Axiomatic System — a system derived from indisputable mathematical axioms or rules. Examples include the Pythagorean system, generated entirely by the ratio 3:2 (pure perfect fifth), and the equal temperament systems, built from the x-th root of 2. Both are axiomatic: every pitch follows inevitably from the founding rule. This keyboard reveals the Pythagorean comma — the audible gap between C♯ (278.437 Hz) and D♭ (274.690 Hz) that equal temperament erases.
Tonal System — a mixed, positional system rooted in a chosen fundamental (key). Its seven core pitches are axiomatic; alterations (♭ lower, ♯ higher) may shift each degree toward the next without crossing it. Importantly, a natural (♮) note can itself act as an alteration — either descending or ascending — depending on its tonal context. For example, D♮ in G♭ major is a chromatic alteration: a chromatic semitone above D♭ and a diatonic semitone below E♭. Seven keys × three alterations (♭ ♮ ♯) = 21 tonal contexts.
The Keyboard — each white key is divided: upper 4/5 = perfect pitch · lower 1/5 = equal-tempered reference. A4 is undivided — the universal reference, identical in all systems. Between every pair of white keys — including E→F and B→C — sits a key with 3 zones: bottom = tempered (cyan) · middle = ♭ | ♯ · top = ♭♭ | ♯♯. E→F and B→C keys are ivory-coloured with turquoise tempered strip (¼-tone).
Tonal Position — the artificial horizon shows the function of each note within the active tonal context. Center line = axiomatic · horizon rising = tonally higher (rose) · horizon falling = tonally lower (blue) · gold = quarter-tone.
Recorder — captures exact Hz frequencies and real-time durations. No MIDI, no quantization, no 12-tone grid. Every recording is a pure sequence of Pythagorean frequencies in time.
QWERTY: z/a/q rows = octaves · Shift = flats · Number row = sharps