Prismatic Harmony · Vladan's Matrix
The Diminished
Family
One diminished chord sits at the center of all harmony. Everything else is a step outward.
Barry Harris' family · Tymoczko's Möbius geometry · one method
The wheel
Twelve relatives, one seed
A four-note diminished chord splits the octave into equal parts — the gold diamond at the center. Nudge any note a half-step and you land on a relative: down → a dominant, up → a minor-6, a pair → a major-6.
- dominant 7
- minor 6
- major 6
The idea, in three beats
The idea, in three beats
01 · THE CENTER
No home key
A diminished 7th is four notes a minor-third apart — perfectly symmetric. It divides the octave evenly, so it has no root to pull toward. In Tymoczko's geometry of chords, that puts it at the exact center of the space.
02 · THE FAMILY
Move one note
Shift a single note a half-step and the center resolves into a real chord: down gives a dominant 7, up gives a minor-6; move a pair and you get a major-6. One seed, twelve relatives — Barry Harris called it the family.
03 · THE TWIST
The tritone is a fold
The four dominants pair off a tritone apart, sharing the very same guide tones. Travel that tritone once and you're on the mirror side; twice and you're home. A tritone substitution is the family's Möbius half-twist.
In a real tune
Reframes you can hear
The engine keeps a chord's function and swaps it for a family relative — a resolved, singable recolor, not a random substitution.
D7Am6
keeps F♯ + C · sixth-diminished
Cmaj7Am6
the relative-minor-6 tonic sub
Gmaj7Em6
keeps B + G · tonic recolor
G7Dm6
rootless G13 · keeps B + F
The Möbius twist
The Möbius twist
The tritone substitution isn't a trick you memorize. It's the half-twist of the diminished family — the same guide tones, seen from the other side of the strip.
B7 and F7 hold the identical tritone, A and E♭. Rotate the seed a tritone and every dominant swaps with its partner while the guide tones stay fixed — go around once, you're flipped; twice, you're back. That return-flipped-then-restored is the mathematical signature of a Möbius strip.