This paper presents a novel navigation system for 24-tone equal temperament (24TET), derived from the theory of well-formed scales (Carey Krantz & Douthett, 2000). We demonstrate that the units modulo 24, U (24) = 1, 5, 7, 11, 13, 17, 19, 23, form four self-dual pairs of interval generators: +1, +23, +5, +19, +7, +17, and +11, +13. For the diatonic region (d = 7), the equation dg ≡ ±1 (mod 24) yields two solutions: g = 7 and g = 17. These generators induce four independent unidirectional proximity cycles, enabling smooth modulation between two interlocked 12TET subsystems (even and odd pitch classes). We introduce the Harmonic Compass — a structural framework comprising these four cycles, seven well-formed modes of region R (24, 7), nine triad classes, and six structural anchors (diminished seventh chords). We also introduce a two-level harmonic architecture: (1) navigational scales (well-formed modes for modulation) and (2) functional modes (including a 10-step microchromatic mode S10 = 0, 3, 6, 8, 11, 13, 15, 18, 20, 23 derived from Jędrzejewski’s limited transposition sets). This work extends the mathematical theory of musical scales to microtonal temperaments while preserving the perceptual smoothness of diatonic structures.
Ilia Prikhodko (Fri,) studied this question.