The L15 Transform
L15 on C, voice by voice: the chords it pairs, the voice that steps, the source's own words, and every chord each side spells.
On C, L15 pairs C7 (C E G Bb) with C#dim7 (C# E G Bb): 1 voice steps, 3 hold, and the lead costs 1 semitone. The paper also names this step Q15, QQ51, N51.
| Voice | Becomes | Step |
|---|---|---|
| C | C# | +1 |
| E | E | holds |
| G | G | holds |
| Bb | Bb | holds |
What each side spells
- C7: C7
- C#dim7: C#dim7, Edim7, Gdim7, Bbdim7
The paper's printed cycle through all 51 chords takes this step 4 times.
As the source prints it
The most parsimonious transformations exchange two types of sevenths moving just a note by a semitone or a whole-tone.
L15 : [x, x + 4, x + 7, x + 10] ↔ [x + 1, x + 4, x + 7, x + 10]
Q15 : [x, x + 4, x + 7, x + 10] ↔ [x + 1, x + 4, x + 7, x + 10]
QQ51 : [x, x + 3, x + 6, x + 9] ↔ [x, x + 2, x + 6, x + 9]
N51 : [x, x + 3, x + 6, x + 9] ↔ [x, x + 3, x + 5, x + 9]The paper groups these names as one step:
P35 ∼ R35 ∼ R53 ∼ RR35 and L15 ∼ Q15 ∼ QQ51 ∼ N51.
— A Generalized Dual of the Tonnetz for Seventh Chords: Mathematical, Computational and Compositional Aspects, pp. 301–308; Sonia Cannas and Moreno Andreatta. The source, read 2026-10-25.
Rank every pairing of C7 and C#dim7 in the calculator · Every chord one voice from C7