#1
So, I'm having a lot of trouble grasping the concept of ratios and fractions being used in music. Like the ratios here. Can anyone explain these to me? I'm needing to learn it for a school project.
#2
The ratios refer to the relationship in frequency between the two notes.

For example, the most common reference for tuning is a note "A", which is 440Hz - the note an octave higher would be exactly double or "2:1" the frequency - 880hz.
#3
With any ratio X:Y just take the first number and divide it by the second number and you'll get how many times bigger the answer will be. This works because a ratio is really just another way of writing a fraction, i.e. X:Y = X/Y

The octave ratio is 2:1. In this case we have 2 divided by 1 = 2, so anything in the ratio 2:1 is twice as big, e.g. you get 880Hz from 440Hz for an octave on A.

The perfect fifth ratio is 3:2. In this case 3 divided by 2 = 1.5, so any note which is a perfect fifth up must have a frequency 1.5 times higher. For our A at 440Hz this will be 440 x 1.5 = 660 Hz.
#4
The article then uses a four-term ratio (12:9:8:6) to describe four different string lengths of a hypothetical musical instrument.

This means the lowest-sounding string is 12 units long, the next string up is 9 units, the next is 8 units long, and the highest string is 6 units long. (The units can be anything - inches, centimetres, etc.)

On this imaginary instrument the strings will be at equal tension and will be of the same thickness as each other. This means only their different lengths will give them different pitches. The pitch of each string will be inversely proportional to its length. Eg. a string half as long will be twice as high pitched.

The lowest two strings' pitches are in the ratio 12:9 = 12/9 = 4/3 (which is a perfect fourth apart in pitch).

The middle two strings are in the ratio 9:8, a major second apart.

The highest two strings are 8:6 = 4/3 apart, another perfect fourth.

If you plucked the lowest sounding string and the second-highest their pitch ratios would be 12:8 = 3/2, our old friend the perfect fifth.

The four-term ratio looks harder to understand but it can be thought of as a series of fractions:

12/9, 9/8, 8/6

When you multiply these three fractions together you'll get the answer 2 because the denominators and numerators cancel each other out leaving only 12/6 =2. This means from the lowest string to the highest string there's an octave.

To understand all this better it's important to find out how the intervals (perfect fifth, etc.) can be derived from a single vibrating string.
Last edited by Jehannum at Nov 29, 2011,
#5
Where the whole string vibrates we get the fundamental note.

Where the string vibrates in two halves we get the first harmonic, the octave. Frequency ratio = 2/1

These two notes form the start and endpoint of the scale we are constructing.

Where the string vibrates in three equal lengths we get a note with three times the frequency. If we lower this note by an octave (divide frequency by 2) it will fit into our scale and we get the perfect fifth above the fundamental. Ratio = 3/2

Where the string vibrates in four equal lengths we just get a note an octave above the octave.

Where the string vibrates in five equal lengths we get five times the frequency of vibration. To fit this into our scale we have to go two octaves lower (divide by 4). We thus get the major third above the fundamental. Ratio = 5/4

Where the string vibrates in six equal lengths we get a note an octave above the perfect fifth.

Where the string vibrates in seven equal lengths we get an interval that is not used in Western music.

Where the string vibrates in eight equal lengths we get a note an octave above an octave above the octave(!)

Where the string vibrates in nine equal lengths we have nine times the frequency. Going three octaves lower (dividing by 8) gives us a major second above the fundamental. Ratio = 9/8

10 lengths = 5 lengths = major third

11 lengths = not used in Western music

12 lengths = perfect fifth

13 lengths = not used in western music

14 lengths = 7 lengths = not used

Where the string vibrates in fifteen equal lengths, fifteen times the frequency. Three octaves lower gives us a major seventh above the fundamental. Ratio = 15/8

Now consider the two major scale intervals we haven't derived: the perfect fourth and the major sixth with frequency ratios are 4/3 and 5/3 respectively. They can’t be derived by "octaving down" i.e. dividing by powers of 2, like the others. Their respective notes are not in the harmonic series of a string.