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Afrika + 1
An art-ificial history of
mathematics
An art-ificial history of mathematics
Dirk HuylebrouckDepartment for Architecture Sint-Lucas Brussels BELGIUM
"Show" of African music and mathematics
Afrika + 2
Kunst der Füge
1) Douglas Hofstadter: “Gödel, Escher, Bach, an eternal golden braid”
M.C. Escher's "Reptiles," modified for quilts by Kay Parker in herbook "Contem-porary Quilts".
Kunst der Füge
1) Douglas Hofstadter: “Gödel, Escher, Bach, an eternal golden braid”
Kunst der Füge
1) Douglas Hofstadter: “Gödel, Escher, Bach, an eternal golden braid”
Kunst der Füge
Picture of the UN in NY inspired by Escher
1) Douglas Hofstadter: “Gödel, Escher, Bach, an eternal golden braid”
Afrika + 3
Footprints in ash and pumice dated to 385,000 to 325,000 years ago on the slopes of the Roccamonfina volcano in southern Italy.
2) Dawn
43000 - 82000 years old
Neanderthal flute, Slovenia
2) Dawn
43000 - 82000 years old
Neanderthal flute, Slovenia
Tiny 77000-years-old stone from the Lebombo Mountains in South Africa
2) Dawn
43000 - 82000 years old
Neanderthal flute, Slovenia
France's "Grotte Chauvet" up to 32,000 years ago (horses).
2) Dawn
43000 - 82000 years old
Neanderthal flute, Slovenia
25000 years old “key-pattern” from Mezin, Ukraine
Afrika + 4
3) From Africa over the Middle East to Greece
Structural propertiesmusic inAfrica:1 Hemiola2 Additive rhythm3 Gestalt
RujindiriRwanda
3) From Africa over the Middle East to Greece
Structural propertiesmusic inAfrica:1 Hemiola2 Additive rhythm3 Gestalt
RujindiriRwanda
Proportions?
3) From Africa over the Middle East to Greece
Structural propertiesmusic inAfrica:1 Hemiola2 Additive rhythm3 Gestalt
RujindiriRwanda
3) From Africa over the Middle East to Greece
Structural propertiesmusic inAfrica:1 Hemiola2 Additive rhythm3 Gestalt
RujindiriRwanda
“Ostrakas" in what is now called ‘Iraq’
Afrika + 5
Delphi stone with music notation of the hymn.
doremifasollasido
1:19:8
81:644:33:2
27:16243:128
2:1
4) Greece: Euclide, Parthenon, Cithar
Delphi stone with music notation of the hymn.
doremifasollasido
1:19:8
81:644:33:2
27:16243:128
2:1
4) Greece: Euclide, Parthenon, Cithar
Delphi stone with music notation of the hymn.
doremifasollasido
1:19:8
81:644:33:2
27:16243:128
2:1
4) Greece: Euclide, Parthenon, Cithar
Delphi stone with music notation of the hymn.
doremifasollasido
1:19:8
81:644:33:2
27:16243:128
2:1
4) Greece: Euclide, Parthenon, Cithar
Roger Herz-Fischler 1987; George Markowsky, 1992
Afrika + 6
deffisgabccisd'
1:19:8
32:2781:64
4:33:2
27:1616:9
243:1282:1
5) Fibonacci, cathedrals, kyries
Stacking up tones in the medieval tone "system" or … working without it.
deffisgabccisd'
1:19:8
32:2781:64
4:33:2
27:1616:9
243:1282:1
5) Fibonacci, cathedrals, kyries
Stacking up tones in the medievaltone "system" or … working without it.
deffisgabccisd'
1:19:8
32:2781:64
4:33:2
27:1616:9
243:1282:1
5) Fibonacci, cathedrals, kyries
Stacking up tones in the medieval tone "system" or … working without it.
deffisgabccisd'
1:19:8
32:2781:64
4:33:2
27:1616:9
243:1282:1
5) Fibonacci, cathedrals, kyries
Stacking up tones in the medieval tone "system" or … working without it.
