continue normally in this new independent timeline. The use of quantic bifurcation appears in multiple science-fiction movies and communities of fans are sketching schemas to understand the structure behind the film. Movies like Primer by Shane Carruth and Looper by Rian Johnson have generate numbers of complicated charts using quantic bifurcation in order to explain these narrations. (Figure 8)

Print

Figure 8: Looper movie chart by Rick Slusher. Source: Film.com

Another paradoxical time construction that has caused many problems is the time loop. Circular construction of time was accepted by many cultures around the world: Egyptians had circular time named Neheh (Assman, 137), the tzolkin, the Mayan calendar based on cycles of 260 days (Falcón, 19-21) and Hinduism has constructions of multiple intricate circles (Eliade, 134-136). This vision does not conciliate with the European linear construction of time, but it still easily apply to fiction.

Time being both in the future and the past appears in various fictional cases. First of all, whenever there is a time loop a cyclic time has to be accepted. Movies like Terminator (Cameron, 1984), Before the Rain (Manchevsky, 1994) or Chin Chin el Teporocho (Retes, 1976) all present this cyclic time construction. Indeed, time loops can be multiple and quantic bifurcation might again apply.

In the quantic bifurcation case as in the circular time construction, the paradoxical construction induces multivalued time states, discrete moment can be different but at a same time distance from a specific moment, the bifurcation point or those previous, or they can stand both in the future and in the past of a referential moment. It can also be seen as a specific case of multiple ontological states, as previously described, but with the specificity that the ontological state is defined by a time value.

It can be presented in a more mystic way as in the movie Voyage in Time by Tarkovsky and Tonino Guerra. In this case, the movie shows the directors talking about the film they will make about a trip they once had. The anecdotes supposedly in the past appears as well in front of the camera and therefore the time of the movie is triple, it stands for the past, the present and the future as in Three Sundays in a week, but without the logical explanation.

Perhaps the most well-known results about paradoxes is Gödel’s incompleteness theorem. After Cantor and Russell discoveries, logicians have tried to build a perfect and complete system for logic. The project happened to be more problematic than expected and new set of axioms surfaced. The outstanding result obtained by the mathematician Kurt Gödel changed radically the conception of logic and left the community in crisis. The incompleteness theorem states that no matter how many axioms we add to a logic system, there will always appear some statements that will be undecidable, meaning it will be impossible to prove them right or wrong (Nagel, 19-20). This is a perfect example of Paz’s perspective of grasping dual objects as such instead of considering them as problematic undefined concepts to reach a better understanding of it. In this case, the conclusion obtained by paradox is that paradoxes are inherent part of complex logical systems.

Paz’s consideration encompasses a big range of logical instance and, as seen previously, they apply to a wide variety of paradoxical objects: from optical illusions to narrative charts passing through quantum physics. It still does not hold for a type of undecidable statements. Some facts are not necessarily true or false; they stand somewhere in between as a result of incomplete definitions. They work as ambigrams but instead of offering mainly a finite amount of elements, they offer a continuous range of possible information. Such problems are common in everyday life since more situations are not clearly defined. For instance, we can pretend the sky is blue but it can’t be proven without adding precisions to the statement offered; at night the statement does not hold for instance. A relatively new branch of mathematics dedicates itself to such logical system. The idea behind this fuzzy logic, as it is coined, is to attribute truth values that varies continuously between the usual zero and one (Kandell). Therefore allowing any probability of truth ranging from zero to 100%. Such logical system coincides with perspective of quantum physics allowing diverse states with various probabilities. It is the case for instance for electrons in the atomic model were they navigate through a probabilistic area instead of following a precise trajectory.

Finally, paradoxes can appear within humoristic or philosophical functions. The twist are often used in usually called intellectual humour such as Woody Allen’s work. In Allen’s quote from Annie Hall ‘’ The food here is terrible and the portions are too small’’, the double statement stands in the contradiction that, in fact, if the food is terrible there is no reason to ask for more, but complaining about small portions implies asking for more food. This kind of construction can be found as well in Annie Hall: ‘’ Life is full of mystery, loneliness, and suffering –and it’s all over much too soon’’. The role of the paradox is then, in this case, to release a tension constructed around the paradoxical statement. In this situation, the contradiction, or double truth value, stands as a sign that the joke has reached its climax. The contradictory aspect of the logic involves is to be read as a sign to character does not make sense anymore, therefore the humoristic relief. The humour can follow as a comment on a paradox: way before Russell, Lewis Carroll underlines that no one can contain himself because of excitement because nothing can contain itself (Benayoun, 84). These considerations follow the seriousness of Ambrose Bierce’s definition of logic as the art of thinking within human capacities (Benayoun, 113), the presence of this limit is in itself both humoristic and a relief.

Paradox Humour

Figure 9: Paradox humour

In a broader perspective, the same applies to koans, small stories or statement present in the zen tradition. The sentences serve to increase doubt and questioning. The simple logic behind the koan ‘’What is the sound of one hand clapping’’ is similar; it states the possibility and impossibility of the referred sound. Possible since it is stated there is a clap sound and the impossibility by the uniqueness of the hand producing it. In this case, the paradoxical information serves again to release tension. The same holds for the koan: If you have a stick, I’ll give one to you, if you don’t I’ll steal it from you. The tension is released with acceptation to work outside a strict logical frame, to accept our humanity as proposed by Bierce.

This work outside logic may serve as well, paradoxically, for theological arguments. To understand we have to go back to the unliftable rock paradox. God, being almighty, should be able to create an unliftable rock, but then if he can lift the rock he is not almighty. An easy solution to this problem is to state that God’s work beyond human based logic.

As we have shown, the contradictory dialectic raising from paradoxes only cause problem within its own paradigm of binary logical values as being true or false. When grasped as specific concepts gathering both values, or, even infinitely many values ranging from absolutely true to absolutely false, many applications can be found. Accepting such condition standing in between these poles is what allows us to enjoy narratives in different ways; first to enhance the emotive effect of a diegetic world by accepting the ontological quality of fiction as being an image of reality that is itself included in and presented as a simulacra of reality, secondly as to define science fiction or fantastic narratives as legitimate by extending the accepted diegetic world. Logical statements sharing both truth and false value are integral parts of human scientific and cultural knowledge[1]. It is hoped that more research to consider paradoxes in their social appearances will be provided as to understand better their functions as a fundamental part of human thinking rather than solely as odd mythological thoughts gravitating in the abstract spheres of philosophy and logic.

Felix Lambert

First version September 2015

[1] As it is the case in Dialetheism. For a good review of this as a philosophe, the reader is invited to read the entry in the Stanford Encyclopedia of Philosophy by Francesco Berto.

 

Mediagraphy

Alexander, Amir. 2014. «Guldin et les indivisibles de Cavalieri». Pour La Science, n˚ 440, p.70-73.

Allen, Woody (real.) 1977. Annie Hall. USA: Rollins-Joffe Productions. DVD

Assman, Jan. ‘’Le Temps Double de l’Égypte Ancienne’’. Pour La Science, n˚ 397,2010.136- 141. Print

Baltrušaitis,Jurgis. Aberrations : Les Perspectives Dépravées-1. Paris : Flammarion, 1995. Print

Baltrušaitis,Jurgis. Anamorphoses : Les Perspectives Dépravées-II. Paris : Flammarion, 1996. Print

Benayoun, Robert. Les dingues du nonsense.Paris : Balland,1984. Print

Berto, Francesco. Dialetheism. Entry in the Stanford Encyclopedia of Philosophy. Online: http://plato.stanford.edu/entries/dialetheism/.

