Math and Music Jiyou Li lijiyou@sjtu.edu.cn School of Mathematical Sciences, SJTU 2016.07.21
Outline 1 Mathematics 2 Music 3 Math and Music
Mathematics Mathematics is the study of topics such as numbers, structure, space, and change.
Mathematics Mathematics is the study of topics such as numbers, structure, space, and change. Mathematicians seek out patterns and use them to formulate new conjectures.
Mathematics Mathematics is the study of topics such as numbers, structure, space, and change. Mathematicians seek out patterns and use them to formulate new conjectures. Mathematics is to establish truth by rigorous deduction from appropriately chosen axioms and definitions.
Mathematics Mathematics is the study of topics such as numbers, structure, space, and change. Mathematicians seek out patterns and use them to formulate new conjectures. Mathematics is to establish truth by rigorous deduction from appropriately chosen axioms and definitions. Gödel s incompleteness theorem.
Mathematics Mathematics is the study of topics such as numbers, structure, space, and change. Mathematicians seek out patterns and use them to formulate new conjectures. Mathematics is to establish truth by rigorous deduction from appropriately chosen axioms and definitions. Gödel s incompleteness theorem. Key words: abstraction, logic, counting, pattern, measurement.
Music Music is an art form and cultural activity whose medium is sound and silence. The common elements of music are pitch, rhythm, dynamics, timbre, form and style.
Music Music is an art form and cultural activity whose medium is sound and silence. The common elements of music are pitch, rhythm, dynamics, timbre, form and style. Creating music, listening to music, and dancing to the rhythms of music are practices cherished in cultures all over the world.
Music Music is an art form and cultural activity whose medium is sound and silence. The common elements of music are pitch, rhythm, dynamics, timbre, form and style. Creating music, listening to music, and dancing to the rhythms of music are practices cherished in cultures all over the world. Music satisfies a deep human need.
Music Music is an art form and cultural activity whose medium is sound and silence. The common elements of music are pitch, rhythm, dynamics, timbre, form and style. Creating music, listening to music, and dancing to the rhythms of music are practices cherished in cultures all over the world. Music satisfies a deep human need. Universal definition: music is any combination and of any kind of sounds that someone enjoys listening to.
Activities in music: from The Topos of Music
Sciences in music: from The Topos of Music
Quotes about music Wö, U/ƒÚ ; rö, U/ƒS. «rp WP WØ» Ñö, )<%ö. œäu, /u(. (, ƒñ. «rp WP W» ƒñs±w, ÙÚ; Ï ƒñ±ä, Ù#; IƒÑH±g, Ù (. (у, Ï. «rp W P W»
Favorite quotes about music Music is a moral law. It gives soul to the universe, wings to the mind, flight to the imagination, and charm and gaiety to life and to everything. Plato Without music life would be a mistake. Nietzsche Music is the mediator between the spiritual and the sensual life. van Beethoven Music is a higher revelation than all wisdom and philosophy. van Beethoven
Favorite quotes about music If I were not a physicist, I would probably be a musician. I often think in music. I live my daydreams in music. I see my life in terms of music. Albert Einstein No school would eliminate the study of language, mathematics, or history from its curriculum, yet the study of music, which encompasses so many aspects of these fields and can even contribute to a better understanding of them, is entirely ignored. Daniel Barenboim
What does music bring us? Huge amount of treasure in human history
What does music bring us? Huge amount of treasure in human history Story of Confucius
What does music bring us? Huge amount of treasure in human history Story of Confucius Health
What does music bring us? Huge amount of treasure in human history Story of Confucius Health The Mozart effect and spatial temporal reasoning
What does music bring us? Huge amount of treasure in human history Story of Confucius Health The Mozart effect and spatial temporal reasoning Scientists and literary masters: Einstein, Tagore, Qian
What does music bring us? Huge amount of treasure in human history Story of Confucius Health The Mozart effect and spatial temporal reasoning Scientists and literary masters: Einstein, Tagore, Qian Self-balance
What does music bring us? Huge amount of treasure in human history Story of Confucius Health The Mozart effect and spatial temporal reasoning Scientists and literary masters: Einstein, Tagore, Qian Self-balance Social functions: Music does bring people together
Structure of a music piece Pitch: basic element in music; A=440 Hz; Music: interval->chord-> motive->sentence->chapter-> Jingle Bell
Math and Music Music: pitch, rhythm, timbre, dynamics, scale. Math: number, structure, spectrum, change, space.