Afrika + 7
doremifasollasido
1:19:85:44:33:25:3
15:82:1
6) Pacioli, Brunelleschi, simplified ratios
Simplifying ratios in tackling the Pythagorean commaproblem
Filippo Brunelleschi, San Lorenzo, 1440.
doremifasollasido
1:19:85:44:33:25:3
15:82:1
6) Pacioli, Brunelleschi, simplified ratios
Simplifying ratios in tackling the Pythagorean commaproblem
Dutch artistsVredeman de Vries ±1550
doremifasollasido
1:19:85:44:33:25:3
15:82:1
6) Pacioli, Brunelleschi, simplified ratios
Simplifying ratios in tackling the Pythagorean commaproblem
Borromini gallery, as a transportable"blow up 'play' castle"
doremifasollasido
1:19:85:44:33:25:3
15:82:1
6) Pacioli, Brunelleschi, simplified ratios
Simplifying ratios in tackling the Pythagorean commaproblem
Afrika + 8
Rembrandt, The Night Watch, 1642 (Amsterdam).
7) Newton, Vermeer, Bach
Music studies in centre of attention:mean tone scales, etcmusic geometry
doremi…
11.118… = √5/2 1.25 = (√5/2)2
7) Newton, Vermeer, Bach
Music studies in centre of attention:mean tone scales, etcmusic geometry
doremi…
11.118… = √5/2 1.25 = (√5/2)2
Aldert Mantje & Jan Maris: Night Watch Research Project
www.verbekefoundation.com/
7) Newton, Vermeer, Bach
Music studies in centre of attention:mean tone scales, etcmusic geometry
doremi…
11.118… = √5/2 1.25 = (√5/2)2
Johannes Vermeer 1632 - 1675 Girl with a pearl earring
7) Newton, Vermeer, Bach
Music studies in centre of attention:mean tone scales, etcmusic geometry
doremi…
11.118… = √5/2 1.25 = (√5/2)2
The Anatomy Lesson of Dr. Nicolaes Tulp(Rembrandt)
Influence science & technology:linseed (flax seed) replaces egg tempera paint; lenses and mirrors: chiaroscuro (clairobsur), geometric composition ...
Afrika + 9
doremifasollasido
11.122…=22/12
1.259…=24/12
1.334…=25/12
1.498…=27/12
1.681…=29/12
1.887…=211/12
2
8) Gauss, J-F Millet, 21/12 relation.
One mathematical rule explains all music notes
John Constable, Dedham Vale 1802
doremifasollasido
11.122…=22/12
1.259…=24/12
1.334…=25/12
1.498…=27/12
1.681…=29/12
1.887…=211/12
2
8) Gauss, J-F Millet, 21/12 relation.
One mathematical rule explains all music notes
Jacques-Louis David, La mort de Socrate 1787
doremifasollasido
11.122…=22/12
1.259…=24/12
1.334…=25/12
1.498…=27/12
1.681…=29/12
1.887…=211/12
2
8) Gauss, J-F Millet, 21/12 relation.
One mathematical rule explains all music notes
Adolph von Menzel, Flötenkonzert 1850-52
doremifasollasido
11.122…=22/12
1.259…=24/12
1.334…=25/12
1.498…=27/12
1.681…=29/12
1.887…=211/12
2
8) Gauss, J-F Millet, 21/12 relation.
One mathematical rule explains all music notes
Jean-François Millet, The Gleaners (1857), a realistic painting, applying all known techniques before photography.
Afrika + 10
doremifasollasido
261.63 Hz293.66 Hz329.63 Hz349.23 Hz
392 Hz440 Hz
493.88 Hz523.25 Hz
9) Cantor, Seurat, Hertz.
Some label AntonWebern’s “Concerto for Nine Instruments” pointillist.Georges Seurat, Sunday Afternoon on the Island of La
Grande Jatte 1886
doremifasollasido
261.63 Hz293.66 Hz329.63 Hz349.23 Hz
392 Hz440 Hz
493.88 Hz523.25 Hz
9) Cantor, Seurat, Hertz.
