Maths and music

Music and rational numbers
The just scale
The well-tempered scale
Summary

Additional notes on reading music and musical theory

The mathematics of musical scales

A scale is a regular sequence of notes. In most music in the west since the seventeenth century, such scales are commonly major or minor scales. The scale of C major (which uses just the white notes on a keyboard) is illustrated on the right, both in musical notation, and on a keyboard.

However, if the notes are to sound right when played, the instrument must be tuned correctly. For a stringed instrument, such as a violin or guitar, the player controls this, but with a keyboard, the notes have to be tuned in advance. Tuning means that the strings or the notes of the instrument vibrate with the correct frequency. Higher notes have higher frequencies than lower notes.

The image to the left shows the organ at Ely cathedral, UK. It has four keyboards which are played with the hands and one keyboard (pedals) which is played by the feet. The white knobs on the right and left enable the player to choose different lengths of pipe, so that notes at different pitches can be chosen. Middle C (the left-hand C shown above) has a pipe of length 8 feet. A pipe of 16 feet will play a note an octave lower, a pipe of 4 feet will play a note an octave higher (C' shown above). This organ has pipes from less than 1 foot right up to 32 feet.

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The connection between music and rational numbers

If you work through the project on Numbers, you will discover that the ancient Greeks were interested in fractions and one of the main reasons they were so interested was because they were interested in music. The Greeks had stringed instruments which played notes when you plucked them. What they observed was that if a string sounded with a particular note, then when the length of the string was reduced, it sounded with a higher note. The Greeks noticed that certain fractions of the length of the string seemed to give particularly pleasing notes when you played them together with the original note, whereas others sounded quite unpleasant with the original note.

The following fractions were thought to be the best. Firstly, if you halve the length of the string you basically get the same note, but higher. This gives us the octave interval. They also noticed that if you divide the string in the ratio 2/3 you got another nice note which is a perfect fifth above the original and leads to a very nice chord. This observation was not only made by the Greeks, it was made by many other cultures as well. Another ratio which gives a nice note is 4/5 giving a note which is a third above the original.

In the 17th century, Galileo observed that the frequency of the note on a string is inversely proportional to the length of the string.

Q1 Take a minute to think about what that last statement means. Can you express it in your own words?

• What will happen if you halve the length of the string. What effect will that have on the frequency?
• What about if you divide the string by 2/3, what effect will that have on the frequency?
• What about if you divide the string by 4/5, what effect will that have on the frequency?
• Use the strings on a stringed instrument to try out Galileo's observation with some of these ratios.

Answers

The just scale

The Greek mathematician, Pythagoras (c580-500 BCE), was part of a school of mathematicians called the Pythagoreans. Amongst much else, they noticed that when the frequencies of notes are in a ratio which is a rational number, the notes they heard were related. They noticed that if they vibrated two strings, one double the length of the other, then the note sounded the same, apart from pitch. The two notes are an octave apart (such as C and upper C). If the frequencies were in the ratio 3:2, the two notes were a fifth apart, and if they were in the ratio 4:3, they were a fourth apart. This is perhaps the oldest quantitative law known. Having found the octave, fifth and fourth, the Pythagoreans then looked at the interval between the fourth and the fifth. They defined this as a tone.

Early music and musical instruments in the west were based on these intervals of rational numbers, and so, up to the seventeenth century, musical instruments were tuned to the just scale, the scale based on these ratios. All the intervals in the just scale are rational fractions, and once you know a few basic facts, you can work out all the frequency ratios in the just scale.

For instance, if you know that the interval from C to G has a frequency ratio of 3/2 and the interval from C to F has a frequency ratio of 4/3, you can use these facts to work out the frequency ratio of the interval from F to G. We work from F to C to G. The frequency of F is 4/3 that of C, and so the frequency of C is 3/4 that of F (the higher note has the higher frequency). Suppose the frequency of the F note is f. Then that of the C below it is 3f/4, and that of the G above is

This means that G has a frequency 9/8 times that of F, or that the frequency ratio of G to F is 9:8. The fact that we don't necessarily know what the value of f is doesn't matter, since we are just working with ratios.

Q2 Use the facts below to work out the frequency ratios of all the notes in a just scale based on 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).

• 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 C' 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.

