Algorithms In Music Outreach to Students Interested in the Arts - - PowerPoint PPT Presentation

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

2 4 5 3 1 7 9 11 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

f f f f f 2 , 2 , , 2 , 2 ,

12 11 12 2 12 1



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

• f 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
• ctaves 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.

2 4 5 3 1 7 9 11 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

• f 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

• nce in each column and each row.

Construct a Musical Latin Square (called a Serial Matrix)

2 5 3 4 1 4 1 3 2 5 3 1 2 5 3 4 5 2 5 2 1 4 4 1 5 3 1 4 2 3

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 1 5 3 R4 P1 1 4 7 3 6 5 8 11 9 10 2 R1 P10 10 1 4 3 2 5 8 6 7 11 9 R10 P2 2 5 8 4 7 6 9 10 11 3 1 R2 P11 11 2 5 1 4 3 6 9 7 8 10 R11 P0 3 6 2 5 4 7 10 8 9 1 11 R0 P9 9 3 11 2 1 4 7 5 6 10 8 R9 P6 6 9 8 11 10 1 4 2 3 7 5 R6 P8 8 11 2 10 1 3 6 4 5 9 7 R8 P7 7 10 1 9 11 2 5 3 4 8 6 R7 P3 3 6 9 5 8 7 10 1 11 4 2 R3 P5 5 8 11 7 10 9 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
• f tone rows randomly selected
• Option: Include rhythmic structure or not
• Option: Select pitches from multiple octaves

1 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 2

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 2 4 3 5 6

Diatonic Scale Rhythm

• Randomly choose amongst 4 rhythmic

structures with accompanying stresses to make music sound like it is in 4/4 time

2 4 5 3 1 7 9 11 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