Maths and music

Musical theory

Reading music

On the right, you can see the scale of C major written out in musical notation and with the corresponding notes on a keyboard shown.

The letters A, B, C, D, E, F, G are the only ones used for musical notes, and these repeat every 8 notes. The two C notes are an octave (8 notes) apart, and sound the same, except that one is higher (the C on the right) than the other.

The position of the note on the lines or between them determines which note a symbol stand for. So for instance, a note in the space between the two bottom lines is always F; a note on the middle line is always B, and so on. Middle C (the left-hand C in the diagram on the right) has a little line drawn through it. The symbol to the left of the notes (called a treble clef) means the notes are played with the right hand on a keyboard.

Whether the notes are open or closed, whether they have stems or not, determines how many beats they are held for. The open notes shown are called semi-breves, and last for a count of 4 beats. On the left you can see notes of different lengths. The open note with a stem, which is called a minim, is lasts for a count of 2 beats. The closed note with a stem, a crotchet, lasts for a count of 1 beat, and the closed notes with stems and joined together are quavers, and last for a count of 1/2 a beat.

On the left you can see a bar in each of four-four time and three-four time. Four-four time means there are four beats in a bar (the line signifies the end of a bar), and such music is counted 1 2 3 4, 1 2 3 4, .... Three-four time means there are three beats in a bar, and is counted 1 2 3, 1 2 3, .... You have to have notes giving the correct number of beats in each bar. So in the four-four bar, we have two crotchets (worth 1 beat each) and a minim (worth 2 beats). In the three-four bar, we have a minim (worth 2 beats) and a crotchet (worth 1 beat). The sign on the left in front of the time signature indicates that the notes are to be played on a keyboard with the right hand.

In Composing music with numbers one of the suggestions you might like to follow-up is that of using the ratio 1:2:4:8:16 to determine the lengths of notes. The diagram on the right shows how to show this in musical notation, where there are four beats in a bar. The four filled in notes with stems are crotchets and are worth one beat each. The two open notes with stems are minims and are worth two beats each. The open note without a stem is a semibreve and is worth four beats. The second and third lines of music demonstrate how musicians show notes longer than four beats. The second line shows two semibreves tied together with a line - this means you do not sound the second note, but count eight beats for the first note. The third line shows four semibreves tied together with lines - this means you only sound the first note, and count 16 beats in total. Any note can be tied with another, provided they are of the same pitch (those shown here are all A's).

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Some musical theory

If you are unfamiliar with musical scales, look at the keyboard on the right. The scale of C major starts on the left-hand note labelled C, then moves up one white note at a time, through D E F G A B and back to C. This C is an octave higher than the C we started from. The C in the middle of the keyboard is known as middle C.

The distance from a white note to the black note just above or below it is called a semitone, and the distance between two white notes which have a black note between them is called a tone. There are two occasions when there is no black note between two white notes - E to F and B to C - these intervals are both semitones also.

It will help you in the work below if you know something of musical intervals. An interval is just the distance from one note to another. A third is the third note in the scale, so starting from C this gives us E, and this interval is made up of 4 semitones. A fourth is the fourth note in the scale, so starting from C this gives us F, and this interval is made up of 5 semitones. A fifth is the fifth note in the scale, so starting from C this gives us G; this interval is made up of 7 semitones. The other intervals in the major scale are the second (eg. C to D), the sixth (eg. C to A), and the seventh (eg. C to B). Some intervals sound better than others when the notes are played together, but which you think sound good will depend on the type of music you like listening to.