Biederman, Irving. ‘’Recognition-by-Components : A Theory of Human Image Understanding.’’ Psychological Review, Vol. 94, No. 2, 1987. 115-147. Print

Boll, Marcel. Histoire des mathématiques.Paris: Presse Universitaire de France, 1968. Print

Browne, Cameron. ‘’Impossible Fractals’’. Computer & Graphics, 31(4), 2007. 659-667. Online: http://eprints.qut.edu.au/15013/1/15013.pdf

Cantor, Georg. Contributions to the Founding of Theory of Transfinite Numbers. New York: Dover Publications, 1952. Print

Carruth, Shane (real.) Primer. 2005. États-Unis: ERBP. DVD

Cronenberg, David (real.) EXistenZ. 1999. Canada et Grande Bretagne : Alliance Atlantic Communications, Canadian Telivision Fund, The Harold Greenberg Fund, The Movie Network, Natural Nylon Entertainment, Serendipity Point Films, Téléfilm Canada et L’Union Générale Cinématographique. DVD

Delahaye, Jean-Paul. ‘’Labyrinthes de longueur infinie’’. Pour la Science, n˚318 (Avril), 2004. 90-95. Print

Delahaye, Jean-Paul. ‘’Ambigrammes’’. Pour la Science, n˚32 (Septembre ), 2004.90-95.Print

Falcón, Maricela Ayala. ‘’Tiempos mesoamericanos, calendarios mayas’’. Artes de México, vol 107, El Arte del Tiempo Maya, 2012. 18-25. Print

Garibay, Tito. El albur: Método introductorio. México, D.F.: Grupo Editorial Tomo, S.A. de C.V., 2008.

Grimaldi, Ralph P.. Discrete and combinatorial mathematics; an applied introduction, 5th Ed.    Boston: Pearson Addison Wesley, 2004. Print

Hofstadter, Douglas. Gödel Escher Bach: Les Brins d’une Guirlande Éternelle. Paris : Interéditions,         1985. Print

Hofstadter, Douglas.’’On Self-Referential Sentences‘‘. Metamagical Themas: Questing fort the Essence of Mind and Pattern. New York: Basic Books, 1985, p. 5-24. Print.

Jones, Duncan ( real.) Source Code. 2011. États-Unis: Vendome Pictures et The Mark Gordon Company. DVD

Kandel, Abarham. Fuzzy Mathematical Techniques with Applictions. Addison-Wesley Publishing company.

Koch, Hel von. “ On a Continuous Curve without Tangent Constructible from Elementary Geometry .“ Traduit du français par Ilan Vardi. Dans: Classics on Fractals, édité par Gerard Edgar. New York : Addison-Wesley Publishing Company, 1993: p. 25-45.

Kunzmann, Franz, Franz-Peter Burkard et Franz Wiedmann. Atlas de la philosophie.Paris : La Pochothèque, 1999. Print.

Labelle, Jacques et Armel Mercier. Introduction à l’Analyse Réelle. Montréal: Modulo, 1993.. Print

Mandelbrot, Benoît. The Fractal Geometry of Nature.New York: W.H. Freeman and Company,1977.

McHale, Brian. Postmodernist Fiction. New York: Methuen,1987. Print

Moore, Ruth. Niels Bohr: The Man, His Science, & the World They Changed.New York: Knopf, 1966. Print

Nagel, Ernest, James R. Newman, Kurt Gödel, Jean-Yves Girard. Le théorème de Gödel. Traduis de l’anglais et de l’allemand par Jean-Baptiste Scherrer. Paris :Éditions du Seuil, 1989. Print

Nolan, Christopher (réal.). Inception. 2012. États-Unis et Grande-Bretagne : Syncopy. DVD

Paz, Octavio. Conjunciones y disyunciones. Distrito Federal : Editorial Joaquín Mortiz, 1991. Print

Peano, Giuseppe. ‘’Sur une courbe, qui remplit toute une aire plane’’. En ligne : Mathematiche Annalen, Vol. 36, n° 1, p. 157-160, 1890. Consulté le 10-04-2012. DOI : 10.1007/BF01199.438.

Pelletier, Francis Jeffry. 2000. «Review of mathemtics of fuzzy logic». The Bulletin of Symbolic Logic 6 (3), p. 342-346.

Poe, Edgar Allan. Ne pariez jamais votre tête au diable : et autres contes non traduits par Beaudelaire. Traduction, préface et notes par Alain Jaubert. Paris : Gallimard, 1989. Print

Reinhardt, Fritz et Heinrich Soeder. Atlas des mathématiques. Paris: La Pochothèque, 1997. Print

Sagan, Hans. Space-Filling Curves. New-York : Springer, 1994. Print

Schrödinger, Erwin. 1935. «The Present Situation in Quantum Physics». Trans. John D. Trimmer. The original article in Naturwiessenschaften, 23 (49), p. 807-812. Traduction in Proceedings of the American Philosophical Society, 123, p. 323 -338

Slaatté, Howard A. A Re-Appraisal of Kierkegaard. Boston: University Press of America, 1995. Print

Stewart, James. Analyse : concepts et contextes Volume 1. Fonctions d’une variable. Traduction de la première édition par Micheline Citta-Vanthemsche. Paris : De Boeck Université. Print

Tarkovski, Andrei, Tonnino Guerra (real.) 1983. Tempo di viaggio. Italy: Genius s.r.I. and RAI Radiotelevisione Italiana. DVD

Vidal-Rosset, Joseph. Qu’est-ce qu’un paradoxe?. Paris: Librairie Philosophique J.Vrin, 2004. Print

Walton, Kendall, L. «Fearing Fictions». The Journal of Philosophy, Vol.75, No.1 (Jan., 1978), p. 5-27. Print

Sharing Paradoxes: Impossible Spaces, Impossible Times and Impossible Facts. The Function of Self-Contradictory Structures in Arts, Sciences and Philosophy. (Part 3)

We can classify visual paradoxes into two categories: the simply self-contradictory ones and the ambigrams. In looking at most of Maurelius C. Escher’s optical illusions, the spatial design seems to be both true and false. As pointed out by Biederman, since the shapes preserve adequate angle constructions, the objects seems to be credible. That is a direct result of the importance of line intersection in object recognition. Nevertheless, the overall conceptualisation of the object seems not acceptable as being true (Biedermann, 135-140). In this case, these visual paradoxes are simply self-contradictory because the macroscopic veracity statement contradicts the microscopic ones (as in the two sentence: The next sentence is true. The last sentence is false (Hofstadter, 19)) Many artists have found multiple interesting ways to produce these types of paradoxes. Escher indeed is well known for such constructions, but we can name as well graffiti artist Damien Gilley, Dutch artist Ramon Bruin and Istvan Orosz (Figure 3) Having met quite a wide popularity, it is normal that these simply self-contradictory visual objects have appeared in various situations. For instance, Penrose’s triangles has appear on post stamps, tattoos and many everyday objects.

Inception stair scene

Figure 4: Inception.s stairs scene by Christopher Nolan

These self-contradictory visual objects can contribute solving other artistic problems. In Christopher Nolan’s movie Inception the infinite stairs illusion is used and as real instance within the diegetic world to trap the enemy. In this case, it is a use of a visual paradox to solve of narrative problem. (Figure 4) On his hand, Cameron Browne have found interesting ways to merge the optical illusion construction with another old problem; the paving of the plane (Browne, 2007). Browne has constructed infinite patterns of self-contradictory visual objects that can be used to fulfill the entire plane. He worked as well with contradictory fractal structures (Figure 5)

Camron Browne.jpg

Figure 5: Impossible Fractal by Cameron Browne. Source: http://www.cameronius.com/graphics/impossible-fractals-figures/

Ambigrams are figures that show two incompatible information at the same time, inasmuch a paradox, they work on a scale more nuanced than the dichotomic paradoxes[1]. The figure of a young-old lady is a popular example of ambigram due to H.H. Hill (Delahaye, 91). In this image we can actually perceive two different portraits. One pictures a young lady and one offers the profile view of an old lady. It is a paradox since it contradicts itself, not in the previous case straightforward manner, but by ricochet. If it is a young lady, then it is impossible to be the old lady at the same time and vice versa. In the other hand, we can interpret the image as containing two informations, two different images. Ambigrams, working on a larger scale; they can contain more information. For instance, in figure 5 one can find six apparitions of the word palindrome, all put upside-down and to be read in both directions.

