Lesson 1
Beat
This course covers the other axis of the piano roll, and it opens by giving that idea away: there is no other axis. Rhythm and pitch are one continuum, and the only thing dividing them is the listener.
The same axis, twice
Foundations lesson 1 established that pitch is frequency, measured logarithmically. A rhythm is also a frequency. A tempo of 120 beats per minute is two beats per second, which is 2 Hz. Sixteenth notes at that tempo are 8 Hz.
Those are the same units as a pitch, and the numbers are only small. So the obvious question is what happens if you keep going, and the answer is not that the rhythm gets impossibly fast. It stops being a rhythm.
From rhythm to pitch, without a seam
One click, repeating faster and faster. Listen for where you stop counting.
- Repetition rate
- 1.0 Hz
- In tempo terms
- 60 bpm
- Gap between clicks
- 1000 ms
- Heard as
- events
The click never changes. Only how often it happens changes, and yet somewhere around twenty a second it stops being a fast rhythm you could in principle count and becomes a low, buzzing note with a pitch. Nothing marks that moment in the signal, and there is no boundary in the physics to mark. Rhythm and pitch are the same axis, and the line between them is drawn by the listener.
What that figure shows
One click, unchanged throughout, repeating at a rate that climbs from once a second to a hundred times a second. At the start you can count it. By the end you cannot, and there is nothing to count, because it has become a low note with a definite pitch. A hundred events a second is a pitch near G2.
Somewhere around twenty a second the individual events stop being separable and fuse. In tempo terms that is 1,200 beats per minute, and it is only about three and a third octaves above a comfortable 120. The whole of rhythm occupies the bottom few octaves of the same scale that the whole of pitch sits on.
The fusion point is a zone rather than a line, which is why the figure marks it as a band. It shifts with the listener, with the spectrum of the click itself, and with how hard you are attending; between roughly twenty and thirty a second most people move from hearing flutter to hearing a tone, but nobody crosses a threshold at a defined instant. What matters for the argument is that no such instant exists in the signal either.
Now the part that is not physics
That first figure was arithmetic. This next one is not, and the difference is the reason this course needs to be careful.
A beat is a regular pulse you feel running through music. The natural assumption is that it is the thing being played: that somewhere in the sound there are evenly spaced events and the beat is those events.
It is not, and this is straightforward to demonstrate. The pattern below has six onsets and not one of them lands on a beat.
Where is the beat?
Nothing sounds on it, and you can still point at it.
Six onsets, none of them on a beat. The four outlined squares are the beats. Click any square to change the pattern.
- Onsets
- 6
- Onsets on a beat
- 0 of 6
- Step length
- 156 ms
- Click position
- silent
Start with the click off and let the pattern loop until you are tapping along. Then turn the click on. If it lands where you were already tapping, you inferred the beat from a pattern that never once played on it. Now shift the click. Three of the four positions are equally steady, equally regular, and sound wrong, which is the part that matters: you were not merely accepting any pulse, you had committed to a specific one. The beat is a place in time you are supplying, not a sound you are receiving.
What that figure shows
You can find the beat in a pattern that never plays on it. That already rules out the beat being a sound you receive, because there is no sound in that position to receive.
The offset control is the real evidence, though, and it is worth being clear about why. Turning the click on and finding that it fits proves very little on its own: a steady click sounds plausible against almost any loop, and it is easy to convince yourself afterwards that you knew. What is harder to explain away is that three of the four positions sound wrong. They are equally regular and equally steady. The only thing separating them is that you had already committed to one of them.
So the beat is a place in time rather than an event, it is supplied by the listener, and different listeners can supply different ones. Everything the rest of this course measures is measured against something the audience brings with them.
The window it works in
Beat inference does not work at any speed. Below roughly thirty events a minute the gaps get long enough that you stop feeling a pulse and start noticing separate events waiting for each other. Above three hundred or so, the pulse is too fast to move to and you begin hearing it as subdivision of some slower beat instead.
In between there is a preference. Asked to tap at a comfortable rate, or to choose which level of a piece is the beat, people cluster around 100 to 120 a minute, which is a gap of about half a second. That is not a law and the spread is wide, but it is consistent enough that it shows up in tempo distributions across repertoires that share nothing else.
Which is a convenient place to note what this course can and cannot do. The ramp figure is arithmetic and would hold if nobody were listening. The tempo window is a fact about people, established by experiment, and quoted here rather than derived. Lesson 6 has to be considerably more careful still. The three kinds of claim are not equally strong, and this course will say which is which.