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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 3rd and 4th 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
0 2 4 5
3 1
7 9 11 0
8 6 10
The Problem of Tuning
“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
must be:
12 2
fffff 2,2,,2,2, 1211
122
121
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 (18th 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
-4 4 8 12 16 20 24 28 32 36
-2
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
t, time
pitch (distance in semitones relative to middle C)
Passage from The Art of the Fugue, by J.S. Bach
D#
F#
G
E
D
G#
A#
C#
A
C
F
B
Inversion of Melody
-4 4 8 12 16 20 24 28 32 36
-2
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
t, time
pitch (distance in semitones relative to middle C)
Passage from The Art of the Fugue, by J.S. Bach
D#
F#
G
E
D
G#
A#
C#
A
C
F
B
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)
5th movement = RI of 1st movement:
• 1st Movement: http://www.youtube.com/watch?v=cBm9TE2Lcyg
• 5th 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.
0 2 4 5
3 1
7 9 11 0
8 6 10
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
20th 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
Aesthetic Objectives
• Highly Organized: Compose structured
music in a way that maintains continuity
with earlier musical traditions
• Highlight Intervals: Make the interval
content of the tone row the unifying thread
of the music
Use of Latin Squares in
12-Tone Serialism
Definition: An n by n Latin Square is an
arrangement of n symbols into an n by n
array so that each symbol appears exactly
once in each column and each row.
Construct a Musical Latin Square
(called a Serial Matrix) 0 2 5 3 4 1
4
1
3
2
5
0 3 1 2 5
3 0 4 5 2
5 2 0 1 4
4 1 5 0 3
1 4 2 3 0
Latin Square for 12 Tone Row
I4 I7 I10 I6 I9 I8 I11 I2 I0 I1 I5 I3
P4 4 7 10 6 9 8 11 2 0 1 5 3 R4
P1 1 4 7 3 6 5 8 11 9 10 2 0 R1
P10 10 1 4 0 3 2 5 8 6 7 11 9 R10
P2 2 5 8 4 7 6 9 0 10 11 3 1 R2
P11 11 2 5 1 4 3 6 9 7 8 0 10 R11
P0 0 3 6 2 5 4 7 10 8 9 1 11 R0
P9 9 0 3 11 2 1 4 7 5 6 10 8 R9
P6 6 9 0 8 11 10 1 4 2 3 7 5 R6
P8 8 11 2 10 1 0 3 6 4 5 9 7 R8
P7 7 10 1 9 0 11 2 5 3 4 8 6 R7
P3 3 6 9 5 8 7 10 1 11 0 4 2 R3
P5 5 8 11 7 10 9 0 3 1 2 6 4 R5
RI4 RI7 RI10 RI6 RI9 RI8 RI11 RI2 RI0 RI1 RI5 RI3
Computer Generated Music:
Outline of Algorithm
• Select one of four different sets of pitch classes
• Randomly choose the prime tone row (permute the
pitch classes)
• Build the serial matrix and then play some number
of tone rows randomly selected
• Option: Include rhythmic structure or not
• Option: Select pitches from multiple octaves
1 0 3 2 4
Pentatonic Scale
Pentatonic Scale Rhythm
• Randomly choose amongst 3 rhythmic
structures with accompanying stresses to
make music sound like it is in ¾ time
4 3 5
Whole Tone Scale
1 0 2 0
Whole Tone Scale Rhythm
• Randomly choose amongst 4 rhythmic
structures with accompanying stresses to
make music sound like it is in 4/4 time
• One rhythmic structure includes a tie across
a bar line
Diatonic Scale
1 0 2 4 3 5 6 0
Diatonic Scale Rhythm
• Randomly choose amongst 4 rhythmic
structures with accompanying stresses to
make music sound like it is in 4/4 time
0 2 4 5
3 1
7 9 11 0
8 6 10
12 tone Chromatic Scale
12 tone Chromatic Scale
• Randomly choose note duration
• Randomly (and independently) choose
accompanying stress
• No sense of rhythmic pulse