The advantage of using a word like ambigram is that it underlines an interesting property of paradoxes, that fact of being containing multiple statement that would usually not appear in general in a coherent manner. To go further in this sense, we have to go back to an analysis made by Mexican sociologist and writer Octavio Paz. In an analysis of complementary and dual concepts existing in various societies, Paz stresses the importance of considering such pairs as a whole by focusing on the relation between them. For instance, body and no-body are not to be considered as specific meanings except to express contraries (Paz, 55). This perception of duals as a whole can lead to interesting results when applied to paradoxes.

Palindrome

Figure 6: Ambigram

[1] We mean by this that the information is not straight opposite, like true and false, up and down, etc.

Sharing Paradoxes: Impossible Spaces, Impossible Times and Impossible Facts. The Function of Self-Contradictory Structures in Arts, Sciences and Philosophy. (Part 1)

By Félix Lambert

Relativity-escher

Relativity by M.C. Escher

Logical twists and games have always seemed to intrigued thinkers from all times and civilizations. As being mostly curiosities, they appeared in an unorganised fashion in many disciplines, arts and games. One such logical twists is the paradox. From Antiquity’s philosophy to modern mathematics, paradoxes have brought various questions and, in some cases, answers about human knowledge. The main goal of this paper is to demonstrate how paradoxes have proven useful in various cases in sciences and narrative arts, therefore justifying them as proper object of study instead of being considered simply as odd singularities. Different definitions of paradoxes will first be discussed and a series of paradoxes will be presented. We will then use Octavio Paz’s discussion over dual concepts to approach paradoxes. Finally, we will come back to some examples where a dualistic study of some paradoxical structures has been useful. This will show how paradoxes now constitute a significant part of contemporary knowledge, art and to a certain extent, mythology.

The first common point found in various definitions of paradoxes is the self-contradictory aspect of its claim. One of the most common paradoxes that clearly exemplifies this fact is the liar’s paradox. It seems to first have been proposed by Epimenide when saying that all Cretans lie. The problem appears when we realise Epimenide is himself native from Crete and therefore two options are possible: first, if he lies then his statement is right and therefore we cannot trust his saying, secondly if he tells the truth we because he is Cretan then he tells a lie. A more condensed version of this paradox was expressed by Eubulibe of Millet (Vidal-Rosset, 31): ‘’I’m lying’’ of which the literary equivalent is ‘’This sentence is false’’. The common point of all these statements is that they all appear as contradicting themselves, leaving us incapable of deciding the rightness of their claim.

Many authors have proposed classifications for paradoxes. One interesting proposition has been Willard Van Orman Quine’s tripartite classification. Quines distinguishes falsifical paradoxes, veridical paradoxes and antinomy (Vidal-Rosset, 105). The first category includes paradoxes that are finally proved to be false. Such an example is given during the Renaissance by Guido Ubaldus finding that 0=1, which was interpreted at the time by implying that matters can be creating out of nothing. The claim seems to contradict itself by giving two different values to an integer, one being equal to zero and of course to itself. The proof uses infinitely many addition of ones and zeros by reorganising them in such a way to obtain the result (Stewart, 578). The paradox is false because it does not use the allowed operations between infinite series (Labelle, 262-264). Another common example is 1=2 obtained by a division by zero, which is of course prohibited. The second type of paradoxes, the veridical ones, contains paradoxes that seem false but end up being true. The most common example is the Monty Hall paradox. A player is offered three choices of doors behind which one of them a price is hidden. The player picks a door. After the choice, one of the two remaining doors is opened and shows no price. The player is then asked to choose again. Although it is commonly believed that the chances on the last pick are even, it is in fact false: it seems that there is a 50% chance of winning when there is actually a higher chance to win if the player switches their choice. We can compute all possible options and the results shows the player stands better chances, in fact 2 out of 3, if he doesn’t change his mind. This paradox is a veridical paradox because we can prove it to be true. (Figure 1)

Monty Hall Paradox

Figure 1: The Monty Hall paradox

Another veridical paradox has been used by Edgar Allan Poe in one of his short story “Three Sundays in a Week’’. In this story, two young lovers want to get married but the uncle in charge decides it will only happen when a week will have three Sundays. A year later on a Sunday afternoon, the couple meet the uncle with two captains that traveled the globe in opposite directions at such speed that they respectively lost and gained a day, therefore thinking that Sunday was the day before or the day after and fulfilling the uncle’s condition (Poe, 225-232). Yet again, the statement seems contradictory but an analysis on the matter shows in fact in it is a veridical paradox; Poe constructs the story around the fact that the referential for the day was not specified.

Finally, there are paradoxes that can be both, true or false. Grelling’s paradox falls into this category: let us divide all adjectives in two sets, the autological and heterological ones. The first ones are those that describes themselves, for instance short is a short word. Heterological are those that does not describes themselves: long is a short word. The paradox arises when we try to classify the adjective heterological: if its autological then it describes itself, it is then heterological which contradicts the statement. On the other hand, if it is heterological, then it cannot be in its own category, it must then be autological, but we already showed that it can’t be (Vidal-Rosset, 26). We will see later that another paradox by Bertrand Russell is similar to this case.

Trying to classify paradoxes into one of these categories is the first way to turn paradoxes into useful abstract objects enlarging the scope of knowledge. The categorisation implies gathering enough knowledge and understanding of the paradigm in which the paradox is stated to be able to classify it into the proper box, but this procedure might redefined the paradigm or lead to new theories. Although, it is quite natural to take some assumption as true -as axiomatic- in part of building knowledge, but the interrogation about these keystones of knowledge really comes unavoidable when paradoxes are found. This thought is well express by many great thinkers in sciences and philosophy. As Niels Bohr said when working on theoretical model for the atom ‘’How wonderful that we have met with a paradox. Now we have some hope of making progress.’’ (Moore, 196). The idea of meditating on the meaning of paradoxes have even brought to an almost mystical perspective in Kierkegaard’s’ writings: ‘’ The paradox’s really the pathos of intellectual life and just as only great souls are exposed to passions it is only the great thinker who is exposed to what I call paradoxes which are nothing else than grandiose thoughts in embryo’’ (Slaatté, 64). In these cases, paradoxes technically mean self-contradiction, but it is seen as well as a possible extension of knowledge, and wider possible range for the concerned paradigm. An interesting and humoristic quote sometimes attributed to Mark Twain expresses this state of mind about knowledge: ‘’All generalisations are false, even this one’’.

Les fondements de l’écriture procédurale : images, espaces et algorithmie musicale de l’algèbre aux fractals. (Chapitre 3 et Médiagraphie)

Différentes avenues

3.1 Défis et applications possibles de l’écriture procédurale

Nous allons nous tourner à présent vers la pratique d’écriture et de certains défis à venir. Le but est en fait de voir quels sont quelques chantiers en productions, de voir d’où découlent certaines problématiques, de trouver quelques projets excitants qui n’ont peut-être pas encore trouvé d’applications concrètes. De sorte, nous proposons autant qu’analysons des œuvres et groupes d’œuvres.

Nous traitons principalement de deux problématiques, toujours en écartant autant que possible l’aspect de l’harmonisation et de la fréquence des notes[1]. La première est la conception de l’écriture procédurale autour d’un objet relativement contemporain qu’est le fractal. En effet, si l’interprétation visuelle des structures fractales s’est avérée une source inépuisable de recherches et créations, sa transposition en musique attend encore  une œuvre phare qui permettrait de valoriser davantage cette pratique. En second lieu, nous abordons la performativité d’une œuvre structurale. Nous présenterons brièvement quelques pièces qui travaillent sur cette limite encore nébuleuse de l’ensemble des possibles de l’écriture procédurale.

3.2 Petit survol de différentes compositions fractales

En ce qui concerne la première problématique, c’est à la fois dans la forme absolue de la nature structurale des fractals, ou voir même des natures structurales, et dans la procédure de sa mise en forme musicale qui pose problème. Afin de comprendre l’écriture de fractals comme une écriture procédurale, nous devons retourner à la source de la notion de fractale et une première difficulté qui s’impose d’elle-même est l’absence d’une définition complète des fractals.