Some label AntonWebern’s “Concerto for Nine Instruments” pointillist.
Theodoor van Rysselberghe,
Portrait of Alice Sethe, 1888
doremifasollasido
261.63 Hz293.66 Hz329.63 Hz349.23 Hz
392 Hz440 Hz
493.88 Hz523.25 Hz
9) Cantor, Seurat, Hertz.
Some label AntonWebern’s “Concerto for Nine Instruments” pointillist.Georges Seurat, Sunday Afternoon on the Island of La
Grande Jatte 1886Georges Seurat, La Parade, 1889
doremifasollasido
261.63 Hz293.66 Hz329.63 Hz349.23 Hz
392 Hz440 Hz
493.88 Hz523.25 Hz
9) Cantor, Seurat, Hertz.
Some label AntonWebern’s “Concerto for Nine Instruments” pointillist.
After Georges Seurat, La Parade, 1889
Afrika + 11
10) Einstein, Picasso, Grisey.
Pablo Picasso, The Bull lithographies, 1945.
Helmholtz: consonance = overlap overtones,in ear of observer.
Applied by composer Gérard Grisey founder of spectralism.
10) Einstein, Picasso, Grisey.
Pablo Picasso,, Les Demoiselles d'Avignon, 1907
Helmholtz: consonance = overlap overtones,in ear of observer.
Applied by composer Gérard Grisey founder of spectralism.
10) Einstein, Picasso, Grisey.
Georges Braque, Violin and Candlestick, 1910.
Helmholtz: consonance = overlap overtones,in ear of observer.
Applied by composer Gérard Grisey founder of spectralism.
10) Einstein, Picasso, Grisey.
Helmholtz: consonance = overlap overtones,in ear of observer.
Applied by composer Gérard Grisey founder of spectralism.
Einstein formuleerde de bijzondere relativiteitstheorie (1905) weliswaar vóór het kubisme tot bloei kwam (1910-11), maar deze bloei werd uiteraard voorafgegaan door een ontluiking. De nieuwe, relativistische wijze van zien tekent zich in de schilderkunst reeds twintig jaar eerder af, met name in het werk van Cézanne. Bovendien maakt de bijzondere relativiteitstheorie de werkelijkheid nog niet kenbaar als een complex energetisch veld. Deze opvatting van de werkelijkheid, die in het kubisme van 1910 gestalte krijgt, vindt haar wetenschappelijke formulering pas in de algemene relativiteitstheorie, die Einstein in 1916 voltooide.
Uit “Aldo Van Eyck”: […] Hieruit hoeft echter niet geconcludeerd te worden dat het kubisme beïnvloed werd door de relativiteitstheorie.
Afrika + 12
Vasarely, Stri Neu. Some speak of colloïdal art, and make comparisons with chemical topics
11) Russell, Vasarelly, sound illusion.
Sound illusions
perception
Bridget Riley,Movement in Squares, 1961.
11) Russell, Vasarelly, sound illusion.
Sound illusions
perception
11) Russell, Vasarelly, sound illusion.
Sound illusions
perception
11) Russell, Vasarelly, sound illusion.
Sound illusions
perception
Afrika + 13
Xenakis’ musical pavilion in Brussels (1958) and stochastic music.
Jack The Dripper.
12) Schrödinger, Pollock, Xenakis.
Xenakis’ musical pavilion in Brussels (1958) and stochastic music.
Jackson Pollock, Lavender Mist 1950.
12) Schrödinger, Pollock, Xenakis.
Xenakis’ musical pavilion in Brussels (1958) and stochastic music.
CD collection jazz Pollock allegedly listened to while painting. Is it art?
12) Schrödinger, Pollock, Xenakis.
Xenakis’ musical pavilion in Brussels (1958) and stochastic music.
12) Schrödinger, Pollock, Xenakis.
Afrika + 14
13) Finite geometry, minimal art, discrete-tones
Jean-Marie Dendoncker (Belgium) is a carpenter – mathematician, who makes wooden models, for, for instance, an arrangement of 7
points on 7 lines, 3 by 3.
