Art, Math, Music and Computer Programming Languages

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Art, Math, MusicandComputer Programming
Languages

• Creativity with Patterns

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Programming with Functions
Everyone understands functions f(x) x 1
g(x,y) sine(x) - y abs(x) if x lt 0 then
-x else x Functional Programming uses functions
(and nothing else!) to write computer
programs. We use functions to create some simple
but interesting programs that explore things like
math or music or art
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Programming languages

• If you use languages like Java or C you have to
  tell the computer a lot about how to do
  something. You need to learn a lot to do a
  little but you can do almost anything if you work
  really hard
• We are interested in languages where you dont
  have to say how to do things - anyone can write
  programs in these languages! Our languages deal
  with specific domains - they are not meant to do
  everything.

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Everything is a Function
Why functions? Because we can represent just
about everything with them Pictures are
functions from points onto colors Music is a
function from time to sets of notes Math is a
study of functions In all of these areas, we
want to capture patterns and give them names
Reflection Shapes Scales Rotations
Repetition
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Notation

• Using a computer requires precise notations.
• f x y -- function f applied to arguments x
  and y
• (x,y) -- A 2-dimensional point (or vector)
• f x x1 -- A function definition
• f (x,y) (y,x) -- f has one argument, a point

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Functional Images

• We use a language called Pan (Picture Animation)
  to create pictures.

A picture is a function from a coordinate (place
on the X-Y plane) to a color. Computer geeks
write things like this a Picture has type
Point -gt Color You should know about
coordinates but ... Whats a color?
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Colors
We describe colors using 3 numbers, each between
0 and 1 Amount of Red Amount of Green
Amount of Blue Example Yellow Red Green
rgb 1 1 0 We can also add Transparency 0
clear, 1 solid rgba 1 1 0 0.5 --
semi-transparent yellow
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A Picture!

• We can define a simple picture like this
• pic (x,y) if abs x abs y lt 100
• then black else white
• What does this define?
• The coordinate system is in pixels and the origin
  is at the center of the window.

Test1
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Another Picture!

• More colors! Lets change the black to a color
  that varies with location
• pic (x,y) if abs x abs y lt 100
• then rgb 0.5 (abs x/100) (abs y/100)
• else white
• rgb defines a color from the red, green, and blue
  components.

Test2
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Adding Controls

• r lt- slider red (0,1) 0
• pic (x,y) if abs x abs y lt 100
• then rgb r (abs x/100) (abs y/100)
• else white
• slider uses a label, a range, and an initial
  value to define an interactive controller

Test3
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Transformations
A function that maps one point onto another is a
transformation A Transformation has type
Point -gt Point moveIt (x,y) (x1, y1) What
is moveIt (2,3)? twistIt (x,y) (y, -x) What
is twistIt (2,3)
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Visualizing Transformations
So how can we see a transform on points? One
way is to use it as a lens to view an
image. Heres a transformation f (x,y) (xx,
yy) f (0,0) (0,0) f (1,1) f(-1,1) f(1,-1)
f (-1,-1) (1,1) f (2,2) (4,4) So, if we
look at (2,2) well see whats at (4,4)
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Heres what happens
y
x
Test4
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Symmetry
The image symmetric about both the x and y
axis f(-x,y) f(x,y) Can you prove
this?? f(x,-y) f(x,y)
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Polar Coordinates
A different way to locate points on a plane Use
distance and angle to the origin
d
angle
Simple transformation in polar coordinates have
interesting effects
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Swirling
This transformation adds extra rotation as
points move out from the origin f (distance,
angle) (distance, angle k distance) k is a
parameter we can adjust this and watch how the
picture changes.
Test5
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Altering the Distance
f (r, theta) (g r theta, theta) What if g
r theta r k (k is a parameter) g r
theta r ktheta g r theta rrk
Test6
Test7
Test8
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Rippling
Move further or closer to the origin depending on
the angle g r theta r (1 ksine (n
theta)) The sine function generates a ripple.
Test9
Sine tells you how high up you are on the unit
circle given an angle.
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Ramps
A periodic function repeats itself. One of the
simplest periodic functions is a series of ramps
from 0 to 1, repeated over and over ramps x
x when 0 lt x and x lt 1 x ramps
(x1) when x lt 0 ramps (x-1) when x gt 1
A more useful version of this function is
parameterized by its period pramps p x p
ramps (x/p)
y ramps p x
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Using Ramps
p lt- slider Size of square (10,300) 100 f (x,y)
(pramps p x, pramps p y) This replicates the
image in the square of size s over the entire
plane (tiling)
Test10
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A Sawtooth
These function is symmetric teeth(x) x when
0 lt x and x lt ½ 1-x when
½ lt x and x lt 1 ..... plus rest of
ramps We can also parameterize by the period
(sawtooth) Tiling the angle in polar coordinates
kaleidoscope f (distance, angle)
(distance, sawtooth(width, angle))
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Using the Sawtooth
Tiling f (x,y) (pteeth p x, pteeth p
y) Circular reflections f (r, theta)
(pteeth p r, theta) Kaleidoscope f
(t, theta) (r, pteeth p theta)
test11
test12
test13
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The Grand Finale
Define factor to take apart a number x as
follows (big, little) factor p x
where big is a multiple of p ,
-p/2 lt little lt p/2, and big
little x Now, apply a warping t such that
slicer p t (x,y) (x bigx, y bigy)
where (bigx, littlex) factor p x
(bigy, littley) factor p y (x, y) t
(littlex, littley)
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Apply this to the following transformations
f (r, theta) (rk, theta) f (r, theta)
(0, 0)
test14
test15
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Haskore

