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.