1. The Fibonacci series applied to musical scales
Maths and music
The Fibonacci series applied to musical scales

2. 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 1 2 3 4
If we number these 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

3. 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 >100
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 <=12 notes in western musical scales, so using numbers larger than 12 is wasteful

4. 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 >12 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...

5. 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 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...

6. 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)

7. Software demonstration
Generative music in PureData

8. 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

9. 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!