dividing an octave to $7$ instead of $12$

Question

Usually an octave is divided into $12$ parts based on the harmonic series(basic zeta function).

how can I calculate the frequency of a note if I divide the octave into $7$ parts?

$N_1=A_4(440Hz)$

$N_8=N_1*2=A_5(880Hz)$

$N_2....N_7???$

Thanks guys!

Answer 1

If you want equal musical intervals, then you need to multiply each frequency by a factor of $2^{1/7}$ to get the next higher frequency. Thus, your frequencies will be $440, 440\times 2^{1/7}, 440\times 2^{2/7}, \dots, 440\times 2^{7/7}=880$.

Edit:

The reasoning goes like this:

$$N_1 = 440 \\ N_2 = 440r \\ N_3 = 440r^2 \\ \ldots \\ N_8 = 440r^7 = 880$$ So that $$r^7 = 2$$

I warn you that the harmonies will sound terrible!