A musical note is often described entirely by its pitch, which is measured in hertz, written Hz for short.
Although the Hz of a note can be any positive real number, musically, if two notes differ in Hz by a factor of two, they sound very similar.
Therefore, we can model the "moduli space of a note's pitch" by the topological group $\mathbb{R}/\mathbb{Z}$.
In particular, we interpret any $x \in \mathbb{R}/\mathbb{Z}$ as the equivalence class of notes of $2^{(x + n)}$ Hz for all $n \in \mathbb{Z}$.
A "triad" is a set of three notes played together.
(A set of notes played together is often called a "chord".)
The order of the notes does not matter, although conventionally triads are written from least to greatest Hz.
Additionally, we consider the triads $\{x, y, z\}$ and $\{x + w, y + w, z + w\}$ to be the "same" chord for the purposes of music.
This is because we only care about the differences in Hz between each note played.
Therefore, one wonders: "what is the moduli space of triads?"
It turns out to be homeomorphic to a unit disk.
Let's parameterize the unit disk by $r$ and $\theta$ using polar coordinates. This is my explicit homeomorphism:
$f(r, \theta) = \{0, \frac{1 - r}{3}, \frac{2}{3} (1 - r) + \frac{\theta}{2 \pi} r\}$
Note that when $r = 0$, the value of $\theta$ becomes irrelevant, ensuring this function is well-defined.
One can check that it's, in fact, a homeomorphism, although the details are messy and whether I write them up is a function of how much people care.
I should at least provide the inverse map:
$g(\{x, y, z\})_r = 1 - 3 \min(|x-y|, |x-z|, |y-z|)$
(where $|x-y|$ is the shortest distance from $x$ to $y$.
I'm aware that, by default, this has no meaning in $\mathbb{R}/\mathbb{Z}$.
However, take our equivalence class representatives to be between $-\frac{1}{2}$ and $\frac{1}{2}$ for this to work.)
$g(\{0, x, y\})_\theta = \frac{y - 2x}{1 - 3x}$
(where $2x \leq y < 1-x$ if we take $0 \leq x, y < 1$ as equivalence class representatives.)
Note that when $g(\{x, y, z\})_r = 0$, it's because $\{x, y, z\} = \{0, \frac{1}{3}, \frac{2}{3}\}$, which is the only edge case for $g_\theta$.
Also note that $\{0, x, 2x\} = \{0, x, 1-x\}$ via adding $x$ to the elements of the latter set.