Let me give a try for tempered scales. (In a tempered scale, the ratio of frequencies
for adjacent halftones is always $2^{1/(12)}$.)

In this case, we want a distance on $\mathbb R/\mathbb Z$ (with corresponding frequencies
given by $2^x/2^{\mathbb N}$, invariant under rotations
such that $\frac{5}{12}$ is close to $0$ (in $\mathbb R/\mathbb N$).
Say we want it to be closer to $0$ than any other element among 
$\frac{1}{12},\frac{2}{12},\frac{3}{12},\frac{4}{12},\frac{6}{12}$.
One possible recipee: Choose strictly positive rationals
$\alpha_1,\dots,\alpha_6\in (0,1/12]$ encoding pleasantness of intervals
(say $\alpha_5<\alpha_4<\alpha_3<\alpha_2<\alpha_1<\alpha_6$)
and set $d(y,x)=d(x,y)=\min(y-x,\alpha_i+\vert y-x-i/12\vert)$ if $y>x$ with $y-x\leq 1/2$.

The graph of the distance function is a symmetric 
function modelling the skyline of a symmetric island consisting entirely of
montains with slope $1$. Pleasant intervals correspond to passes.

Remark: The requirement $\alpha_i\leq 1/12$ is slightly to strong: we need
only $\alpha_i\leq \alpha_j+\vert i-j \vert/{12}$.