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You don't have to know any music theory to
do this question as all you need to know is explained here. You
only need to know about logarithms. The actual problems to solve
are given in bold italic text.
Diagram showing
part of a piano keyboard.
Wherever you are on the keyboard,
the interval between one note and the next with the same letter
name is always an octave. In the same way, the interval between C
and the G above it, or D and the A above it, or anything similar
(seven steps on the keyboard, including both black and white
notes), is always a fifth. Each of these intervals, the octave, the
fifth, and others, corresponds to a particular ratio of string
lengths which produce the notes. The octave corresponds to 2/1,
halving the length of the string. The fifth corresponds to 3/2,
taking two thirds of the length of the string. The interval from C
to F, called a fourth, has the ratio 4/3.'Fourth' and 'fifth' etc.
are musical terms and do not refer to the fractions 1/4 and
1/5.
The interval from F to G, between
the fourth and the fifth, has the ratio ${3/2 \over 4/3} = 9/8$.
This interval is called a tone and it is also the the interval from
C to D. Notes tuned in these ratios produce pleasant harmonies but
compromises have to be made in tuning the other notes because the
ratios do not relate to each other exactly. This problem is about
discovering what the compromises might be. The table below shows
some musical intervals and the corresponding ratios of the notes
with respect to the note C.
C 
D 
E 
F 
G 
A 
B 
C 
1/1
unison

9/8
tone

81/64 
4/3
fourth

3/2
fifth

27/16 
243/128 
2/1
octave

The Pythagorean Scale
You might expect there to be three tones in the interval from C to
F, changing the ratio by 9/8 then 9/8 again and then 9/8 a third
time, that is ${(9/8)}^3$.
How does this compare with
the ratio 4/3?
Pythagorean tuning kept the
ideal ratios for the octaves, fifths and fourths and tuned the note
E with a ratio ${(9/8)}^2$ and the notes A and B with the ratios
$(3/2)\times (9/8)$ and $(3/2)\times {(9/8)}^2$. This system
preserves the ratios of the fourth and fifth but produces a major
third, from C to E and a major sixth, from C to A, which sound
unpleasant. The Greeks were not interested in these intervals, but
when composers in the middle ages and Renaissance wanted to use
them, a new system of tuning was needed.
Just intonation (shown in
the table below) has much better major thirds and major sixths, of
5/4 and 5/3. It tunes the B so that the interval from E to B is a
perfect fifth and the interval from G to B is a perfect third. But
it does this by having two different sizes of tone: the intervals
CD and DE are different here, but they were the same in the
Pythagorean scale.
Find the ratio
corresponding to B. Also compare the DA ratio with the ideal fifth
(3/2).
C 
D 
E 
F 
G 
A 
B 
C 
1/1
unison

9/8
tone

5/4
third

4/3
fourth

3/2
fifth

5/3
sixth

? 
2/1
octave

Just Intonation
You might expect there to be six tones and twelve semitones in an
octave (because there are 12 notes) but again that is not exactly
so.
Find exactly how many tones
there are in an octave by finding what power of $9\over 8$ gives 2.
Find also exactly how many major thirds there are in an
octave.
In the
equal tempered
scale, the standard tuning nowadays, these mismatches
between different ratios are removed by defining the intervals
differently. The tone is made a little smaller, so that there are
exactly six of them in an octave. The semitone (half a tone) is a
twelfth of an octave: so its ratio is ${(2)}^{1/12}$. If, for
example, the note C is tuned with a string length of 32 units then
the C# is tuned with a string of length ${32\over
{(2)}^{1/12}}=30.2$ (to 3 significant figures).
This applet by Benjamin
Wardhaugh demonstrates the relationship between the string lengths
and the notes. The ratio ${(2)}^{1/12}$ is used to build up the
other intervals, so that each interval is a whole number of
semitones, and the ratio between its frequency and the frequency of
the lowest note in the scale is given by a power of ${(2)}^{1/12}$.
For example the fifth is ${(2)}^{7/12}$.
Instrument tuners customarily use a logarithmic unit of measure,
the cent, where 1200 cents are equal to one octave, a frequency
ratio of 2/1, so that a cent is a 1200th root of 2. The table below
shows the Equal tempered, Pythagorean and Just systems of tuning
given in cents showing how many cents the note lies above the
starting C.
Fill in the
table.

C 
D 
E 
F 
G 
A 
B 
C 
Equal tempered scale 
0 
200 
400 
500 
700 
900 
1100 
1200 
Pythagorean scale 
0 



702 


1200 
Just intonation 
0 

386 

702



1200 