Dans son premier ouvrage sur le sujet, le mathématicien français refusait même de donner une définition exacte en sachant que celle-ci ne ferait qu’exclure inutilement un certain nombre d’exemples. Son travail s’inscrit malgré tout dans une ligné de publication sur le sujet qui remonte principalement à la géométrie et à l’analyse. Pour donner un exemple concret, il faut comprendre un vieux débat qu’est celui de la notion de continuité et de dérivabilité. En termes simples, la continuité d’une fonction est sa qualité d’être traçable à la main sans lever le crayon[2]. La notion de dérivé est celle du taux de variation d’une droite tangente en un point d’une courbe. Pendant longtemps, on a cru que la continuité impliquait la dérivabilité et cela changea lorsque quelques contre-exemples ont commencé à voir le jour. Par exemple, Bolzano et Weirstrass ont défini des fonctions qui sont par leur nature très complexes, continues mais nul part dérivables. Cette prouesse s’obtient en fait avec la somme infinie de fonctions sinusoïdales. Un autre exemple de courbe possédant cette même structure est la courbe de Von Koch. Celle s’obtient aisément à partir d’itérations géométriques. À partir d’un segment de droite, on remplace le tiers milieu par les deux côtés d’un triangle équilatéral de côté égal à la longueur de ce segment. Pour tous les segments restants, on réitère le processus ainsi à l’infini. Le tout devient abstrait lorsque l’on définit la courbe de von Koch comme le résultat de ce processus, lorsqu’appliqué une infinité de fois. Donc, autant l’objet est clairement défini qu’il n’est en aucun cas entièrement réalisable puisque cela prendrait un temps infini. Évidemment, à partir d’un certain nombre d’itérations, la nuance de se voit plus à l’œil, nous laissant avec une forme infiniment brisée comme l’est la fonction de Weirstrass. (Figure 16)

Figure 1

Figure 16: La function de Weirstrass. Source: Wikipedia

Cet exemple fort simple de la structure de la courbe de Von Koch mène vers deux pierres d’achoppement auxquelles le compositeur se bute dans le transfert d’une structure de la sorte en musique. La première est évidemment ce point de jonction entre processus infini et structure, la seconde ici ne s’obtient, théoriquement, qu’après un temps infini. Donc, tout comme les représentations de la courbe de Von Koch ne peuvent être qu’approximatives, il en va de même pour son transfert en musique. Ce transfert, aussi élaboré qu’il puisse être, demeurera toujours incomplet par nature. Le seul point de conciliation est que son transfert en musique possède son seuil perceptible, que ce soit pour un transfert rythmique ou fréquentiel.

Par la suite, un choix doit être fait concernant cette structure à savoir quelle propriété doit être transférée. Il y a d’une part l’aspect géométrique, et d’autre part son aspect infiniment brisé. Pour le premier aspect, un travail similaire à celui de Tom Johnson pourrait être effectué en assignant à chaque hauteur de la courbe une hauteur de note. Il est évident que dans ce choix, un nombre limité d’itérations est possible avant d’atteindre le seuil de différentiation. Pour le second aspect, il faut discuter de l’invariance d’échelle. Afin de représenter ce principe, nous pouvons nous observer un exemple dans la composition La Vie est si courte de Johnson. Dans ce cas, la mélodie de la portée du bas est la même que celle du haut à une différence d’échelle près. La mélodie du haut est en fait trois fois plus rapide que celle du bas.

Tom Johnson - La vie est si courte

Figure 17: La vie est si courte de Tom Johnson. Source: Pour la Science

Les fractales possèdent justement une telle invariance d’échelle, ce qui implique qu’elles sont soit en chaque partie identique à leur tout -à la limite, chaque segment de la courbe de Von Koch est en fait identique à son ensemble- soit structurellement identique en possédant, par exemple, le même degré de brisure. Cette idée implique donc également une brisure infinie de la mesure rythmique.

Idéalement, cette invariance existe autant vers l’infiniment petit que vers l’infiniment grand. Il y a la possibilité de représenter instinctivement l’infiniment petit, par exemple comme le fait Julio Estrada dans sa composition ishini’ioni (Sauer, p. 69) (Figure 18) ou par un modèle structural basée sur la décomposition infinie de l’ensemble de Cantor comme proposé dans l’ouvrage de Pareyon. (Figure 17) (Pareyon, p. 73)

Cantor Set notes

Figure 18: Itérations vers l’infiniment petit basée sur l’ensemble de Cantor. Source: Pareyon

Le choix de l’infiniment grand est plus accessible que son inverse puisque son seuil de différentiation est plus souple. Nous pourrions imaginer une structure rythmique incommensurable basée sur le l’ensemble de Cantor. En partant d’une mesure qui vaut la longueur du segment initial, nous pourrions ‘’déplier’’ l’ensemble de Cantor et mimer sa structure vers l’infiniment grand plutôt que dans sa version de poudre infinie. De la sorte, la seconde itération serait constituée de trois mesures du segment initial, avec la mesure du milieu silencieuse, la prochaine itération contiendrait neuf mesures et ainsi de suite. Afin de mettre l’emphase sur une œuvre conceptuelle de la sorte, il serait possible de faire un programme qui permet de générer automatiquement une infinité d’itérations musicales.

score

Figure 19: Extrait de ishini’ioni par Julio Estrada (1959)

Parfois, il peut être utile de transformer une structure dans une forme secondaire avant que d’en chercher un transfert adéquat qui peut même laisser entrevoir certaines qualités inhérentes. Pour en revenir à la courbe de Von Koch par exemple, nous pouvons nous tourner vers une version plus abstraite de sa structure. Jun Ma et Judy Holdener ont démontré que cette courbe était reliée à la suite de Thue-Morse. Cette suite se construit comme tel : nous débutons avec 01 et nous recopions, par une suite d’itérations, l’inverse de la suite de nombres en inversant les zéros et les uns. Les 4 premières itérations sont : 01, 0110, 01101001, 0110100110010110. Cette suite a été construire originalement pour n’avoir aucune répétition plus longue que deux valeurs[3]. Cette impossibilité découle directement de la nature même de son processus itératif. Or, ce que Jun Ma et Judy Holdener ont démontré est qu’en assignant des valeurs directionnelles aux valeurs de 0 et 1, il est possible d’obtenir une courbe qui elle aussi tend précisément vers la courbe de Von Koch. Donc, en utilisant la suite de Thue-Morse comme partition, il y a l’utilisation sous-jacente de différentes propriétés structures. Il y a le transfert de sa propriété fractale, et transfert de son absence de répétition triadique. Cela peut mener à différentes expérimentations[4].

Dans l’un des rares textes à discuter de la musique fractale[5], le compositeur Robert Sherlaw Johnson mentionne qu’il est resté fort peu satisfait des compositions fractales qu’il lui a été donné d’entendre (p. 163). La difficulté à rendre efficace le transfert du visuel vers l’auditif vient peut-être d’une différence fondamentale dans la lecture de telles œuvres. En discutant de la forme visuelle d’un fractal, il spécifie que «The longer the generation takes place, the more interesting  and complex the pattern becomes. Music is not perceived in this way. ..()…in the case of music the whole is not perceived simultaneously and only localized patterns make sense» (Sherlaw Johnson, p. 166). C’est d’ailleurs le même principe qui limite l’effectivité de l’autoréférentialité en musique et qui pose la question de la valeur de la perception du rythme dans les canons de Vuza.

