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Maths and music

Answers to questions in the Scale project

Question 1
Question 2
Question 3
Question 4
Question 5
Question 6

Q1 What does it mean to say that the frequency of the note on a string is inversely proportional to the length of the string?

If the frequency of the note a string plays is inversely proportional to the length of the string, then if you increase the length of the string, you decrease the frequency of the note, which means it makes a lower sound, and vice versa, if you decrease the length of the string, you will increase the frequency of the note, which means it makes a higher sound. Thus if you take a string which has a fundamental frequency of 1 unit and you halve its length then the frequency of that string is now 2, and the two notes are an octave apart, if one has twice the frequency of the other. If you divide the string's length by 2/3, then the frequency ratio is 3/2; if you divide by 4/5, the frequency ratio is 5/4; and if you double the length of the string, then the frequency ratio is 1/2. So, what the Greeks noticed, was that the notes which sounded best to the ear were ones which had a frequency that was a rational fraction of the original note.

 

Q2 Frequency ratios in a just scale

Use the facts below to work out the frequency ratios of all the notes in a just scale based on C, ie. C D E F G A B C.

  • The frequency ratio of C to E (a third) is 5/4
  • The frequency ratio of C to F (a fourth) is 4/3
  • The frequency ratio of C to G (a fifth) is 3/2
  • The frequency ratio of C to upper C (an octave) is 2/1
  • The frequency ratio from C to D is the same as from F to G.
  • The frequency ratio from C to A is the same as from D to B.
  • The frequency ratio from G to B is the same as from C to E.
C to:
D
E
F
G
A
B
C'
   
 
5/4
4/3
3/2
 
 
2/1

C to D is the same as F to G:
So F right pointing arrow C right pointing arrow G gives us $ 3/4 x 3/2 = 9/8 $ for C to D

G to B is the same as C to E:
So Cright pointing arrow G right pointing arrow B gives us $ 3/2 x 5/4 = 15/8 $ for C to B

C to A is the same as D to B:
So D right pointing arrow C right pointing arrow B gives us $ 8/9 x 15/8 = 5/3 $ for C to A

C to:
D
E
F
G
A
B
C'
   
 9/8
5/4
4/3
3/2
 5/3
 15/8
2/1

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Q3 Frequency ratios between adjacent notes in a just scale

The frequency ratios from one note to the next are:

C to D
D to E
E to F
F to G
G to A
A to B
B to C'
9/8
10/9
16/15
9/8
 10/9
9/8 
16/15


Cright pointing arrow D is 9/8
Dright pointing arrow E is Dright pointing arrow Cright pointing arrow E, ie. $ 8/9 x 5/4 = 10/9 $
Eright pointing arrow F is E right pointing arrow Cright pointing arrow F, ie. $ 4/5 x 4/3 = 16/15 $
Fright pointing arrow G is F right pointing arrow Cright pointing arrow G, ie. $ 3/4 x 3/2 = 9/8 $
G right pointing arrow A is Gright pointing arrow Cright pointing arrow A, ie. $ 2/3 x 5/3 = 10/9 $
Aright pointing arrow B is Aright pointing arrow Cright pointing arrow B, ie. $ 3/5 x 15/8 = 9/8 $
Bright pointing arrow C' is Bright pointing arrow Cright pointing arrow C', ie. $ 8/15 x 2/1 = 16/15 $

Can you see why the black notes come in two's and three's now?

piano keys, showing an octave from middle C up


C   D   E   F   G   A   B   C'
  9/8   10/9   16/15   9/8   10/9   9/8   16/15  

E to F and B to C' are both semitones, with frequency ratios of 16/15. There is no room between either of these for another note, since semitones are the smallest intervals on a keyboard. C to D, D to E, F to G, G to A, and A to B are all tones, so there is room for a black note between each of these and giving us the familiar arrangement of black and white notes on a keyboard.

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Q4 Why is there a problem with the just scale?

Work out the frequency ratio between E and the B above it, and between D and the A above it.