Math and Music Music: patterns, logic, chosen axioms, piece. Math: patterns, logic, chosen axioms, proof.
Math and music There is geometry in the humming of the strings, and there is music in the spacing of the spheres. Pythagoras All nature consists of harmony arising out of numbers. Pythagoras May not music be described as the mathematic of sense, mathematic as music of the reason? The soul of each the same! Sylvester Music is the pleasure of the human mind experiences from counting without being aware that it is counting. Music is nothing but unconscious arithmetic. Leibniz In math and music, there is an intense study part and a performance part. Robert Bryant (President of AMS)
The understanding of consonance (harmony)??-900s: melodic sense 900s-1300s interval: octave and perfect fifth/fourth 1300s-1700s interval: sixth and third, relativity 1800s Rameau: root theory 1900s Helmholtz: beat/roughness theory 1965: Plomp and Levelt: critical bandwidth
Why do we have consonance? Mathematics and music! The most glaring possible opposites of human thought! And yet connected, mutually sustained! It is as if they would demonstrate the hidden consensus of all the actions of our mind, which in the revelations of genius makes us forefeel unconscious utterances of a mysteriously active intelligence. Hermann von Helmholtz, "On The Physiological Causes of Harmony in Music," 1857
Consonance and dissonance Neuroscience
Consonance and dissonance Physics
Vibrating strings Basic things about sound; Taylor(1685-1731); D. Bernoulli(1700-1782); D Alembert(1717-1783); Euler (1708-1783); Fourier (1768-1830).
Vibrating strings From Wiki
D Alembert: 1-d wave equation, 1746; Mersenne, 1636 2 y x 2 = c2 2 y, y(0, t) = 0, y(l, t) = 0; y(x, 0) = f (x) t2
D Alembert: 1-d wave equation, 1746; Mersenne, 1636 2 y x 2 = c2 2 y, y(0, t) = 0, y(l, t) = 0; y(x, 0) = f (x) t2 y(x, t) = A k sin( kπ L k=1 x) cos(kπ cl t).
D Alembert: 1-d wave equation, 1746; Mersenne, 1636 2 y x 2 = c2 2 y, y(0, t) = 0, y(l, t) = 0; y(x, 0) = f (x) t2 y(x, t) = A k sin( kπ L k=1 x) cos(kπ cl t). f k = kc 2L = k T 2L ρ
Overtone
Fourier analysis
Musical instrument: Flute
Musical instrument: Violin
Musical instrument: Flute and Violin
Harmony
Musical instrument: Drum
Musical instrument: Drum
Can we hear the shape of a drum?
Can we hear the shape of a drum? From Wiki
Music scale A scale is any finite set of musical notes ordered by pitch. Question: how to build a nice set (scale)?
