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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!