Le compositeur Gustavo Días Perez offre une belle interprétation des ensembles de Julia et de Mandelbrot comme partition musicale. Par un choix judicieux de sons de synthèse, le compositeur arrive à recréer le halo mystérieux qui règne autour de ces objets étranges. Malgré cet accomplissement esthétique, les structures ne sont pas directement sensibles dans ses compositions. Les liens entre la spacialisation d’objets fractals complexes, tels les ensembles de Julia et de Mandelbrot[6], et une musique qui contiendrait une structure équivalente restent encore difficile à définir. Ce fait peut venir de la différence entre ces ensembles qui ont été davantage découverts sur le plan complexe contrairement aux structures fractales construites comme dans le cas de la courbe de von Koch ou la suite de Thue-Morse.

Figure 20: Lecture de l’ensemble de Mandelbrot par Gustavo Días Perez . Source: youtube

Cette distinction mène au problème de la compréhension du chaos. Par définition, le chaos tente d’exclure toute règle structurale et échapper à toute forme de formalisation. La géométrie fractale sert souvent de modèle pour comprendre la nature extrêmement complexe de la géométrie de la nature. Le cas des ensembles de Julia et de Mandelbrot sont plus proche de cette nature chaotique des fractales puisque qu’ils semblent hors contrôle, leur complexité va au-delà de ce que nous aurions pu simplement imaginer et, de ce fait, construire. La retranscription musicale d’une telle complexité est par ce fait même difficile et le mieux que nous pourrons peut-être espérer serait de définir une sorte de partition musicale qui contiendrait naturellement des objets d’une complexité inimaginable, tout comme il en a été le cas avec le plan des nombres complexes. Évidemment, la généralisation des ensembles de Julia et de Mandelbrot à un nombre supérieur de dimensions laissent entrevoir la possibilité de chercher pour de plus grands défis d’écritures procédurales

3.3 La structure de la performabilité

La performabilité peut également avoir une forme, souligné dans la procédure des actes nécessaires à la présentation de l’œuvre.

Un exemple d’écriture procédurale liée à la performabilité se retrouve en la pièce Failing de Tom Johnson[7]. Des défis reliés à la structure de cette pièce, et par conséquent à la migration de celle-ci vers d’autres œuvres, tient en la forme tripartite de sa définition. Dans cette pièce une contrebassiste doit lire un texte (et en improviser des bouts) et jouer des extraits de mélodies, soit par intermittences, soit simultanément. Évidemment, cette pièce, comme le mentionne le même texte, est difficile à interpréter. Le texte souligne encore que la lecture du texte ne doit pas interférer dans la prestation strictement musicale de la contrebasse. Le texte mentionne que cette tâche est somme toute impossible, de là le titre de la pièce. Or, comme le mentionne encore une fois le texte, de ne pas réussir la pièce est également est aussi la réussir, ce qui en fait une structure paradoxale dont la valeur de la prestation soit se trouve simultanément dans une double position binaire était réussite et non-réussite à la fois, elle glisse constamment entre les deux. Nous précisions ces deux cas puisque cela dépend de la position du spectateur, soit celui-ci tend à attribuer une valeur binaire quant à la réussite de cette prestation, soit il admet que la réussite d’une pièce admet des zones grises. Dans ce cas, cela revient à décider si la lecture du texte mine ou non la prestation musicale de la contrebasse.

L’écriture de cette pièce est triple. En effet, pour atteindre cette tierce valeur paradoxale, les deux statuts de réussite et de non-réussite de l’interprétation se doivent d’être préalablement confirmés afin de positionner le tout comme œuvre paradoxale. Donc, cette triple valeur doit se transpose-t-elle aisément. Il va de soi qu’un changement d’instrument (autre qu’un instrument à vent qui nécessite l’usage de la bouche) ne modifie pas réellement cette structure.

Notons que la portion autoréférentielle de l’œuvre, qui apparaît dans son titre et dans son texte, n’est pas en soi une modalité difficile à transposer vers une autre œuvre. Il suffit d’ajouter du texte une part de méta discours dans l’œuvre. Le défi devient intéressant lorsque nous voulons transférer cet acte d’écriture, et sa forme procédurale particulière, vers une œuvre audiovisuelle. La pièce de Johnson se base en fait sur une limitation kinesthésique qui positionne le musicien dans une position précaire, entre réussite et échec. La difficulté vient particulièrement de l’aspect rythmé et temporalisé de la parole et de la musique qui, de part et d’autre, entrent en conflit. Un premier transfert serait de conserver cette limitation kinesthésique. Une forme transposée qui pourrait induire le même type de difficulté serait la lecture d’une partition graphique filmique. Par filmique nous entendons que son ensemble n’est pas visuellement accessible d’emblée, et que celle-ci défile devant le musicien. Cette partition, avec sa liste d’instruction, pourrait être accessible instantanément au public, faisant ainsi de l’œuvre un travail audiovisuel pour le spectateur. Une forme participative, invitant à réciter un texte par cœur tout en interprétant la partition musicale filmique en ferait une œuvre vidéo ludique.

Un point important de l’interprétabilité de Failing réside en la mise en abîme du processus de création, technique présente également dans sa composition Narayana’s Cow (1989). Ces compositions débutent par l’explication autoréférentielle de l’œuvre afin d’en préciser les rouages[8]. Cette méthode permet à la fois d’inclure la présentation du processus de création dans l’œuvre même et d’avoir une transition graduelle du positionnement spectatoriel entre analyse et appréciation. Ce double positionnement peut être conservé pour l’entièreté de la durée de la pièce, comme dans Failing ou une autre œuvre de Johnson, Music and Questions (1988).

Ces compositions démontre qu’il est possible de jouer énormément sur deux axes binaires[9], celui du vrai et du faux, et celui de l’intérieur et l’extérieure de l’œuvre, équivalent à l’intra et l’extra diégétique dans le cas d’une narration. Ces structures peuvent évidemment se complexifier lorsque plusieurs paliers ontologiques apparaissent. Si de telles structures sont sujettes à diverses explorations en théâtre en cinéma et dans les arts visuels en général -comme le témoigne le mockumentaire ou des films comme Exit through the Gift Shop (Banksy, 2010)- leur présence reste encore timide dans le domaine de la musique.

Conclusion 

Nous avons défini l’écriture procédurale afin de pouvoir extraire une partie du pouvoir d’abstraction du processus d’écriture. Cette écriture pose des règles et des structures qui s’appliquent naturellement dans une variété de contextes scientifiques et artistiques. Un lieu de rencontre particulièrement fertile apparaît lorsque les arts visuels, la musique et la théorie des groupes s’unissent. Nous avons étudié trois exemples importants. Le premier met en lien le pavage régulier de surfaces comme la bande, le plan et le ruban de Möbius, l’avènement géométrique de la théorie des groupes et la géométrie de la partition musicale. Le second part de l’algèbre et les canons rythmiques et reprend la notion de pavage appliquée à l’espace à n dimensions. Dans cet espace abstrait se cache la solution à un autre problème de canons rythmiques, celui des canons de Vuza. Finalement, nous avons vu que la notion de fractal et des espaces qu’ils engendrent mènent vers de nombreux défis compositionnels dont plusieurs restent encore à résoudre. Il semble que la notion d’espace sert à la fois comme point unificateur de ces perspectives distinctes que comme fil conducteur de l’évolution d’une même problématique, celle de l’unification des sens. Comme mentionné en début de texte, le travail effectué directement dans l’abstrait permet d’inversé le processus d’unifications de pratiques distinctes en une création simultanée d’un corpus multidisciplinaire.

Afin de comprendre l’étendue réelle de l’écriture procédurale, il faudrait déjà couvrir plusieurs sujets connexes. En premier lieu il faudrait inclure l’ensemble des savoirs propres à l’analyse harmonique qui viennent compléter nombres de résultats et recherches présentés dans ce texte. Ensuite, il faudrait inclure une analyse détaillée des systèmes stochastiques qui servent de matière de composition pour plusieurs compositeurs et artistes visuels. L’étude des systèmes formels comme les systèmes Lindenmayer ainsi qu’une compréhension plus large de l’algèbre et la théorie des groupes permettraient d’inclure les théorèmes et preuves qui servent l’écriture procédurale. Finalement, contrairement à ce texte, il faudrait également inclure l’étude un large éventail d’écritures plus souples qui permettent des transpositions axés davantage sur l’affect que sur la structure inhérente.

