Mathematical Symmetry and Algorithms In Music Outreach to Students Interested in the Arts Lisa Lajeunesse Capilano University
Liberal Studies Bachelor of Arts (LSBA) at Capilano • Program is broad-based like a traditional Liberal Arts degree • Has an interdisciplinary component with semester “themes” • Request for 3 rd and 4 th year courses in sciences that would be relevant and interesting to students with primarily a humanities/social sciences background
Math and the Creative Arts Course To explore interdisciplinary connections between math and: • Visual art • Music • Literature • Theatre? Dance? Math puzzles?
Practical Constraints • Want the course to be accessible, with prereqs no higher than Grade 11 Math • Course topics that relate to LSBA semester theme of Creation have preference • Assume students’ backgrounds differ in math, music, visual art and literature
Overall Course Objectives • Foster an appreciation for the role that math can play in art • Help students see mathematics as beautiful • Spark interest in further math study • Empower students to use math
Mathematical Objectives • Exercise numeracy: Work with ratios and proportion, modular arithmetic, geometry, basic algebra, logic, counting and enumeration, sequences and series • Recognize and explain mathematical patterns • Recognize and execute algorithms Time permitting: • Graph theory, encryption • Mathematical proofs as works of art
Artistic Objectives • Use math to solve artistic problems • Use math to develop/enhance a technical skill and provide technical mastery • Use math to direct artistic form • Generate discussion and debate on merits of using mathematics in artistic creation
Music • Long history of connection with math • Musical problems have fuelled mathematical research • Math and/or physics found on many levels • Broad appeal
Music and Math Connections Properties of sound: • Physics of waves etc. Relationship between pitch and frequency; relationship between loudness and intensity: • Logarithms and Exponentials Musical intervals, consonance/dissonance: • Metric, arithmetic and geometric means, sequences • Ratios and proportion
Symmetry in musical scales and chords: • Divisibility of integers Rhythms and recurring patterns: • Least common multiple Octave equivalence: • Modular arithmetic, equivalence classes Musical Timbre: • Addition of functions, Fourier analysis Sound envelope: • Multiplication of functions
Change ringing and other examples: • Permutations and Combinations Composition with chance: • Stochastic processes eg . Minuet and Trio (1790) (unknown composer, perhaps Haydn or Mozart) use dice to choose amongst a variety of bars of music http://Sunsite.univie.ac.at/Mozart/dice Computer composition: • Design and use of algorithms
The Problem of Tuning 1 3 6 8 10 0 2 4 5 7 9 11 0
“Distance” between pitches can be measured For a sequence of pitches to sound evenly spaced (perceived by ear as in arithmetic sequence), the frequencies must be in a geometric sequence. For 12 evenly spaced pitches in one octave, the common ratio 12 2 must be: 1 2 11 , 2 12 , 2 12 , , 2 12 , 2 f f f f f
Tuning issues: • Exponents, irrational numbers Historically solutions have involved: • Geometry, equations of lines, intersections • Diophantine approximation • Dominant eigenvalue and eigenvectors of a 12 by 12 matrix (18 th c. Christoph Gottlieb Schröter)
Symmetry in Music http://www.youtube.com/watch?v=MuWUp1M-vuM Bach’s Crab Canon : http://strangepaths.com/canon-1-a-2/2009/01/18/en/ http://www.youtube.com/watch?v=xUHQ2ybTejU& feature=player_embedded#
Melody as a Function • There is an inherent “height” to the pitches that we hear (determined by frequency). • Higher frequencies are perceived as “higher” pitches. • Most people can identify the (relative) difference in height between two pitches.
Melody as a Function • Rhythm determines how pitch changes with time. • Melody is comprised primarily of pitch and rhythm.