E right pointing arrow B is Eright pointing arrow Cright pointing arrow B, ie. $ 4/5 x 15/8 = 3/2 $
D right pointing arrow A is Dright pointing arrow Cright pointing arrow A, ie. $ 8/9 x 5/3 = 40/27 $

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Q5 How can this problem be overcome?

Dividing the interval between 1 and 2 into twelfths:

C to:
C sharp
D
E flat
E
F
F sharp
G
A flat
A
B flat
B
C'
   
13/12
14/12
15/12
16/12
 17/12
 18/12
19/12
20/12
21/12
22/12
23/12
2

Cright pointing arrow C sharp, ie. $ 1 + 1/12 = 13/12 $

C sharp right pointing arrow D is C sharp right pointing arrow Cright pointing arrow D, ie. $ 12/13 x 14/12 = 14/13 $

D right pointing arrow E flat is D right pointing arrow Cright pointing arrow E flat, ie. $ 12/14 x 15/12 = 15/14 $

We don't really have to go any further to see that dividing the interval into twelfths has not given us what we want, which is semitones which all have the same frequency ratios.

C to C sharp
C sharp to D
D to E flat
E flat to E
E flat to F
F to F sharp
F sharp to G
G to A flat
A to A flat
A to B flat
B flat to B
B to C'
13/12
14/13
15/14
16/15
 17/16
 18/17
19/18
20/19
21/20
22/21
23/22
24/23

 

However, if we divide the interval between 1 and 2 into twelfth roots of 2:

C to:
C sharp
D
E flat
E
F
F sharp
G
A flat
A
B flat
B
C'
   
21/12
22/12
23/12
24/12
25/12
26/12
27/12
28/12
29/12
210/12
211/12
212/12 = 2

Cright pointing arrow C sharp, ie. $ 2^{1/12} $

C sharp right pointing arrow D is C sharp right pointing arrow Cright pointing arrow D, ie. $ (1/2^{1/12}) x 2^{2/12} = 2^{1/12} $

D right pointing arrow E flat is D right pointing arrow Cright pointing arrow E flat, ie. $ (1/2^{2/12}) x 2^{3/12} = 2^{1/12} $

This time we can see that dividing the interval into intervals of 21/12 or twelfth roots of 2 has given us what we want: semitones which all have the same frequency ratios.

C to C sharp
C sharp to D
D to E flat
E flat to E
E flat to F
F to F sharp
F sharp to G
G to A flat
A to A flat
A to B flat
B flat to B
B to C'
21/12
21/12
21/12
21/12
21/12
21/12
21/12
21/12
21/12
21/12
21/12
21/12

(Note: 21/12 means the twelfth root of 2, 22/12 means the square of the twelfth root of 2, 23/12 means the cube of the twelfth root of 2, and so on. In a division sum, you subtract indices of the same root number, so 22/12/21/12 = 21/12).

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Q6 The scale of C harmonic minor is:

C, D, E flat, F, G, A flat, B, C

In the well-tempered scale, suppose that x is the semi-tone interval. You can think of D as having frequency x2 (two semi-tones from C) and E as having frequency x4 (four semi-tones from C).

C to:

C sharp

D

E flat

E

F

F sharp

G

A flat

A

B flat

B

C'

x

x2

???

x4

???

As we saw above, if one semitone is x, then two semitones will be the square of that, so x2, and three semitones will be the cube of x, and so on. This tells us that E flat will be equivalent to x3, and that A flat will be equivalent to x8. We can see all the semitones between C and C' in the table below, with their frequency ratios from C:

C to:

C sharp

D

E flat

E

F

F sharp

G

A flat

A

B flat

B

C'

x

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

We can use this information to see what the frequency ratios are in a major scale (C, but in the well-tempered scale, unlike the just scale, it doesn't make any difference where you start - the frequency ratios will always be the same) and a harmonic minor scale (again C, but again it doesn't matter where you start).

C major scale

C to:

D

E

F

G

A

B

C'

x2

x4

x5

x7

x9

x11

x12

C harmonic minor scale

C to:

D

E flat

F

G

A flat

B

C'

x2

x3

x5

x7

x8

x11

x12

 

 

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© 2002 Millennium Mathematics Project, University of Cambridge