The Harmonic Lattice: Compositional Use of the Harmonic Compass in 24TET

This paper constructs the Harmonic Lattice: a compositional system for 24-tone equal temperament (24TET) that extends the Harmonic Compass and the Harmonic Sphere into a complete tonal language. The Harmonic Compass is formalised as a pair of rings (E, O) partitioning Z24, with a free and transitive action of Z12 on the rotated odd ring. From this action arise harmonic lattices L(a, b), classified by three repetition rules (base, overlap, concatenation) into periodic, rational, and spiral types. The principal lattice L2(3, 5) is locally diatonic (7 notes per octave) and globally chromatic (24 pitch classes over 25 octaves); its dual character distinguishes a lattice from a scale. Every stretched lattice in the family projects onto all 24 pairs of the compasses of its own steps, establishing a universal projection principle. The map x ↦→ 14−x is an involution with two fixed points, 7 and 19, which are the poles of the compass symmetry. The modes S10 and S′10, together with the amplituhedronlike structure generated by their step combinations, provide a combinatorial template for building lattices. Chord cascades are constructed by anchor-bridge embedding operators Ep,q, whose types (5, 5), (5, 7), (7, 5), (7, 7) correspond to the midpoint bridges and asymmetric anchors of the Harmonic Sphere. Harmony and voice leading become two views of a single operator system. The Harmonic Lattice is offered as both a theoretical framework and a compositional instrument, integrating the harmonic resources of 12TET with the microtonal resources of 24TET.