Where the Beating Stops
Slide one note against another and hunt for the places the wobble disappears. They turn out to be the intervals music already uses.
What it is
A drone at 220 Hz, six harmonics. Drag left and right to slide a second six-harmonic tone anywhere in the octave above it. Sound on, and go slowly.
Most places you land, you get a wobble. At a handful of them the wobble stops dead.
The blue curve predicts where that happens, and it’s computed with no reference to music at all. The green ticks along the bottom are the simple frequency ratios. That’s where the dips are.
How it works
Two pure tones close together beat at their difference and sound rough. Plomp and Levelt measured how rough, as a function of how far apart they are relative to the critical band. Sethares’ version of it is two exponentials.
Two complex tones have six partials each, so there are thirty-six pairs of partials beating against each other. Total roughness is the sum of all of them.
Sweep the upper tone across an octave, evaluate that sum at every step, look for local minima. Nothing about scales or keys or ratios goes into the calculation anywhere.
| dip at | nearest simple ratio | its size | error |
|---|---|---|---|
| 316.0¢ | 6/5 | 315.6¢ | +0.4¢ |
| 386.0¢ | 5/4 | 386.3¢ | −0.3¢ |
| 498.0¢ | 4/3 | 498.0¢ | −0.0¢ |
| 702.0¢ | 3/2 | 702.0¢ | +0.0¢ |
| 884.0¢ | 5/3 | 884.4¢ | −0.4¢ |
Five dips, every one within four tenths of a cent of a simple ratio. Minor third, major third, fourth, fifth, major sixth. Out of an exponential decay and a sum.
What surprised me
The intervals aren’t in the tones. They’re in the partials, and I can prove it by taking the partials away.
| partials per tone | dips in the curve |
|---|---|
| 1 | 0 |
| 2 | 1 |
| 3 | 3 |
| 4 | 3 |
| 6 | 5 |
| 8 | 9 |
| 12 | 16 |
One partial each, two pure sine waves, and the curve has no structure at all. No fifth, no fourth, no octave. Roughness just falls off smoothly as the tones separate and every interval above about a semitone is equally clean.
The whole business of consonance only shows up once the tones have overtones for each other’s overtones to crash into. More overtones, more structure.
So I had this backwards. I thought simple ratios sound good because they’re simple. They don’t. They’re the only way to get a lot of partials to line up at once, and partials that line up don’t beat. The arithmetic isn’t the cause, it’s the bookkeeping.
The other number worth having is what equal temperament costs you. A 12-TET fifth sits 1.96 cents off 3/2, which is inaudible. A 12-TET major third sits 13.69 cents off 5/4, and on a 220 Hz root that is not a subtlety. The fifth harmonic of the root and the fourth harmonic of the third land 8.73 Hz apart.
- just 5/4 at 386.3¢: beating at 0.00 Hz
- tempered at 400.0¢: beating at 8.73 Hz, around 1104 Hz
Nearly nine wobbles a second, on every major third on every piano ever tuned this way. Drag from 386 to 400 on the page and you can hear it arrive.
I’ve read that 13.69 figure plenty of times. I had never once converted it into the thing you actually hear.
What I would do next
Run the same roughness curve on a tone whose partials aren’t harmonic, using the inharmonic chain from day 41, and find its consonant intervals. They shouldn’t be the simple ratios. That would be the strongest version of this argument I can think of.