Fractals in Music

1
Fractals in Music

• It has been suggested that some classical music
  is fractal in nature.
• Misconceptions
• What are fractals?
• How do they relate to music?
• How is fractal music composed?

2
Misconceptions

• Fractal music is not as easy to notice as fractal
  geometry/graphics!
• Listening to a fractal music composition is not
  like looking at a fractal shape
• Fractals appear in music most in the general
  structure of what we as humans consider music
• Fractal music is about self-similarity!
• All music exhibits some degree of fractality

3
Fractals The Basics

• A fractal is an object or shape that exhibits
  self-similarity
• Usually created by a recursive function
• A magnified part of the fractal will appear the
  same as (or very similar to) the whole
• eg. Von Koch curve

4
Music Basics Note Pitch

• Each note has a specific frequency, or pitch
• In a scale, each note has a specific letter
  assigned to it, referring to the note's pitch
• For instance, the C scale notes are C D E F G A B
  C, in that order
• The D scale notes are D E F G A B C D
• To keep things simple, we'll use the C scale for
  our examples to avoid the use of sharps/flats

5
Music Basics Note Duration

• Each note in a piece of music is played for a set
  amount of time
• The fraction of a note refers to how much of a
  bar or measure the note is played through
• Eg. a whole note is played for an entire bar,
  and an 8th note is played for 1/8 of a bar
• Therefore it takes eight 8th notes (or rests) to
  play through a bar, four quarter notes, etc

6
How do Fractals Relate to Music?

• On a very basic level, we can see that all
  musical pieces are divided into bars
• Each bar usually exhibits a degree of
  self-similarity to other bars
• This property is fractal in nature the larger
  piece can be broken down into small pieces, each
  piece resembling each other and the whole.

7
Ex Beethoven's Moonlight Sonata
8
How do Fractals Relate to Music?

• We can map a numerical fractal function to a
  scale.
• eg. our function can be the sum of each digit in
  a binary number 0, 01, 10, 11, 100, 101, 110,
  111, 1000, 1001, 1010, 1011, 1100, 1101, 1110,
  1111 becomes
• 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3,
  4.
• Mapping this to a C-scale gives us the note
  sequence
• C, D, D, E, D, E, E, F, D, E, E, F, E, F, F, G
• Note that if we remove ever other note, we are
  left with C, D, D, E, D, E, E, F
• Repeating this, we are given C, D, D, E
• This sequence is self-similar!
• This technique is known as pitch-scaling.

9
Pitch Scaling

• Occurs when there is a logarithmic relationship
  between the number of occurences of each note
  pitch in the piece
• For instance, if the number of occurences of
  notes G, F, E, D, C, B, A are 64, 32, 16, 8, 4,
  2, 1 times each (respectively), there is clearly
  a logarithmic relationship (6, 5, 4, 3, 2, 1, 0)
• This means removing half the notes will maintain
  the general sound of the piece, as we will be
  left with 32, 16, 8, 4, 2, 1, 0 of the above note
  pitches! A piece with this pitch spread
  exhibits self-similarity

10
Duration Scaling

• Occurs when there is a logarithmic relationship
  between the number of occurences of each note
  duration in the piece
• For instance, if the number of occurences of 16th
  notes, 8th notes, quarter notes and half notes
  are 32, 16, 8, and 4, respectively, there is a
  logarithmic relationship
• Removing half the notes will exhibit
  self-similarity

11
Furthering Self-Similarity

• We can take this a step further and make each
  measure similar to the next
• For instance, keep the same order of note
  durations the same in each bar
• We could also keep the same order of note pitch
  changes the same in each bar

12
1/f Noise

• 1/f noise is a naturally occurring signal process
• It is found in heart beat rhythms,
  electromagnetic radiation from stars, and
  electronic devices
• Also known as pink noise, its name arises from
  having a frequency spectrum between white noise
  (1/f0) and red noise (1/f2)
• 1/f noise is found to be fractal frequency
  (pitch) scaling naturally occurs in 1/f noise
• This means the logarithmic relationship between
  1/f amplitude and frequency is linear, just like
  the scaling processes reviewed earlier

13
1/f Noise cont'd
14
1/f Noise cont'd

• 1/f noise can be approximated by a mathematical
  formula although it is often generated by
  filtering white noise
• We can use 1/f noise signals to generate fractal
  music!
• Mapping 1/f noise signals to pitches and
  durations will naturally scale our music in a
  logarithmic fashion, creating fractal music
• Richard Voss J. Clark claim that almost all
  music, when plotted in terms of pitch, show the
  tendencies of the 1/f noise spectrum

15
Examples
16
Examples (cont'd)
Examples (cont'd)
17
If we have time (and ample technology)...

• I'll play a sample of mainstream music that is
  obviously fractal in nature.

18
Bibliography

• 1/f noise in music Music from 1/f noise
• Richard F. Voss and John Clark
• 30 January 1976
• Fractal Geometry of Music
• Kenneth Hsu and Andreas Hsu
• 31 October 1989
• Has Classical Music a Fractal Nature? - A
  Reanalysis
• Brian Henderson-Sellers and David Cooper
• Making Music Fractally
• Dietrick Thomsen
• Self-Similarity of the 1/f Noise Called Music
• Hsu and Hsu