What does the space induced by this unusual metric(?) on R/Z look like? The motivation for this question comes from music theory.  Dmitri
Tymoczko models "good" voice leading as minimizing distance between
pitches in successive chords.  While this theory works well for upper
voices, it does not work so well for the bass, which tends to move by
4ths and 5ths quite frequently.  (Tymoczko explicitly excludes the
bass from his model.)
Taking log (all logs are base 2 in this question) of frequency and
taking pitches which differ by an octave as equivalent, we get
$\mathbb{R}/\mathbb{Z}$.  For the upper voices, we want the standard
metric on this.
For the bass, we want moving by a fifth or a fourth - meaning by $\pm
\log (3/2)$ to be small.  So we want $d(x,x\pm\log(3/2))=k_1$, where
$k_1$ is probably somewhere around $0.05$.  To make this a metric
space, let's declare that $d(x,y)$ should be the minimum of $|x-y|$,
$k_1+||x-y|-\log(3/2)|$, and $k_1+||x-y|-\log(4/3)|$.
We probably want moving by a major third - meaning by $\pm\log(5/4)$
to also be small, but not as small.  I suspect $2k_1$ would make the
most mathematical sense, but any constant of roughly that magnitude is
fine.  Ditto for minor thirds - this would be movement by
$\pm\log(6/5)$, with a slightly larger constant.
If we do this, we might as well make all movements by $\pm\log(p/q)$
small if $q$ is small.
CLARIFICATION: I also want the standard metric to be one of the options for getting from $x$ to $y$.  So the distance between $0$ and $\sqrt{1/500}$ should be $\sqrt{1/500}$, while the distance between $0$ and $7/12$ should be $k_1+|\log(3/2)-7/12|$.  (Musically, $|\log(3/2)-7/12|$ is how far off an even-tempered 5th is from Pythagorean tuning.)
Question 1: Can one actually define something along these lines that
satisfies the triangle inequality?  (I don't think I actually have; I
probably need to take the minimum (or infimum) of some infinite
sequence, but am not entirely sure that works.)
Question 2: Assuming the answer to (1) is yes, what does this metric
space look like?  Can someone help me with a picture that seems less
exotic, perhaps comparing it to the Hawaiian Earring or something of
that sort?  In particular, what might the fundamental group look like?
My background: I'm a combinatorialist and algebraic geometer who happens
to be the one least unqualified here to be supervising an
undergraduate independent study on mathematics in music theory.  I did
the standard first year graduate courses in point set topology and
algebraic topology, but that was almost a dozen years ago.
 A: I doubt that you can do this in a way which addresses the intended application. For instance, you say you want bass movement by a fifth to be "small" because it is frequent in traditional music. You also want stepwise movement by a full or half step to be "small." But you can combine a fifth and a half step to make a tritone, and the triangle inequality would then force that to be "small." But this is not something you want. It seems to me that the triangle inequality is not an adequate assumption to impose on what you are trying to model.
A: 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 $\alpha_i\leq \alpha_j+\vert i-j \vert/{12}$.
