Download 1. The Musical Scale and Fibonacci 2. Fibonacci

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FIBONACCI NUMBERS AND MUSIC
There is too much here for you to do everything in the time available. I suggest you do the
first section, and then choose one of the others to work on. Musical Frequencies is harder, and
involves working with fractions – good practice!
1. The Musical Scale and Fibonacci
If you have a keyboard available, look at the scale from one C
to the C an octave above it (or use the image on the right).
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Count how many white notes there are, and how many
black notes.
How many notes are there in total between one C and the other?
How are the black notes grouped?
Your answers will depend on whether you counted the C twice – what does this say about
the assertion that musical scales are based on Fibonacci Numbers?
2. Fibonacci Compositions
If we label the notes with numbers, we can use Fibonacci Numbers to compose a tune, using
some ‘clock’ arithmetic.
On a 12-hour clock, we don’t use any numbers higher than 12 for the hours, so if we want to
know what the time will be in 10 hours if it is 11 now, we add on 10 to 11, but then take off 12:
11 + 10 = 21, 21 – 12 = 9
Work out what the time will be in each of these examples:
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13 hours
25 hours
14 hours
29 hours
after 9
after 7
before 6
before 10
Note that 0 and 12 are equivalent on a 12-hour clock, and so are 1 and 13, 2 and 14, and so on.
For our Fibonacci composition, we are going to use clock arithmetic, but with an 8-hour clock.
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Write out the sequence of the Fibonacci Numbers as far as you know
them.
Give each number a note, using the table on the right – where the
numbers are greater than 7, use clock arithmetic to get a value
between 0 and 7.
Now play your tune! You might want to get some more Fibonacci
Numbers to make it a bit longer.
You can improve your tune by thinking about the lengths of the notes as
well as their pitch, so that you have a more interesting rhythm.
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Note
Number
Middle C
0
D
1
E
2
F
3
G
4
A
5
B
6
Upper C
7
Decide on a method to use Fibonacci Numbers to create a rhythm for your tune.
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3. Musical frequencies
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If you have a violin or other stringed instrument available, and someone who knows where
the notes are, measure the lengths of string which produce notes such as C, A, G, F, … and
see how they are related to each other.
The Greek mathematician, Pythagoras (c580-500 BCE), was part of a school of mathematicians
called the Pythagoreans. They made notes with lengths of string and noted the connection
between the length of the string and the note produced. They also discovered that the
relationship between different sound frequencies is the inverse of the relationship of the
lengths of the strings.
Suppose a particular length of string vibrates at middle C.
Length of string Note produced
1
1
2
2
3
3
4
Middle C
Upper C
G
Frequency of note
(relative to middle C)
1
2
1
3
2
F
E
5
4
G has a frequency 3/2 times that of middle C. F has a frequency 4/3 times that of middle C.
So how are the frequencies of F and G related?
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3
×
×
4
2 →G
F 
→ C 
3 3 9
× =
4 2 8
If F has a frequency 4/3 times that of middle C, then the frequency of middle C is its reciprocal
or inverse, 3/4. Multiplying the fractions shows us that G has a frequency 9/8 times that of F.
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Complete the table above and then use it to work out the frequencies of the notes in
the octave from middle C, ie. C to D, C to E, ... C to B, and C to C' (C' is the upper C one
octave above the first C).
Highlight the numbers in the Fibonacci sequence. How many of the relationships
involve Fibonacci Numbers?
You will also need to know that C to D is the same relationship as F to G, G to B is the same as
C to E, and C to A is the same as D to B.
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Now try finding the frequency ratios of adjacent notes, ie. C to D, D to E, E to F, and so
on, up to B to C'.
What do you notice?
Can you see why black notes on a keyboard are grouped in two’s and three’s?