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Final Projects
Some simple ideas
Composition
(1) program that "learns"
some aspect of musical
composition
(2) fractal music that sounds
musical
(3) program that creates
engaging new styles
(4) vivaldi music maker
(scales, arps, sequences, etc.)
(5) program that sets some of
Messiaen's ideas into code
(6) real-time transformation of
drawing to music
(7) improvisation program
(8) accompaniment program
(9) re-write masterpieces
according to some plan
(10) logically replace one of
the elements of known music
Analysis
(1) performance attributes of
given performers
(2) mapping rhythm, texture,
harmonic rhythm, etc.
(3) reduction by mathematics
(4) analysis using 2D cellular
automata
(5) statistical representation
and comparison
(6) analysis of chromatic
versus diatonic content of
music
(7) tension analyzing program
(Hindemith theories?)
(8) relevance of dynamics to
pitch, etc. (i.e., cross
dependency)
(9) compare some aspect of
music to some aspect of nonmusic
(10) a composer's use of
some attribute over an
extended period
Short Paper
Well-Documented Code
Five Sample Outputs
Presentations
due
Thursday June 12, 8-11am
Determinacy
versus
Indeterminacy
Sir Isaac Newton
1726
Principia
“Actioni contrarium semper et
equalem esse reactionem”
“to every action there is always
opposed an equal and opposite
reaction”
Richard Feynman
“it is impossible to predict which
way a photon will go”
Murray Gell-Mann
“there is no way to predict the
exact moment of disintegration”
Werner Heisenberg
uncertainty principle
“the act of observation itself may
cause apparent randomness at the
subatomic level”
Albert Einstein
“God does not play with dice.”
Cope
“Observation alone cannot
determine indeterminacy.”
Ignorance?
Too complex?
Too patternless?
Too irrelevant?
Discrete Mathematics
Study of discontinuous numbers
Logic, Set Theory,
Combinatorics Algorithms,
Automata Theory, Graph Theory,
Number Theory, Game Theory,
Information Theory
Recreational
Number
Theory
Power of 9s
9 * 9 = 81
8+1=9
Multiply any number by 9
Add the resultant digits together
until you get one digit
Always 9
e.g.,
4 * 9 = 36
3+6=9
Square Root of
Palendromic Numbers
Square Root of
123454321
=
11111
Square Root of
1234567654321
=
1111111
Pascal’s Triangle
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
1
11
1 1
1111
1
1
11 11
1 1 1 1
11111111
1
1
11
11
1 1
1 1
1111
1111
1
1
1
1
11 11 11 11
1 1 1 1 1 1 1 1
1111111111111111
1
1
11
11
1 1
1 1
1111
1111
1
1
1
1
11 11
11 11
1 1 1 1
1 1 1 1
11111111
11111111
1
1
1
1
11
11
11
11
1 1
1 1
1 1
1 1
1111
1111
1111
1111
1
1
1
1
1
1
1
1
11 11 11 11 11 11 11 11
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
11111111111111111111111111111111
1
1
11
11
1 1
1 1
1111
1111
1
1
1
1
11 11
11 11
1 1 1 1
1 1 1 1
11111111
11111111
1
1
1
1
11
11
11
11
1 1
1 1
1 1
1 1
1111
1111
1111
1111
1
1
1
1
1
1
1
1
11 11 11 11
11 11 11 11
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1111111111111111
1111111111111111
1
1
1
1
11
11
11
11
1 1
1 1
1 1
1 1
1111
1111
1111
1111
1
1
1
1
1
1
1
1
11 11
11 11
11 11
11 11
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
11111111
11111111
11111111
11111111
1
1
1
1
1
1
1
1
11
11
11
11
11
11
11
11
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1111
1111
1111
1111
1111
1111
1111
1111
• The sum of each row results in increasing powers of 2 (i.e., 1, 2, 4, 8,
16, 32, and so on).
• The 45° diagonals represent various number systems. For example,
the first diagonal represents units (1, 1 . . .), the second diagonal, the
natural numbers (1, 2, 3, 4 . . .), the third diagonal, the triangular
numbers (1, 3, 6, 10 . . .), the fourth diagonal, the tetrahedral numbers
(1, 4, 10, 20 . . .), and so on.
