tempo

Rhythm as a concept of temporal pattern is ancient and universal; tempo as a specific concept measuring beat speed is a later refinement within that broader domain. Which existed first? Rhythm predates tempo historically, and tempo builds on rhythm's concept.

Tempo is the measured speed or rate of a rhythmic pattern. Remove rhythm as a concept and tempo has nothing to measure the speed of — it ceases to operate. The removal test passes cleanly: tempo needs rhythm to function as a concept.

Law 8b removal test on the pinned senses per both accepted definitions: tempo is carved as 'the measured speed or rate at which a musical composition or passage is performed, conventionally quantified in beats per minute,' its parameter (1) being 'the speed of the underlying pulse or beat.' beat is carved as 'a regularly recurring pulse of emphasis in time, the temporal unit that organizes rhythmic experience.' Remove beat now: tempo stops OPERATING — BPM has no pulse to count, the metronome mark (quarter note = N) loses the beat unit it designates, and the Italian tempo names, which name speed regions of the pulse, lose their referent. Not a naming loss (the Law 8b objection): the measured quantity itself disappears. Reverse test: beat operates fully without tempo — unmarked pulses, free rhythm, and rubato performance all operate without any fixed speed. Board precedent: meter DEPENDS_ON beat is ACCEPTED — a framework depending on the unit it measures; tempo is the RATE of that same unit, so this is the same accepted pattern, not a leap. The accepted tempo DEPENDS_ON rhythm is complementary, not subsuming: rhythm organizes events into pattern; tempo fixes the speed of the beat itself; no nearer rung (pulse, beat-unit) exists on the board. Pinned sense: tempo as the rate of beats; beat as the temporal unit whose rate tempo fixes.

tempo IS a specific kind of measurement: it quantifies the speed of a beat cycle, measured in beats per minute (BPM). A competent speaker in music would call tempo a type of measurement — specifically, a temporal rate measurement. Ladder: measurement is the nearest general kind.

DERIVED_FROM: which came first? Tempo as a concept of measured beat speed (e.g. BPM, tempo markings like allegro) predates time-signature as a specific notational convention. Time-signature (e.g. 4/4, 3/4) is a later refinement that builds on the prior concept of tempo plus the need to organize beats into measures. The chronology test passes: the concept of tempo existed first and fed into the development of time-signature notation.