• Motivation
• Traditional music notation has many limitations
• Unable to express a composers intentions.
• Biased toward music that is humanly performable.
• Unable to express notions of algorithmic
  composition.Haskore (Haskell Score) is a
  library of Haskellfunctions and datatypes for
  composing music.

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Defining Music

• Can we create music with functions? Of course!
• NoteName Cf C Cs Df D Ds Ef E
  Es Ff F Fs
• Gf G Gs Af A As Bf
  B Bs
• Music
• Note (NoteName, Integer) Duration Attributes
  -- a note
• Rest Duration -- a rest
• Music Music -- sequential
  composition
• Music Music -- parallel
  composition
• Tempo Float Music -- scale the tempo
• Trans Integer Music -- transposition

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C Major Scale
For convenience, c oct duration attributes Note
(C, oct) duration attributes cmaj c 3 0.2
d 3 0.2 e 3
0.2 .. c 4 0.2
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Simple Transformations
Add a parallel line 7 notes higher pfifth m
m Trans 7 m cmaj5 pfifth cmaj
againFaster m m (Tempo 2 m) cmaj5twice
againFaster cmaj5 cmaj5four againFaster
cmaj5twice
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Function Composition
Function composition notation (f . g) x f
(g x) Make a pipeline of transformers. pfifth
. Trans 5 . Tempo 2 -- Music transfomers
means double the tempo, transpose up 5, and add
a parallel part a fifth
higher Such notation makes it easier to work
with functions!
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Types
Types help us understand functions x Music
-- x is a piece of music y Music -gt Music
-- y a function that accepts music
and returns music (a music
transformer) z Integer -gt Music -gt Music --
z has two arguments, an
integer and some
music z 3 -- if passed only one argument, you
get a music transformer Using only some of the
arguments is called currying named after Haskell
Curry
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Simple Iteration
We use functions to capture music structures One
such structure is simple iteration rpt
Integer -gt Music -gt Music rpt 0 m Rest 0
--- a null piece of music rpt n m m rpt
(n-1) m Use recursion to define a function that
iterates n times cmaj4 rpt 4 cmaj What does
rpt 3 . rpt 3 do?
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More Complex Patterns
Why repeat the same thing exactly the same? We
can add a function that changes the music with
each iteration rptm Integer -gt (Music -gt
Music) -gt Music -gt Music rptm 0 f m Rest 0 rptm
n f m m rptm (n-1) f (f m) cmr rptm 4
pfifth cmaj cmr1 rptm 4 (pfifth . Tempo 1.5)
cmaj
This changes the music!
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More Examples
A function to recursively apply transformations f
(to elements in a sequence) and g (to accumulated
phrases) rep (Music -gt Music) -gt (Music -gt
Music) -gt Integer -gt Music -gt Music rep f g 0 m
Rest 0 rep f g n m m g (rep f g (n-1) (f
m)) An example using "rep" three times,
recursively, to create a "cascade" of
sounds. run rep (Trans 5) (delay
tn) 8 (c 4 tn) cascade rep (Trans 4) (delay
en) 8 run cascades rep id (delay sn) 2
cascade waterfall cascades revM cascades
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How Did That Work?
rep f g 0 m Rest 0 rep f g n m m g (rep f
g (n-1) (f m)) At each step, apply f going
down and g coming back Play m at the same time
as the rep part If n 2, we get rep f g 2 m
m g ( rep f g 1 (f m))
m g ( f m g (rep f g o (f (f m))))
m g ( f m g (Rest
0)) run rep (Trans 5) (delay tn) 8
(c 4 tn)
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The Grand Finale
Write a function that plays a piece of music with
the same music played twice as fast an octave
higher, and so on.
m
Higher Faster
m
m
m
m
m
m
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The Program
pyr 1 m m pyr n m m
(rpt 2 . Trans 12 . Tempo 2 . pyr (n-1)) m tune
c 4 0.1 g 4 0.1 e 4
0.1 g 4 0.1 alldone pyr 4 (Tempo 0.3
tune)
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Gloveby Tom Makucevichwith help from Paul
HudakComposed using Haskore, a Haskell DSL,and
rendered using csound.
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Thank You!
Join us tomorrow to get some hands on
experience with tunes and toons. Help us build a
web page. Bring your creativity! Embarrass
your friends. Compose music. Play with
pictures. We will have a camera available