### Rationals Between...

What fractions can you find between the square roots of 65 and 67?

### Root to Poly

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

### Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

# Equal Temperament

##### Stage: 4 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 $0.0016929231 (10\text{dp})$