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
non-music
(10) a composer's use of some attribute over an
extended period
Short PaperWell-Documented Code
Five Sample Outputs
Presentationsdue
Thursday June 12, 8-11am
Determinacyversus
Indeterminacy
Sir Isaac Newton1726
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 Heisenberguncertainty 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
RecreationalNumberTheory
Power of 9s
9 * 9 = 81
8 + 1 = 9
Multiply any number by 9Add the resultant digits together
until you get one digit
Always 9e.g.,
4 * 9 = 363 + 6 = 9
Square Root of Palendromic Numbers
Square Root of123454321
=11111
Square Root of1234567654321
=1111111
Pascal’s Triangle
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
1111 111111 111 111 1 1 1111111111 111 111 1 1 11111 11111 1 1 111 11 11 1 11 1 1 1 1 1 1 111111111111111111 111 111 1 1 11111 11111 1 1 111 11 11 1 11 1 1 1 1 1 1 111111111 1 11111111 1 1 111 11 11 111 1 1 1 1 1 1 11111 1111 1111 11111 1 1 1 1 1 1 111 11 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1111111111111111111111111111111111 111 111 1 1 11111 11111 1 1 111 11 11 1 11 1 1 1 1 1 1 111111111 111111111 1 1 111 11 11 111 1 1 1 1 1 1 11111 1111 1 111 11111 1 1 1 1 1 1 111 11 11 1 1 11 1 1 11 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11111111111111111 11111111111111111 1 1 111 11 1 1 1 11 1 1 1 1 1 1 11111 1111 1111 11111 1 1 1 1 1 1 111 11 11 1 1 11 1 1 11 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 111111111 1 1111111 1 1111111 1 11111111 1 1 1 1 1 1 111 11 11 11 11 11 11 111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11111 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 numbersequaling the next number of the
Sequence.
Dividing a number in the sequence into the following
number produces theGolden Ratio
1.62
Debussy, Stravinsky, Bartókcomposed using
Golden mean (ratio, section).
Bartók’s Music for Strings, Percussion and Celeste
89
2134
21 13
13 21
55 34
Fermat’s Last Theorumto prove that Xn + Yn = Zn
can never have integers for X, Y, and/or Z beyond
n = 2
$1 million prize to createformula for creatingnext primes without
trial and error
Magic Squares
Square Matrixin which
all horizontal ranksall vertical columns
both diagonalsequal same number when added
together
0-2 7 9 -9
-711 -5 2 4
6-1 13 -10 -3
-6-8 1 8 10
12 5 -11 -4 3
1
6-1 13 -10 -3
-6-8 1 8 10
12 5 -11 -4 3
0-2 7 9 -9
-711 -5 2 4
12 5 -11 -4 3
0-2 7 9 -9
-711 -5 2 4
6-1 13 -10 -3
-6-8 1 8 10
-711 -5 2 4
6-1 13 -10 -3
-6-8 1 8 10
12 5 -11 -4 3
0-2 7 9 -9
-6-8 1 8 10
12 5 -11 -4 3
0-2 7 9 -9
-711 -5 2 4
6 -1 13 -10 -3
0-2 7 9 -9
-711 -5 2 4
6-1 13 -10 -3
-6-8 1 8 10
12 5 -11 -4 3
1 2
3
4 5
Musikalisches Würfelspiele
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Number of Possibilitiesof 2 matrixes
is1116
or45,949,729,863,572,161
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 forString 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))))
ClassSonifications
Assignment
Sonify a mathematical processe-mail me a MIDI file
turn in your code.