interval

An interval names the distance between two pitches. Remove the concept of pitch and an interval has nothing to measure — it ceases to operate. The removal test passes: intervals need pitch as their constitutive referent.

Musical and physical intervals are derived from frequency ratios and differences. The concept of interval as a quantified relationship between two frequencies was built upon the pre-existing concept of frequency. Frequency existed first and fed into the interval concept.

Law 9, specific to the nearest accepted kind: octave's own accepted definition declares 'Octave is a human-made musical interval defined by a 2:1 frequency ratio between two pitches' — the kind is stated in the definition itself. interval's accepted def carves the named, measured distance between two pitches within a tuning system (step count, quality classification, size in cents); the octave is exactly such a named interval — the 2:1 distance, perfect in quality, 1200 cents. Pinned senses match both accepted defs (Law 11d): octave as the named 2:1 interval, interval as the musical distance concept. No nearer accepted kind exists on the board (musical-interval is unheld); no existing octave edges, so no ladder conflict.

Law 9 specific to nearest accepted kind; Law 11d pinned senses against both accepted defs. interval's accepted definition carves 'a named, categorized measure of the distance between two pitches within a given tuning system' with parameters step count, quality classification, and size in cents. semitone's own accepted definition declares it 'the smallest named interval of Western chromatic music — the distance between two adjacent pitch classes, named in two qualities' with parameters quality (diatonic/chromatic), size in cents (100 in equal temperament; ~112/~90 in just intonation), and notation. That is exactly interval's carve: a named, measured, quality-classified distance in cents. A competent speaker calls the semitone 'the smallest interval'. No nearer accepted kind exists on the board (no step or half-step entry; musical-interval unheld), so interval is the nearest rung.

Removal test (Law 8b): remove the interval construct — the named, categorized distance between two pitches per its accepted definition — and chord stops operating as a unit of harmonic analysis. Chord's own accepted definition fixes its quality parameter in interval terms: major (root–major third–perfect fifth), minor (root–minor third–perfect fifth), diminished, augmented. Without the interval construct a chord is an undifferentiated mass of simultaneous pitches: major and minor can no longer be told apart, chord symbols can no longer be read, and harmonic analysis — the mechanism by which chord persists (theory, notation, pedagogy) — has nothing to name. A chord's identity is fixed by the intervallic structure among its members, so the dependency is operational, not mere sayability. Pinned sense: chord as a harmonic unit whose quality is determined by intervals, not chord as a bare simultaneous set of pitches.