Equal Temperament
Age 14 to 16
Challenge Level
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
$0.0016929231 (10\text{dp})$