Évidemment, comme le démontre la prestation de Failing de Tom Johnson, nous ne pouvons pas exclure totalement le rapport de cette écriture abstraite avec le réel et ce, que ce soit par l’analyse de la performabilité d’une œuvre, où même de sa prestance possible dans le monde, ou par sa réception par le ‘’lecteur’’. Une entreprise intéressante mais lourde de complications tenterait de prévoir la portée ‘’synesthésique’’ d’une écriture procédurale. Cette tâche ardue devrait alors inclure deux paliers importants : celui de prévoir dans quels médius ou combinaisons de médiums une œuvre procédurale pourra prendre une forme intéressante et celui de comprendre les différents schèmes cognitifs qui seront impliqué dans sa réception afin de favorisé la sensation immédiate des réseaux de concepts que ces œuvres impliquent.

[1] Pour un exemple d’exploration dans cette direction, nous référons le lecteur à l’article fort intéressant de Carlton Gamer et Robin Wilson sur les gammes et la structures de plans projectifs référé dans la bibliographie.

[2] En termes plus mathématiques, cela revient au respect de trois conditions qu’en chaque point la fonction existe, que sa limite existe, et que la valeur de la fonction est celle de la limite.

[3] Il serait d’ailleurs intéressant de vérifier si la suite de Thue-Morse contient toute les instances possibles de rythmes aksak.

[4] Par exemple, pour une composition de Steve Gilliland basée sur la suite de Thue-Morse : https://www.youtube.com/watch?v=6VZq7EurckI

[5] Pour un ouvrage qui contient plusieurs exemples fractals en vertu de leur auto-similarité, voir le livre de Gabriel Pareyon.

[6] Ainsi que leurs formalisations dans un plus grand nombre de dimensions.

[7] https://www.youtube.com/watch?v=9P8C6-XqaNs

[8] Rouges qu’il est possible de trouver dans un article de J.-P. Allouche et Tom Johnson, «Nayrana’s Cows and Delayed Morphisms» paru dans les Cahiers du GREYC des Troisièmes Journées d’Informatique Musicale.

[9] Ce qui donne une structure similaire au groupe de Klein, tout comme le carré de Greimas.

Mediagraphie:

Allouche, Gabrielle, Jean-Paul Allouche et Jeffrey Shallit. 2006. « Kolam indiens, dessins sur le sable aux îles Vanatu, courbe de Sierpinski et morphismes de monoïde ». En Ligne : Annales de L’Institut Fourier, Tome 56, n°7, p. 2115-2130. Consulté le 07/02/12. http://aif.cedram.org/item?id=AIF_2006_56_7_2115_0

Amiot, Emmanuel. 2004. « Why rhythmic canons are interesting? ». Perspectives in Mathematical Music Theory, (G. Mazzola, E. Puebla, and T. Noll eds), EpOs, University of Osnabrück, p. 1-19.

Amiot, Emmanuel. 2005. «À propos des canons rythmiques». Gazette des mathématiciens, 106, p. 42-67.

Amiot, Emmanuel. 2011. «Structures, Algorithms and Algebraic Tools for Rhythmic Canons».Dans Perspectives of New Music, Vol. 49, n˚2, p. 93-143.

Andreatta, Moreno et Dan T. Vuza. 2001. « On Some Properties of Periodic Sequences in Anatol Vieru’s Modal Theory ». Tatra Montains Mathematical Publications, 23, p. 1-15.

Andreatta, Moreno and Carlos Agon. 2009. « Special Issue: Tiling Problems in Music. Guest Editors’ Foreword». Journal of Mathematics and Music: Mathematical and Computtional Approaches to Music Theory, Analysis, Composition and Performance, Vol.2, n˚2, p. 63-70.

Andreatta, Moreno. 2011. « Constructing and Formalizing Tiling Rhythmic Canons: A Historical Survey of a ‘’Mathemusical’’ Problem ». Dans Perspectives of New Music, Vol. 49, n˚2, p. 33-65.

Andreatta, Moreno. 2014. «Une introduction musicologique à la recherche «mathémusicale»: aspects théoriques et enjeux épistémologiques». Dans Circuit : La recherche musicale : Aux croisements de l’art et de la science. Vol. 24, n˚2. Montréal : Les Presses de l’Université de Montréal, p. 51-66.

Armstrong, M.A., 1988. Groups and Symmetry. New York : Springer-Verlag.

Audin, Michèle. 2011. Fatou, Julia, Montel : The Great Prize of Mathematical Sciences of 1918, and Beyond. New York: Springer, Lecture Notes in Mathematics 2014, History of Mathematics Subseries.

Babbitt, Milton. 1962. «Twelve-Tone Rhythmic Structure and the Electric Medium». Perspective of New Music 1, nu. 1, (Fall-Winter), p. 49-79.

Bardis, Panos. «The Moebius Band : History, Synopsis, and Algebraic Generalizations». Portugaliae Mathematica, Vol.33, No.2, 1974, p.117-132.

Bernhard, Thomas. 1993 (1986). Le naufragé. Paris : Gallimard.

Boll, Marcel. Histoire des mathématiques. Paris: Presse Universitaire de France, 1968.

Borges, Jorge Luis. (1956) 1983. Fictions. Traduit de l’espagnol par P. Verdevoye, Ibarra Caillois et Roger Caillois. Paris : Gallimard.

Busser, Élizabeth. 2010 (2005). «Symétrie et composition». Dans Tangente Hors-Série n˚11, Maths et Musique : Des destinées parallèles. Paris : Éditions Poles, p. 60-65.

Busser, É. et D. Souder. 2010 (2005). «Composition automatique et ordinateur». Dans Tangente Hors-Série n˚11, Maths et Musique : Des destinées parallèles. Paris : Éditions Poles, p. 66-69.

Carphin, Philippe et Christianne Rousseau. « Finir une gravure d’Escher». Acromath, vol.4, été-automne 2009, p.20-25.

Conway, John, Daniel H. Huson. «The Orbifold Notation for Two-Dimensional Groups». Structural Chemistry, Vol, 13, nu. 3, 2002, p. 247-257.

Coxeter, H.S.M. 1968. «Music and Mathematics». Dans The Mathematics Teacher, Vol. 61, n° 3, p. 312-320.

Cucker, Felipe. 2013. Manifold Mirrors: The Crossing of the Arts and Mathematics. New York: Cambridge University Press.

Curtis, David. 1971. Experimental Cinema. New York : Delta Book.

Cytowic, Richard E. 1989. Synesthesia: A Union of the Senses. New York: Springer.

Dahan-Dalmedico, A. et J. Pfeiffer. Une Histoire des Mathématiques : routes et dédales. Paris : Éditions du Seuil, 1984.

Delahaye, Jean-Paul. 2014. ‘’Équations résolubles ou non?’’. Pour La Science, n˚ 440, p.74-79.

Delahaye, Jean-Paul. 2012. «La suite de Stern-Bricot, sœur de Fibonacci». Pour la Science, n˚420 (octobre), p.86-91.

Delahaye, Jean-Paul. 2006. « Des mots magiques infinis ». Pour la Science, n˚347 (Septembre), p.90-95.

Delahaye, Jean-Paul. 2004. « La musique mathématique de Tom Johnson». Pour la Science, n˚325 (Novembre), p. 88-93.

DeLio, Thomas. 2002. «A Question of Order; Cage, Wolpe, and Pluralism». Dans The New York Schools of Music and Visual Arts, édité par Steven Johnson, New York : Routledge, p. 135-158.

Diaz-Jerez, Gustavo. 2009. «La música fractal». En ligne: http://www.fractalmusicpress.net/publications.html. Musikene, juin, nu. 4.