Twelve tone symmetry
Xenakis's Bohor layout
13) Finite geometry, minimal art, discrete-tones
ISISTihany2004Bukowsky
Twelve tone symmetry
Xenakis's Bohor layout
13) Finite geometry, minimal art, discrete-tones
Carlo Sequin's (Berkeley) Minimal Trefoil
Twelve tone symmetry
Xenakis's Bohor layout
13) Finite geometry, minimal art, discrete-tones
Twelve tone symmetry
Xenakis's Bohor layout
Afrika + 15
14) Romanticism
Young van Beethoven; Beethoven's third symphony = “good romantic style”
J.M.W. Turner, The fighting Téméraire tugged to her last Berth to be broken, 1839.
14) Romanticism
Young van Beethoven; Beethoven's third symphony = “good romantic style”Francisco de Goya y Lucientes, The Third of
May, 1808.
Egide Charles Gustave Wappers, Episode of the Belgian Revolution of 1830, painted in 1834.
14) Romanticism
Young van Beethoven; Beethoven's third symphony = “good romantic style”
Francisco de Goya y Lucientes, The Third of May, 1808.
14) Romanticism
Young van Beethoven; Beethoven's third symphony = “good romantic style”
Théodore Géricault, Raft of the Medusa, 1839.
Afrika + 16
15) Impressionism
In Claude Debussy's time, nocturnes, arabesques and preludes became fashionable.
Edgar Degas, La Classe de Danse, 1874
15) Impressionism
In Claude Debussy's time, nocturnes, arabesques and preludes became fashionable.Pierre-Auguste Renoir, Le Moulin de la
Galette, 1876
15) Impressionism
In Claude Debussy's time, nocturnes, arabesques and preludes became fashionable.Claude Monet, Impression, soleil levant, 1872
15) Impressionism
In Claude Debussy's time, nocturnes, arabesques and preludes became fashionable.
Monet’s Sunset changed to grey values: the disk of the sun had to be added.
Afrika + 17
Tatlin Tower. Model of the Monument to the Third Interna-tional
16) Constructivism
Alexander Mosolov’s “Steel Foundry” introduced mechanical sounds.
16) Constructivism
Alexander Mosolov’s “Steel Foundry” introduced mechanical sounds.
Rodchenko, Stairway, 1930.
16) Constructivism
Alexander Mosolov’s “Steel Foundry” introduced mechanical sounds.
Constructivists designed of posters for every-thing from film to propa-ganda; here: photo-montage by Gustav Klutsis.
16) Constructivism
Alexander Mosolov’s “Steel Foundry” introduced mechanical sounds.
Tatlin’s tribute to the working class.
Afrika + 18
17) Expressionism
Schoenberg’ s opera “Erwartung” (1909) is called expressionist because of “screaming emotions” of the main character.Constant Permeke, The Daily Bread
17) Expressionism
Schoenberg’ s opera “Erwartung” (1909) is called expressionist because of “screaming emotions” of the main character.
Edvard Munch, The Scream, 1893; forerunner of expressi-onism.
17) Expressionism
Schoenberg’ s opera “Erwartung” (1909) is called expressionist because of “screaming emotions” of the main character.El Greco, View of Toledo, 1595-1610 resembles 20th
cent. expressionism (art-historically: “mannerism”).
17) Expressionism
Schoenberg’ s opera “Erwartung” (1909) is called expressionist because of “screaming emotions” of the main character.
Above: painting by Schoenberg’s friend Kandinsky; left: painting by the composer
Schoenberg.
Afrika + 19
18) Continued?* Gödel Escher Bach in recent times?
Popular music?
J. Cage, 1952:4’33”
… of silence
abstract artAd Reinhardt, Painting, 1960
new realism Luc Tuymans The Perfect Table Setting, 2004
Some music is difficult to put in a box and so used here …
* education?
Going by the books? Or …
… learning intuitively, as if it were a game?
18) Continued?
