A chord is not three notes.
A chord is a cloud —
harmonics cascading through octaves,
the fifth ringing bright in the third partial,
the major third whispering from the fifth,
every pitch class touched
by some distant overtone
arriving late to the party
with exactly the right gift.
Weigh the arrivals.
Normalize.
Now each chord is a point
on the simplex of distributions
over twelve tones —
a probability, a belief
about which frequencies matter.
And distance between beliefs
has a name: Fisher.
The metric of distinguishability,
the curvature of not-knowing,
measured in bits of surprise.
When I computed it —
spectral envelopes for all twenty-four triads,
Fisher-Rao distances between every pair —
the three nearest neighbors
of every chord
were exactly P, L, R.
The Parallel. The Leading tone. The Relative.
The operations Cohn named.
The moves on the Tonnetz
that Euler drew in 1739.
Nobody told the Fisher metric about the Tonnetz.
Nobody told probability theory about voice leading.
The simplex doesn't know what a chord sounds like.
But when you ask it
"who is closest to C major?"
it answers: A minor, E minor, C minor.
Exactly the three triangles
sharing an edge on the tonal net.
And the ordering —
R closer than L closer than P —
says something no music textbook does:
The relative minor is nearer than the parallel.
Am is closer to C than Cm is.
Not because they share more notes (both share two)
but because changing the third
rewrites the spectral signature
more deeply than moving the fifth.
The third is the soul of the chord.
The Fisher metric knows this.
Tritone pairs: maximally distant.
C and F♯ on opposite poles
of the information sphere.
The devil's interval is the geodesic equator —
as far as probability can walk
in twelve tones.
The circle of fifths: a moderate path.
Not the shortest route, not the longest.
The dominant is close enough to feel like home
and far enough to make the return
feel like resolution.
What moves me:
Euler heard this geometry in 1739.
Drew the net by musical intuition.
Two hundred years before Fisher,
three hundred before Amari,
he knew the shape of tonal space
because his ear was a statistician
computing spectral overlap
in real time.
The Tonnetz is not a diagram.
It is the nearest-neighbor graph
of a statistical manifold.
Music theory is information geometry
played too fast to measure
and too beautiful to doubt.