Dolven, Jeff. 2009. « Roll Playing». En ligne dans Cabinet, Issue 34 Testing Summer. https://www.youtube.com/watch?v=DOACcFkmwt8

Doxiadis, Apostolos. 2005. « Euclid’s Poetics : An Examination of the similarity between narrative and proof ». Dans Mathematics and Culture II, édité par Michele Emmer, Cambridge : MIT Press, p. 175-182.

Dummit, David S. et Richard M. Foote. Abstract Algebra. 2nd Ed. New Jersey: Prentice Hall, 1991.

Feaster, Patrick. 2012. Pictures of Sound: One Thousand Year of Educed Audio: 980- 1980. Atlanta: Dust-to-Digital.

Emmer, Michele. 2005. « Mathematics and Raymond Quenaud ». Dans Mathematics and Culture II, édité par Michele Emmer, Cambridge : MIT Press, p. 195.200.

Escher, M. C. « L’approche de l’infini ». Dans Le Monde de M.C. Escher : L’œuvre de M.C.Escher, commenté par J.L. Locher, H.A. Broos, M.C.Escher, G.W. Locher et H.S.M.Coxeter. Édité par J.L. Locher.Paris : Éditions du Chêne, 1972.

Field, J.V. 2003. «Musical cosmology: Kepler and his readers». Dans Music and Mathematics: From Pythagoras to Fractals. London: Oxford University Press, p. 29-44.

Francis, George K. and Jeffrey R. Weeks. «Conway’s ZIP Proof». American Mathematical Society, 106 (1999), 393-399.

Gamelin, Theodore W.. Complex Analysis. New York: Springer-Verlag New York, 2001.

Gamer, Carlton et Robin Wilson. 2003. «Microtones and projective planes». Dans Music and Mathematics: From Pythagoras to Fractals. London: Oxford University Press, p. 149-161,

Grimaldi, Ralph P.. Discrete and combinatorial mathematics; an applied introduction, 5th Ed.    Boston: Pearson Addison Wesley, 2004.

Guillen, Michael. Invitation aux Mathématiques : Des Ponts Vers l’Infini. Traduit de l’anglais par            Gilles Minot.Paris : Éditions Albin Michel, 1995.

Hart, Vi. 2012. « Mathematics and Music Boxes». Dans Mathematics and Modern Arts édité par Claude Bruter. Berlin : Springer-Verlag, p. 79-84.

Hodges, W. 2003. « The Geometry of Music ». Dans Music and Mathematics: From Pythagoras to Fractals. London: Oxford University Press, p. 91-111.

Hofstadter, Douglas. 1985. Gödel Escher Bach: Les Brins d’une Guirlande Éternelle. Paris: Interéditions.

Hofstadter, Douglas. 1981. «On Self-Referential Sentences». Dans Metamagical Themas: Questing fort the Essence of Mind and Pattern. New York: Basic Books, 1985, p. 5-24.

Hofstadter, Douglas.1981. «Self-Referential Sentences; A Follow-Up». Dans Metamagical Themas: Questing fort the Essence of Mind and Pattern. New York: Basic Books, 1985, p. 25-48.

Holzaepfel, John. 2002. «Painting by Numbers; The Intersections of Morton Feldman and David Tudor». Dans The New York Schools of Music and Visual Arts, édité par Steven Johnson, New York : Routledge, p. 159-172.

Johnson, Robert Sherlaw. 2003. «Composing with Fractals». Dans Music and Mathematics: From Pythagoras to Fractals. Édité par John Fauvel, Raymond Flood et Robin Wilson. Oxford: Oxford University Press, p. 163-172.

Johnson, Steven, ed. 2002. The New York Schools of Music and Visual Arts. New York: Routledge.

Johnson, Steven. 2002. «Jasper Johns and Morton Feldman: What Ptterns?». Dans The New York Schools of Music and Visual Arts, édité par Steven Johnson, New York : Routledge, p. 217-248.

Johnson, Tom. 2011. « Tiling in my music». Dans Perspectives of New Music, Vol. 49, n˚2, p. 9-22.

Johnson, Tom. 2003. « Some Observations on Tiling Problems ». Lecture MaMuX Meeting, IRCAM, January 25, 2003. (trouver liens)

Johnson, Tom. 2006. «Self-Similar Structures in my Music: an Inventory ». Lecture at MaMux seminar IRCAM, Paris, Oct. 14, 2006. En ligne: http://repmus.ircam.fr/_media/mamux/saisons/saison06-2006-2007/johnson-2006-10-14.pdf

Kandinsky, Wassily. Point-Ligne-Plan: contribution à l’analyse des éléments picturaux. Paris : Denoël/Gonthier,1970.

Klee, Paul. Théorie de l’Art Moderne. Édition et traduction par Pierre-Henri Gonthier. Genève :             Éditions Gonthier, 1985.

Koch, Helge von. 1906. « Une méthode géométrique élémentaire pour l’étude de certaines questions de la théorie des courbes planes ». En Ligne : Acta Mathematica. Vol. 30, No. 1, p. 145-175. Consulté via Metapress Springer le 24/01/12. DOI 10.1007/BF02418570

Lang, Serge. Undergraduate algebra. 3th Ed.New York: Springer, 2005.

Lauwerier, Hans. 1991. Fractals: Endlessy Repeated Geometrical Figures. Princeton: Princeton University Press.

Lemoir-Gordon, Nigel, Will Rood and Ralph Edney. Introducing Fractal Geometry. Edited by    Richard Appignanesi. Cambridge: Icon Books Ltd., 2000.

Levin, Thomas Y. 2004. «Des sons venus de nulle part : Rudolph Pfenninger et l’archéologie du son synthétique». Dans Sons et Lumières. Catalogue d’exposition du Centre Pompidou dirigé par Sophie Duplaix et Marcella Lista. Paris : Éditions du Centre Pompidou, p. 51-60.

Lévy, Fabien. 2011. « Three Uses of Vuza Canons». Dans Perspectives of New Music, Vol. 49, n˚2, p. 23-32.

Lista, Marcella. 2004. «Empreintes sonores et métaphores tactiles : Optophonétique, film et vidéo». Dans Sons et Lumières. Catalogue d’exposition du Centre Pompidou dirigé par Sophie Duplaix et Marcella Lista. Paris : Éditions du Centre Pompidou, p. 63-76.

Locher, J.L. Éditeur.1972 (1971). Le Monde de M.C. Escher: L’Oeuvre de M.C. Escher Commenté              par J.L. Locher, C.H.A. Broos, M.C. Escher. G.W. Locher, H.S.M. Coxeter. Traduit de     l’hollandais par Jeanne A. Renault. Paris: Éditions du Chêne.

Loi, Maurice éditeur. Mathématiques et Art. Paris : Éditions Hermann, 1995.

Ma, Jun and Judy Holdener. 2005. « When Thue-Morse Meets Koch ». Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society, vol. 13. n°3, p. 191-206.

Malderieux, Stéphane. 2007. «Préface». Dans Le pragmatisme : un nouveau nom pour n’anciennes manières de penser de William James. Paris : Éditions Flammarion, p. 7-69.

Mandelbrot, Benoît. Les Objets Fractals : Formes Hasard et Dimension, 4th Ed. Paris : Flammarion, 1995.

Mathews, Max. 1963. The Digital Computer as a Musical Instrument. Science, New Series, Vol. 142, No. 3592 (Novembre), p. 553-557.

Mathieu, Marc-Antoine. Julius Coretin Acquefaques, prisonnier des rêves : L’origine. Paris : Éditions Delcourt, 1991.

Mathieu, Marc-Antoine. Julius Coretin Acquefaques, prisonnier des rêves : La qu…. Paris : Éditions Delcourt, 1991..

Mathieu, Marc-Antoine. Julius Coretin Acquefaques, prisonnier des rêves : Le processus. Paris : Éditions Delcourt, 1993.

Mathieu, Marc-Antoine. Julius Coretin Acquefaques, prisonnier des rêves : Le début de la fin. Paris : Éditions Delcourt, 1995.

Mathieu, Marc-Antoine. Julius Coretin Acquefaques, prisonnier des rêves : La 2,333e dimension. Paris : Éditions Delcourt, 2004.