Music scales Monotonic, Ditonic, Tritonic, Tetratonic Pentatonic Hexatonic Heptatonic Octatonic
Music scales Diatonic scales: five whole steps and two half steps
Music scales Diatonic scales: five whole steps and two half steps 1st-7thõTonic (key note), Supertonic, Mediant, Subdominant, Dominant, Submediant, Leading tone
Music scales Diatonic scales: five whole steps and two half steps 1st-7thõTonic (key note), Supertonic, Mediant, Subdominant, Dominant, Submediant, Leading tone Major scale: C D E F G A B C = 1 2 3 4 5 6 7 1
Music scales Diatonic scales: five whole steps and two half steps 1st-7thõTonic (key note), Supertonic, Mediant, Subdominant, Dominant, Submediant, Leading tone Major scale: C D E F G A B C = 1 2 3 4 5 6 7 1 Natural minor scale: 6. 7. 1 2 3 4 5 6
Music scales Diatonic scales: five whole steps and two half steps 1st-7thõTonic (key note), Supertonic, Mediant, Subdominant, Dominant, Submediant, Leading tone Major scale: C D E F G A B C = 1 2 3 4 5 6 7 1 Natural minor scale: 6. 7. 1 2 3 4 5 6 Melodic scale: 6. 7. 1 2 3 4 5 6
Music scales Diatonic scales: five whole steps and two half steps 1st-7thõTonic (key note), Supertonic, Mediant, Subdominant, Dominant, Submediant, Leading tone Major scale: C D E F G A B C = 1 2 3 4 5 6 7 1 Natural minor scale: 6. 7. 1 2 3 4 5 6 Melodic scale: 6. 7. 1 2 3 4 5 6 Harmonic minor scale: 6. 7. 1 2 3 4 5 6
Music scales Diatonic scales: five whole steps and two half steps 1st-7thõTonic (key note), Supertonic, Mediant, Subdominant, Dominant, Submediant, Leading tone Major scale: C D E F G A B C = 1 2 3 4 5 6 7 1 Natural minor scale: 6. 7. 1 2 3 4 5 6 Melodic scale: 6. 7. 1 2 3 4 5 6 Harmonic minor scale: 6. 7. 1 2 3 4 5 6 Whole tone
Music scales Diatonic scales: five whole steps and two half steps 1st-7thõTonic (key note), Supertonic, Mediant, Subdominant, Dominant, Submediant, Leading tone Major scale: C D E F G A B C = 1 2 3 4 5 6 7 1 Natural minor scale: 6. 7. 1 2 3 4 5 6 Melodic scale: 6. 7. 1 2 3 4 5 6 Harmonic minor scale: 6. 7. 1 2 3 4 5 6 Whole tone Chromatic scale
First scale from the first experiment in history
Pythagorean tuning (Ê݃)Æ) ÊÊl ± û"n Ê o± æ"n Ã Ô ± û"n o l± "n à 8 o± " iê[ 5 P ÆÖ6
Pythagorean tuning (Ê݃)Æ) ÊÊl ± û"n Ê o± æ"n Ã Ô ± û"n o l± "n à 8 o± " iê[ 5 P ÆÖ6 fæ XK ú " f X ê3 " fû XÚ h " fû Xl+" f XW7± Ñ;± " òåêñ Ä kì nƒ om±üê Ê ± ) ƒƒä ± û"n ñ z kl æ"øãkn Ù v ± )û"kn EuÙ ± "kn Ù v ± " +f5/ÿ Ê l6
n Ã{ f/ 0( ÒÐ fârœâ (Ñ" f/ 0( ÒÐ ê" f/û0("òð / pú " f/û0( ÒÐ +" f /0( ÒÐ Ñ/3äþ (Ñq q " ås ÊѺN k(á un ƒ ²Logn íü±ü ÊÊl ƒê dd) ƒñn B û("nø l òù \3l þ z"l Ò ("Ø ŒØ2^nØ3 z"lþ~n ƒ vùêô dd)û("2^nøô \3 êþ dd ) (Ê 8"2^nØ 3Ê 8þ~n ƒ vùê 8 o dd)("
Just intonation: rational approximation by small whole numbers C = 1, D = 9 8, E = 5 4, F = 4 3, G = 3 2, A = 5 3, B = 15 8, c = 2 1
Just intonation: rational approximation by small whole numbers C = 1, D = 9 8, E = 5 4, F = 4 3, G = 3 2, A = 5 3, B = 15 8, c = 2 1 C = 1, C = 16 15, D = 9 8, D = 6 5, E = 5 4, F = 4 45 3, F = 32, G = 3 2, G = 8 5, A = 5 3, A = 9 5, B = 15 8, c = 2
Just intonation: rational approximation by small whole numbers C = 1, D = 9 8, E = 5 4, F = 4 3, G = 3 2, A = 5 3, B = 15 8, c = 2 1 C = 1, C = 16 15, D = 9 8, D = 6 5, E = 5 4, F = 4 45 3, F = 32, G = 3 2, G = 8 5, A = 5 3, A = 9 5, B = 15 8, c = 2 Violin, viola and cello.
