Maths and music

1
Maths and music

• The Fibonacci series applied to musical scales

2
Todays Aim Investigating Generative Music by
mapping the Fibonacci Pisano series onto
musical scales

• We can then use any number sequence with range 0
  to 4, as instructions for which notes to play
• Pisano series is based on Fibonacci rules, but
  Pisano series numbers wrap around within a range
  (I.e. 0 to 4) instead of growing like the
  Fibonacci series numbers

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Overview - Fibonacci series

• Overview of Fibonacci series (0,1,1,2,3,5,etc)
• Take 0 and 1, add them
• Add the result (1) to the previous number (1) to
  get 2
• Add that result (2) to the previous number (1) to
  get 3
• ...and so on
• One problem is that Fibonacci numbers grow very
  rapidly
• Only 12 steps before result gt100
• The Pisano series uses the Fibonacci rules, but
  with the numbers wrapping around using modulo
  arithmetic
• This is useful in musical and computational
  contexts
• Storing large numbers in computers is difficult
• There are lt12 notes in western musical scales,
  so using numbers larger than 12 is wasteful

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Overview - Modulo arithmetic

• The modulo operation gives the remainder after
  division
• A practical analogy is converting 24h time with a
  12h clock
• As you go round the clock nums gt12 wrap round to
  1...
• and 12 maps on to 0
• How does this apply to Fibonacci?
• Fibonacci series is 0,1,1,2,3,5,8,13,21,34,55,etc
  
• In modulo 12 - everything is fine up to
  "13,21,34,55"
• 13 maps to 1,
• 21 maps to 9,
• 34 maps to 10,
• 55 maps to 7, etc...

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Overview - Musical scales

• How do we apply Pisano to musical scales?
• Back to the numbered notes on the piano keyboard
• (C D F G A) with corresponding numbers
  0,1,2,3,4

• Compute the Pisano series in modulo 5
• 0,1,1,2,3,0,3,3,1,4,0,4,4,3,etc...
• Play the notes corresponding to the numbers
• C(0), D(1), D(1), F(2), G(3), C(0),
  G(3),etc...

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Overview Periodicity and scales

• An interesting consequence of modulo arithmetic
  is that the resulting Pisano series becomes
  periodic
• The modulo value is determined by the number of
  notes in the scale, so different scales will
  produce musical phrases with different periods
• Two scales with the same number of notes in will
  certainly produce the same number sequence, but
  the notes themselves may be different (e.g.
  major/minor scales)
• The term scale here is used very loosely and
  can have anywhere between 1 and 12 notes.
• Even the 12 note limitation can be relaxed if we
  venture into other tuning systems (such as the
  quarter-tone system)

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Software demonstration

• Generative music in PureData

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Overview - Worksheet

• Worksheet divided into 2 sections
• Guided exercises
• Guidelines/starting points for investigation
• Worksheet also lists many example scales
• If you have any questions, feel free to ask a
  lecturer

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Overview - Worksheet
Read the worksheet fully and avoid skimming!

• The guided examples are designed to help you
  understand each step, and are therefore quite
  wordy
• Missing out steps can easily lead to confusion!