• All row numbers—row numbers begin at 0—whose contents are
divisible by that row number are successive prime numbers.
• The count of odd numbers in any row always equates to a power of 2.
• The numbers in the shallow diagonals (from 22.5° upper right to lower
left) add to produce the Fibonacci series (1, 1, 2, 3, 5, 8, 13 . . .),
discussed in chapter 4.
• The powers of 11 beginning with zero produce a compacted Pascal's
triangle (e.g., 110 = 1, 111 = 11, 112 = 121, 113 = 1331, 114 = 14641, and
so on).
• Compressing Pascal's triangle using modulo 2 (remainders after
successive divisions of 2, leading to binary 0s and 1s) reveals the
famous Sierpinski gasket, a fractal-like various-sized triangles, as
shown in figure 7.2, with the zeros (a) and without the zeros (b), the
latter presented to make the graph clearer.
Leonardo of Pisa, known as
Fibonacci. Series first stated in
1202 book Liber Abaci
0,1,1,2,3,5,8,13,21,34,55,89. . .
Each pair of previous numbers
equaling the next number of the
Sequence.
Dividing a number in the
sequence into the following
number produces the
Golden Ratio
1.62
Debussy, Stravinsky, Bartók
composed using
Golden mean (ratio, section).
Bartók’s Music for Strings,
Percussion and Celeste
55
34
21
34
21
13
13
21
89
Fermat’s Last Theorum
to prove that
Xn + Yn = Zn
can never have integers for X, Y,
and/or Z beyond
n=2
$1 million prize to create
formula for creating
next primes without
trial and error
Magic Squares
Square Matrix
in which
all horizontal ranks
all vertical columns
both diagonals
equal same number when added
together
1
-2
0
7
9
-9
11
-7
-5
2
4
-1
6
13
-10
-3
-8
-6
1
8
10
5
12
-11
-4
3
2
1
-2
0
7
9
-9
-8
-6
1
8
10
11
-7
-5
2
4
5
12
-11
-4
3
-1
6
13
-10
-3
-2
0
7
9
-9
-8
-6
1
8
10
11
-7
-5
2
4
5
12
-11
-4
3
6
13
-10
-3
4
-1
3
11
-7
-5
2
4
-1
6
13
-10
-3
-8
-6
1
8
10
5
12
-11
-4
3
-2
0
7
9
-9
5
5
12
-11
-4
3
-1
6
13
-10
-3
-2
0
7
9
-9
-8
-6
1
8
10
11
-7
-5
2
4
5
12
-11
-4
3
-1
6
13
-10
-3
-2
0
7
9
-9
-8
-6
1
8
10
11
-7
-5
2
4
Musikalisches Würfelspiele
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Number of Possibilities
of 2 matrixes
is
16
11
or
45,949,729,863,572,161
45 quadrillion
Xn+1 = 1/cosXn2
(defun cope (n seed)
(if (zerop n)()
(let ((test (/ 1 (cos (* seed seed)))))
(cons (round test)
(cope (1- n) test)))))
? (cope 40 2)
(-2 -1 -2 -4 -1 -11 -3 2 -1 10 1 -2 -1 2
-9 -2 1 2 29 1 -7 3 -9 -4 1 2 -2 -1 2
-1 3 1 -2 -1 2 4 1 2 -2 -1)
Tom Johnson’s
Formulas for
String Quartet
Iannis Xenakis
Metastasis
(defun normalize-numbers (numbers midi-low midi-high)
"Normalizes all of its first argument into the midi range."
(normalize numbers
(apply #'min numbers)
(apply #'max numbers)
midi-high
midi-low))
(defun normalize (numbers data-low data-high midi-low midi-high)
"Normalizes its first argument from its range into the midi range.”
(if (null numbers) nil
(cons (normalize-number (first numbers) data-low
data-high midi-low midi-high)
(normalize-number (rest numbers) data-low
data-high midi-low midi-high))))
Class
Sonifications
Assignment
Sonify a mathematical process
e-mail me a MIDI file
turn in your code.