Chromatic scale: geometric progression C = 1, C = 2 1 12, D = 2 2 12,, G = 2 7 12,, c = 2 12 12
Chromatic scale: geometric progression C = 1, C = 2 1 12, D = 2 2 12,, G = 2 7 12,, c = 2 12 12 5WÆÖ6, Á1Ç, 1581, mš
Chromatic scale: geometric progression C = 1, C = 2 1 12, D = 2 2 12,, G = 2 7 12,, c = 2 12 12 5WÆÖ6, Á1Ç, 1581, mš = 1, Œ½= 2 1 12, q, Y, W, ½, mu,, K, H½, Ã, A = 2. Chromatic scale=z 12 or R/12Z.
Chromatic scale: geometric progression C = 1, C = 2 1 12, D = 2 2 12,, G = 2 7 12,, c = 2 12 12 5WÆÖ6, Á1Ç, 1581, mš = 1, Œ½= 2 1 12, q, Y, W, ½, mu,, K, H½, Ã, A = 2. Chromatic scale=z 12 or R/12Z. Symmetry and harmony
Chromatic scale: geometric progression
Diatonic scales
Scales
Rhythm and Euclidean s algorithm Tresillo > Habanera
Chords and transformation A chord is a "nice" subset in Z 12.
Chords and transformation A chord is a "nice" subset in Z 12. C = {C, E, G} = {0, 4, 7}
Chords and transformation A chord is a "nice" subset in Z 12. C = {C, E, G} = {0, 4, 7} G = {G, B, D} = {7, 11, 2}
Chords and transformation A chord is a "nice" subset in Z 12. C = {C, E, G} = {0, 4, 7} G = {G, B, D} = {7, 11, 2} C to G is a scalar transposition
Chords and transformation A chord is a "nice" subset in Z 12. C = {C, E, G} = {0, 4, 7} G = {G, B, D} = {7, 11, 2} C to G is a scalar transposition A m = {A, C, E} = {9, 0, 4}
Chords and transformation A chord is a "nice" subset in Z 12. C = {C, E, G} = {0, 4, 7} G = {G, B, D} = {7, 11, 2} C to G is a scalar transposition A m = {A, C, E} = {9, 0, 4} C to A m is a reflection by axis l(2, 8)
Where are they from?
Chord progression
Geometry of chords Harmony and counterpoint Transpolation Inversion Permutation Geometry, measure and voice leading
A torus
Geometry of chords: T 2 /S 2 (by Tymoczko)
Do we need algebraic geometry? The Topos of Music. by Guerino Mazzola
Do we need algebraic geometry? The Topos of Music. by Guerino Mazzola This is probably already the mathematics of the new age. Grothendieck
Do we need algebraic geometry? The Topos of Music. by Guerino Mazzola This is probably already the mathematics of the new age. Grothendieck If you can t learn algebraic geometry, he sometimes seems to be saying, then you have no business trying to understand Mozart. Dmitri Tymoczko
Review pitch fundamental frequency scales spaces tonality space with a basis harmony overtone timbre spectrum+time rhythm timeline
References Dmitri Tymoczko The Geometry of Musical Chords Science 313, 72 (2006) ÉSŒ ÑW Æ p Ñ (2012) Êp < NÉÑW < ÑWÑ (2007) Guerino Mazzola The Topos of Music, Geometric Logic of Concepts, Theory, and Performance Birkhäuser (2002)
Share and enjoy music
Share and enjoy music Thank you very much for your attention!