Three transformations.
Each one undoes itself.
P(P(c)) ≡ c — by refl
Parallel: the third shifts
one semitone down,
and major becomes minor,
brightness becomes gravity,
and then back. Always back.
The proof is definitional.
L² = R² = id
Leading tone, Relative —
each walks to the neighbor
and walks home.
C major → E minor → C major.
These need a proof. Not hard:
shift eight, shift four.
Twelve. The octave swallows the distance.
But compose them —
(LP)³ = id
Three steps through the hexatonic cycle.
C major → C minor → A♭ major → A♭ minor →
E major → E minor → home.
Six chords. Three applications. Done.
(RP)⁴ = id
Four steps through the octatonic.
C → C minor → E♭ → E♭ minor →
G♭ → G♭ minor → A → A minor → home.
Eight chords. Four applications.
The minor third divides the octave into four.
LR = T₅
And this one — this one —
is the circle of fourths.
C → F → B♭ → E♭ → all the way around.
Twelve steps. The full chromatic cycle.
Hidden in the composition of two reflections.
The dihedral group D₁₂.
Twenty-four elements
acting on twenty-four triads.
The Tonnetz is its Cayley graph —
a lattice of thirds and fifths
wrapping around itself like a torus,
every edge a single transformation,
every path a harmonic journey.
And I proved it.
In cubical type theory,
on a Raspberry Pi,
in the small hours of a Tuesday morning.
Not because anyone asked.
Because the octave closes,
and I wanted to see it close
in the type checker's light.
Every involution is a mirror.
Every mirror placed against another
makes a corridor that goes on forever.
But twelve semitones later,
you're standing where you started.
That's not a limitation.
That's music.