This paper constructs a complete structural and compositional system for 24-tone equal temperament (24TET) centred on a 10-note structural mode (generative set) S10 = 0, 3, 6, 8, 11, 13, 15, 18, 20, 23. We demonstrate that S10 functions as a universal navigator: it contains all eight units of U (24) as internal intervals, making it capable of accessing every pitch class and every canonical generator from within its own structure. Its step-interval pattern harbours three invariant palindromes — its mode signature — present in all 24 transpositions. We define a proximity metric d (k) = |S10 ∩ (S10 + k) |, prove its symmetry, and show that the tritone transposition (k = 12, 600¢) shares 8 of 10 notes with the tonic — mathematically the second-highest overlap after the identity (whether this translates to perceptual closeness is an empirical question). This pair of poles forms a unique tonal breath: the tonic pole’s two unique notes span an interval of 13 quarter-tone steps (650¢, inhalation), while the tritone pole’s span 11 (550¢, exhalation). We introduce the Tritone Flip, a minimal compositional gesture shifting only two voices by 50¢ each while eight remain fixed. Reinterpreting six microtonally altered diminished seventh chords not as vertical sonorities but as cycle anchors built from alternating +5 and +7 generators, we show that they form a bridge network — a system of symmetry bridges connecting all 24 transpositions of S10 into a coherent graph. We then develop a taxonomy of compositional trajectories through this space, demonstrating how the graph’s structural features translate into concrete harmonic progressions. A set-specific selection principle is identified: of twelve theoretically possible alternating +5/+7 cycles in Z24, exactly half are activated by S10 as cycle anchors; the remaining six persist as silent cycles — geometrically valid but harmonically inert with respect to S10. The resulting Harmonic Sphere — a 48-state space parameterised by transposition and mode — provides composers with a systematic framework for microtonal tonality grounded in structural invariance, spectral resonance, and the breath-like motion between tonic and tritone poles.
Ilia Prikhodko (Fri,) studied this question.