Answers

Q3 When you've found the frequency ratios of these intervals, try finding the 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?
• Are all the frequency ratios of the tones the same?
• Can you see that there might be a problem with the just scale?

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The well-tempered scale

The problem with the just scale is that the frequency ratio between tones is not always the same, as you discovered in the last section. This became important once composers wanted to move from one key (we have been talking about the key of C major so far - other keys are based on each of the other notes) to another in a piece of music, and to harmonise their work. As a simple example, imagine you are a composer, and you start your piece of music with a fifth on C (that is, C and G played together), then move to a fifth on E (E and B played together), then to a fifth on D (D and A played together). If you play this on a keyboard which is tuned to the just scale, with the scale of C tuned exactly, the G string will vibrate with a frequency 3/2 times that of the C string.

But what about the intervals between E and B, and D and A?

Q4 Using your answers to Q2, work out the frequency ratio between E and the B above it, and between D and the A above it.

Answers

You should have found that while the intervals between both C and G, and E and B, have frequency ratios of 3/2, the interval between D and A has a frequency ratio of 40/27 (which is approximately 1.48, rather than 1.5, and is therefore a little over 1% out).

You might not think this matters very much, but composers soon discovered that intervals in some keys sounded really peculiar if their instruments were tuned to a just scale, which was correct in just one key. The problem was that their instruments were perfectly tuned for one key, but were badly tuned in others, sometimes so much so that the noise which occurred was called "The Wolf" as it sounded so bad.

Various attempts were made to get round this problem - that if you use the rational frequency ratios discovered by the Pythagoreans, your instruments will only play correctly in one key. The solution that was eventually adopted throughout western classical music, meant making all the notes slightly wrong, but none was so wrong it mattered. This is called the well-tempered scale, and it uses the same frequency ratios for all semitones.

Instead of using the Pythagorean rational frequency ratios, musicians decided that the octave, which consists of 12 semitones, should be divided into 12 equal intervals. (This idea is often attributed to JS Bach, a famous eighteenth century German composer). That means that every semitone will have the same frequency ratio as any other, and so will every tone (1 tone = 2 semitones).

Q5 If we set the interval between C and C' to be equivalent to that between 1 and 2 (C' has a frequency twice that of C), why is the interval between 1 and 2 divided up into units equal to the 12th root of 2, rather than twelfths? To investigate this, you will need to look at the frequency ratios of adjacent semitones from C to C'. This means using the black notes as well as the white notes. Black notes are called sharps and flats - for instance, the black note between C and D can be called either C sharp or D flat. A sharp is the black note above the given white note, while a flat is the black note below a given white note. (The image on the right shows one choice of name for each black note, but they could equally well be called D flat, D sharp, G flat, G sharp and A sharp).

• Work out the frequency ratios from C to C sharp (one semitone), from C sharp to D (the next semitone), from D to E flat (the next semitone), and so on, firstly using ratios in twelfths (so C to C sharp is 13/12; C to D is 14/12; C to E flat is 15/12; and so on).
• What do you notice? Do all the semitone intervals have the same frequency ratio?
• Now try the same thing, but this time use twelfth roots of 2 (so C to C sharp is the twelfth root of 2, or 2^1/12; C to D is the two-twelfths root of 2, or 2^2/12; C to E flat is the three-twelfths root of 2, or 2^3/12).
• What do you notice this time? Do all the semitone intervals have the same frequency ratio?

Answers

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 semitone interval. You can think of D as having frequency x² (two semitones from C) and E as having frequency x⁴ (four semitones from C).

• What do you think the frequency of E flat is?
• What do you think the frequency of A flat is?
• Use this information to work out the relative frequencies of the notes in C harmonic minor.
• Are they the same as in the C major scale?

Answers

Summary

The Pythagoreans based their frequency ratios on rational numbers, and this produced the just scale - perfect in itself, but which could not be applied to more than one key. A compromise, which allowed musicians to play in any key, without needing to retune their instruments (difficult to mid-way through a piece of music!) was to use the well-tempered scale, not perfect in any key, but nearly perfect in all. However, whereas the just scale is based on rational numbers, the well-tempered scale is based on irrational numbers - twelfth powers of 2 are irrational. Musical scales are mathematical objects, which depend very fundamentally on different types of numbers.

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