Equal temperament
Problem
The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. The keyboard is a repeating pattern of seven white keys with a black key in between two whites in five places. If you play any note, twelve points higher will be exactly an octave on.
The piano is made this way to allow a key change in the music without re-tuning the instrument.
C to G is the two to three ratio, explored in the earlier problems. It's called a "fifth", because it's the fifth note along the white keys starting from C .
Is C to G an exact two to three ratio, or just off and, if so, by how much?
Getting Started
Equal temperament means $12$ times you scale up to be equivalent to an overall multiple of $2$
You want to go up seven points to go from C to G
Student Solutions
Thank-you Julian from Wilson's School for this clear explanation :
Consider the chromatic scale with $12$ equal intervals, starting with one note (for example C) and ending with the same note, but an octave above (so C again). We are told that the ratio between each of the notes is the same, and we were told in the previous problems that any note an octave above will be $1/2$ the note one octave below.
Let's use the value $2$ for the bottom note of our scale, so the octave above will be $1$.
We know that there are $12$ equal ratios in between these two in the scale, so to evaluate the note $n$ steps lower on the chromatic scale from any position we use the expression $2^\frac{n}{12}$ . This means take the twelfth root of $2$, which gives the multiplier for one step, and raise it to the power of $n$ to find the multiplier for n steps.
For example, with the bottom note: $2^\frac{12}{12} = 2 $
And with the top note: $2^\frac{0}{12} = 1 $
Therefore, to find out the interval of a fifth, which misleadingly has 7 equal ratios (or semitones), we work out: $2^\frac{7}{12} = 1.498307\ldots $
A perfect ratio of $3:2$ would give the note $1.5$
Therefore, the interval of a fifth is less than $3:2$ by $0.0016929231 (10\text{dp})$
Teachers' Resources
The topic of music can make a good connection between science and mathematics
The nature of sound and the working of the ear are rich areas of applied mathematics.
The ratio emphasis follows from harmonics or overtones and rests on ideas like lowest common multiple.
One teacher has offered the following comment : This Stage 4 work makes practice in multiplying and dividing fractions purposeful. And in school invites collaboration between the music and mathematics departments.