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Maths and music
The Fibonacci series applied to
musical scales
Today’s Aim: Investigating Generative Music by
mapping the Fibonacci & Pisano series onto musical scales
• The black notes on the piano keyboard (C# D# F# G# A#)
form a five note (pentatonic) scale
• If we number these 0,1,2,3,4
0
1
2
3
4
• 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
Overview - Fibonacci series
• Overview of Fibonacci series (0,1,1,2,3,5,etc…)
o Take 0 and 1, add them
o Add the result (1) to the previous number (1) to get 2
o Add that result (2) to the previous number (1) to get 3
o ...and so on
• One problem is that Fibonacci numbers grow very rapidly
o Only 12 steps before result >100
• The Pisano series uses the Fibonacci rules, but with the
numbers “wrapping around” using modulo arithmetic
o This is useful in musical and computational contexts:
o Storing large numbers in computers is difficult
o There are <=12 notes in western musical scales, so
using numbers larger than 12 is wasteful
Overview - Modulo arithmetic
• The modulo operation gives the remainder after division
• A practical analogy is converting 24h time with a 12h clock
o As you go round the clock: nums >12 wrap round to 1...
o … and 12 maps on to 0
• How does this apply to Fibonacci?
o 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"
o 13 maps to 1,
o 21 maps to 9,
o 34 maps to 10,
o 55 maps to 7, etc...
Overview - Musical scales
• How do we apply Pisano to musical scales?
o Back to the numbered notes on the piano keyboard
o (C# D# F# G# A#) with corresponding numbers 0,1,2,3,4
0
1
2
3
4
• Compute the Pisano series in modulo 5
o 0,1,1,2,3,0,3,3,1,4,0,4,4,3,etc...
o Play the notes corresponding to the numbers
o C#(0), D#(1), D#(1), F#(2), G#(3), C#(0), G#(3),etc...
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
o 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.
o Even the 12 note limitation can be relaxed if we venture into
other tuning systems (such as the quarter-tone system)
Software demonstration
Generative music in PureData
Overview - Worksheet
• Worksheet divided into 2 sections
o Guided exercises
o Guidelines/starting points for investigation
• Worksheet also lists many example scales
• If you have any questions, feel free to ask a lecturer
Overview - Worksheet
Read the worksheet fully and avoid skimming!
o
The guided examples are designed to help you
understand each step, and are therefore quite wordy
o
Missing out steps can easily lead to confusion!