The P23 Transform
P23 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, P23 pairs Cm7 (C Eb G Bb) with Cm7b5 (C Eb Gb Bb): 1 voice steps, 3 hold, and the lead costs 1 semitone.
| Voice | Becomes | Step |
|---|---|---|
| C | C | holds |
| Eb | Eb | holds |
| G | Gb | -1 |
| Bb | Bb | holds |
What each side spells
- Cm7: Eb6, Cm7
- Cm7b5: Cm7b5, Ebm6
The paper's printed cycle through all 51 chords takes this step 7 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.
P23 : [x, x + 3, x + 7, x + 10] ↔ [x, x + 3, x + 6, x + 10]
— 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 Cm7 and Cm7b5 in the calculator · Every chord one voice from Cm7