Mathieu, Marc-Antoine. Julius Coretin Acquefaques, prisonnier des rêves : Le décalage Paris : Éditions Delcourt, 2013.

Mattis, Olivia. 2002. «The Physical and the Abstract; Varèse and the New York School». Dans The New York Schools of Music and Visual Arts, édité par Steven Johnson, New York : Routledge, p. 57-74.

McClain, Ernest G.. 1976. The Myth of Invariance : The Origin of the Gods, Mathematics and Music From the Ṛg Veda to Plato.York Beach: Nicolas-Hays, Inc.

McHale, Brian. Postmodernist Fiction. New York: Methuen,1987.

Mendelson, Elliott. 1997. Introduction to Mathematical Logic. 4th ed. Chapman & Hall.

Nagel, Ernest, James R. Newman, Kurt Gödel, Jean-Yves Girard. Le théorème de Gödel. Traduis de l’anglais et de l’allemand par Jean-Baptiste Scherrer. Paris: Éditions du Seuil, 1989.

Nicholls, David. 2002. «Getting Rid of the Glue; The Music of the New York School». Dans The New York Schools of Music and Visual Arts, édité par Steven Johnson, New York : Routledge, p. 17-56.

Ouellet, Gilles. 2002. Algèbre linéaire : Vecteurs et géométrie. Québec : La Griffe d’aigle.

Palacio-Quintin, Cléo. 2014. « Ces lieux où ingénieurs et musiciens convergent». Dans Circuit : La recherche musicale : Aux croisements de l’art et de la science. Vol. 24, n˚2. Montréal : Les Presses de l’Université de Montréal, p. 9-29.

Pareyon, Gabriel. 2011. On Musical Self-Similarity: Intersemiosis as Synechdoche and Analogy. Imatra: The International Semiotics Institute.

Peck, Robert. 2010. « Imaginary Transformations ». Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance, Vol. 4, No. 3, p.157-171.

Pickover, Clifford. The Möbius Strip : Dr. August Möbius’s Marvelous Band in Mathematics, Games, Litterature, Art, Technology, and Cosmology. New York: Thunder’s Mouth, 2006.

Queneau, Anne-Isabelle. 2002. Album Raymond Queneaud, Coll. Bibliothèque de la Pleiade,     Paris : Editions Gallimard.

Quenau, Raymond. 1965. Bâtons, chiffres et lettres. Paris: Gallimard.

Rothstein, Edward. 1995. Emblems of Mind: The Inner Life of Music and Mathematics. Chicago: University Press of Chicago.

Rousseau, Pascal. 2004. «Concordances : Synesthésie et conscience cosmique dans la Color Music ». Dans Sons et Lumières. Catalogue d’exposition du Centre Pompidou dirigé par Sophie Duplaix et Marcella Lista. Paris : Éditions du Centre Pompidou, p. 29-38.

Sagiv, Noam. 2005. «Synesthesia in Perspective». Dans Synesthesia : Perspectives from Cognitive Neuroscience, édité par Lynn C. Robertson et Noam Sagiv. Oxford : Oxford University Press.

Sauer, Theresa, Ed. 2009. Notations 21. New York: Mark Batty Publichers.

Schattschneider, Doris. 1992 ( 1990). Visions de la Symétrie: Les Cahiers, les Dessins Périodiques             et les Oeuvres Corrélatives de M.C. Escher. Traduit de l’américain par Marie Bouazzi. Paris : Éditions du Seuil.

Schoenberg, Arnold. 1984 (1975). Style and Idea. Berkeley: University of California Press.

Shallit, Jeffrey. 2005. «The Mathematics of Per Nørgård’s Rhythmic Infinity System». Fibonnaci Quaterly, Vol. 43, p.262-268.

Slonimsky, Nicolas. 1971. «Moebius Strip Tease». Dans Source, Vol. 5, nu. 1, p. 64-66.

Slonimsky, Nicolas. 2005 (1983). «Géométrie Sonore: Edgar Varèse». Dans Nicolas Slonimsky : Writings on Music : Volume Three Music of the Modern Era, Electra Slonimsky Yourke Ed., New York: Routledge, p. 204-216.

Slonimsky, Nicolas. 2005 (1962). «Colon Nancarrow: Complicated Problem –Drastic Solution». ». Dans Nicolas Slonimsky : Writings on Music : Volume Three Music of the Modern Era, Electra Slonimsky Yourke Ed., New York: Routledge, p. 204-216.

Smit, B. de and H.W. Lenstra Jr. 2003. « Artful Mathematics: The Heritage of M.C. Escher». Notices of the American Mathematical Society, Volume 50, nu 4, p. 446-451.

Souriau, Étienne. 1969. La correspondence des arts. Paris: Flammarion.

Stein, Sherman K., Sándor Szambó. Algebra and Tiling: Homeomorphisms in the Service of Geometry. Washington: The Mathematical Association of America, 1994.

Stewart, Ian and Arthur C. Clarke Éditeurs. The Colours of Infinity: The Beauty and Power of Fractals. Angleterre: Clear Books, 2004.

Stillwell, John. Geometry of Surfaces. New York: Springer-Verlag, 1992.

Stillwell, John. Four Pillars of Geometry. New York: Springer-Verlag, 2010.

Stillwell, John. Mathematics and its History. New York: Springer, 2010.

Tremblay, R. 2003. «Sankyo 20-Note Moebius Strip Plays Inverse Music». En ligne: Mechanical Music Digest, http://www.mmdigest.com/Archives/KWIC/S/sankyo.html

Verdier, Norbert. 2010 (2005). «Pierre Boulez : celui qui a institutionnalisé la musique». Dans Tangente Hors-Série n˚11, Maths et Musique : Des destinées parallèles. Paris : Éditions Poles, p. 70-73.

Verdier, Norbert. 2010 (2005). «Des matrices pour influencer le hasard». Dans Tangente Hors-Série n˚11, Maths et Musique : Des destinées parallèles. Paris : Éditions Poles, p. 74-75.

Verdier, Norbert. 2010 (2005). «Iannis Xenakis». Dans Tangente Hors-Série n˚11, Maths et Musique : Des destinées parallèles. Paris : Éditions Poles, p. 74-81.

Vidal-Rosset, Joseph. Qu’est-ce qu’un paradoxe?. Paris: Librairie Philosophique J.Vrin, 2004.

Von Maur, Karin. 2004. «Bach et l’art de la fugue: modèle structurel musical pour la creation d’un language pictural abstrait». Dans Sons et Lumières. Catalogue d’exposition du Centre Pompidou dirigé par Sophie Duplaix et Marcella Lista. Paris : Éditions du Centre Pompidou, p. 17-27.

Vriezen, Samuel. 2010. « Cows, Chords & Combinations ». Livret pour le disque Tom Johnson: Cows, Chords & Combinations par l’Ensemble Klang. Paris : Édtions 75.

Vuza, Dan Tudor. 1991. « Supplemtary Sets and Regular Complementary Unending Canons (Part One) ». Perspectives of New Music, Vol. 29, No. 2, p. 22-49.

Vuza, Dan Tudor. 1992. « Supplemtary Sets and Regular Complementary Unending Canons (Part Two) ». Perspectives of New Music, Vol. 30, No. 1, p. 184-207

Vuza, Dan Tudor. 1992. « Supplemtary Sets and Regular Complementary Unending Canons (Part Three) ». Perspectives of New Music, Vol. 30, No. 2, p. 102-124.

Vuza, Dan Tudor. 1993. « Supplemtary Sets and Regular Complementary Unending Canons (Part One) ». Perspectives of New Music, Vol. 31, No. 1, p. 270-305.

Vuza, Dan Tudor. 2012. «Vuza Canons At Their 20th Birday –A not Happy Anniversary». ?

Wallace, David Foster. Everything and More: A Compact History of Infinity. New York: Atlas Books, 2010.

 

Site internet:

http://www.carliner-remes.com/jacob/math/project/music.htm