04 · Meter is not tempo

Tempo is the integral. Meter is not — it is arithmetic on the beat axis. A quarter note at 120 BPM lasts 0.5 s in 3/4, 4/4, or 7/8; meter never changes how long a beat takes. What it decides is how beats group:

$$\text{positionInBar}=(({\beta }_{\text{idx}}-\text{segStart})mod\text{beatsPerBar})+1$$$$\text{bar}=\text{barsBefore}+\lfloor \frac{{\beta }_{\text{idx}}-\text{segStart}}{\text{beatsPerBar}}\rfloor +1$$

That is the whole layer — one %, one /, and a +1 shift to keep both bar and positionInBar 1-indexed (matching beat_this's beatInBar convention and how musicians count: bar 1, bar 2, ...; downbeat = position 1). No integration, no calculus. The brief loop in barPositionOf only exists to carry the running bar count across meter changes, since each change can start a run of a different length.

meter.js — the meter layer. No calculus. Integer division and modulo only.

Tempo (chapters 01–02) answers "what SECOND is beat β?" via an integral. Meter answers "what BAR and position-in-bar is beat index n?" via division. They share a timeline but are computed completely independently.

A meterMap is a list of changes, each saying "from this beat index onward, there are this many beats per bar": [{ fromBeat: 0, beatsPerBar: 4 }, { fromBeat: 32, beatsPerBar: 3 }] fromBeat values must be downbeats (a meter change can only begin on a bar).

The accent click is the meter layer made audible

The most concrete demonstration: a metronome plays a higher-pitched accent click on each downbeat and a plainer click on the other beats. The pitch difference is the meter layer in one tone. The grid chooses which click to play by asking exactly one question per beat:

clickForBeat(meterMap, i) === "accent" ? accentBuf : normalBuf

That single classification IS the entire meter layer at work.

Two layers, one timeline

Open the full-screen standalone demo ↗

Try this:

1. Pick the 4/4 → 3/4 at bar 3 preset. The tempo holds steady; the accent pattern changes at beat 8. The accent moves to a new beat without the tempo changing.
2. Drag the middle orange dot. The downbeat seconds shift in the table below while the bar and position columns sit still. This is the tempo↔meter independence the chapter is about — one drag, two layers, only one of them touched.

How beat_this fits

beat_this returns two parallel lists: beats and downbeats. It never emits a declared time signature. The meter is recovered from the spacing between downbeats:

gap = downIndex(k+1) - downIndex(k)         // in beats

When gap changes — say from 4 to 3 — a meter change is recorded. A piece going 4/4 → 3/4 shows up as downbeat gaps switching from 4 to 3 with no separate "time signature changed" event. meterMapFromBeats packages that walk.

"Beats per bar" here means the pulse beat_this tracks; finer distinctions like simple-vs-compound feel (6/8 felt as two pulses versus six) are a labeling convention layered on top, not built into the arithmetic.