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Maths and music
The Fibonacci series applied to
musical scales
Today’s Aim: Map numbers from the Fibonacci series
onto notes in various musical scales
• Overview of Fibonacci series
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 itterations before result >100
o Pisano sequence uses the Fibonacci rules, but with the
numbers "capped" using modulo arithmetic
Overview - Modulo arithmetic
• Modulo operator reports the remainder after division
o x = mod(a,n)
o x will hold remainder after "a" is divided into "n"
• 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 The black notes on the piano keyboard (C# D# F# G# A#)
form a five note (pentatonic) scale
o Number these 0,1,2,3,4
• Compute the Pisano sequence in modulo 5
o 0,1,1,2,3,0,3,3,1,4,0,4,4,3,etc...
o Play the note numbers as they were assigned
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 sequences become periodic depending on the
modulo value
• 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 or 72TET)
Software demonstration
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
• The
Overview - Worksheet
Read the worksheet fully and avoid skimming!
The guided examples are designed to lead you through
each step, and are therefore quite wordy
o Missing out steps will easily lead to confusion!