Graph of Pitch vs. Time pitch (distance in semitones relative to middle C) 13 C 12 B 11 A# 10 A 9 G# 8 G 7 6 F# 5 F E 4 3 D# 2 D 1 C# t, time -4 4 8 12 16 20 24 28 32 36 -1 Passage from The Art of the Fugue , by J.S. Bach -2
Inversion of Melody pitch (distance in semitones relative to middle C) 13 C 12 B 11 A# 10 A 9 G# 8 G 7 F# 6 5 F E 4 3 D# D 2 C# 1 t, time -4 4 8 12 16 20 24 28 32 36 -1 Passage from The Art of the Fugue , by J.S. Bach -2
Transformations in Music • Beautiful music can be created by weaving together multiple occurrences of a single melodic function or “voice” with a variety of transformations of itself • This practice has been used in music from 13 century to present day
Function Transformations Horizontal shifts: (time shift) Musical canon • Row, row, row your boat • Benjamin Britten, Ceremony of Carols , This Little Babe (1942): http://www.youtube.com/watch?v=1wayMn7vUEM&feature=related Horizontal Compressions/Expansions: Mensuration Canon • Josquin Des Prez, Missa l’Homme Arme Agnus Dei (c. 1500) Super Tones Musicales 5. (up to 1:18): http://www.youtube.com/watch?v=kq2693QkTHU&feature=relmfu
• Conlon Nancarrow, Study #30 & #36 for player piano (1940) Horizontal compressions of voices are in ratios of 17:18:19:20; Vertical Shifts: Shifting a sequence of pitches up or down is called Transposition (often tonal rather than strict)
Vertical Reflection: Musical term is Inversion (strict or tonal). Horizontal Reflection: Musical term is Retrograde .
Horizontal and Vertical reflections combined: Musical term is retrograde-inversion (RI) • Paul Hindemith, Ludus Tonalis (1942) 5 th movement = RI of 1 st movement: 1 st Movement: http://www.youtube.com/watch?v=cBm9TE2Lcyg • • 5 th Movement: http://www.youtube.com/watch?v=rxoqD7_Znr0 (9:46) When played in sequence gives odd symmetry
Even Symmetry: When reflection about the middle of the melody gives the same melody. Called palindrome : • Franz Joseph Haydn, Piano Sonata #41 (1773) • George Crumb, Por Que Naci Entre Espejos (1970) Periodicity: • 100 Bottles of Beer on the Wall • Philip Glass (b.1937), Steve Reich (b.1936) and minimalist music
How to represent melody so we can apply mathematical transformations? • Set up 1-1 correspondence between numbers and pitches • use octave equivalence
Octave Equivalence and Equivalence Classes • Pythagoras is thought to be the first to observe that two frequencies in a low integer ratio are pleasing (consonant) to the ear when sounded together. • The most pleasing is the 2:1 ratio which produces a distance or interval of one octave. • Two pitches separated by some integer number of octaves can be grouped into a single equivalence class.
Modular Arithmetic • A single octave is divided into 12 equally spaced intervals giving 12 distinct equivalence classes numbered 0 through 11 mod 12. • 0 generally represents the (equivalence) pitch class for C.
1 3 6 8 10 0 2 4 5 7 9 11 0
Melody represented as a Sequence of Pitch Classes Row, row, row your boat : Row, row, row your boat 0 0 0 2 4 Gent-ly down the stream 4 2 4 5 7 Mer-ri-ly, mer-ri-ly, mer-ri-ly, mer-ri-ly 12 (0) 7 4 0 Life is but a dream 7 5 4 2 0
20 th Century 12-Tone Serialism • Pioneered in the early 1900’s by Arnold Schoenberg (1874 – 1951) • Also by pupils Anton Webern (1883 – 1945), Alban Berg (1885 – 1935) etc. • In vogue for 50+ years amongst composers
Original Rules of 12-Tone Serialism • Choose one of 12! permutations of the 12 pitch classes of the chromatic scale to give the prime form of the tone row. • Compose the musical piece using transpositions of the prime, retrograde, inversion and retrograde inversion of the prime (Note: This gives 48 different tone rows from which to choose). • Each row must be completely used before another one of the 48 tone rows is employed in a given voice. • Pitches may be repeated. Two or more consecutive pitches in the row may be sounded simultaneously as a chord.
Examples: Anton Webern , Concerto, op. 24 : Prime Tone Row -1 -2 2 | 3 7 6 | 8 4 5 | 0 1 -3 Prime | RI | R | I http://www.youtube.com/watch?v=4OPfHfWBZLY In film scores: The Prisoner (Alec Guiness) The Curse of the Werewolf
Algorithm in Musical Composition Gareth Loy in Musimathics Vol. 1 on composition and use of non-deterministic methodologies: The analysis of methodology can reveal the aesthetic agenda of its (the music’s) creator
Aesthetic Objectives Reflect musically the existential angst of Viennese society in the early 20th century: • Atonality : Move beyond a single tonal center (likewise beyond polytonality) • Chromatic saturation : democratic use of 12 tones equally • Emancipation of dissonance : No need for dissonances to resolve into consonance
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