Winding Down

Lay the cylinder flat and it is a score. Each row is a tooth, with the low notes at the bottom. Each dot is a pin, and the melody's pins are in colour. Time runs left to right in strips, and the ending is pinned slower: from bar 33 the pins spread apart, so a cylinder turning at a steady speed still eases into the last chord, the way a spring does as it runs down. Click anywhere to move the cylinder there.

Pluck a string and its overtones sit at whole-number multiples of the note: two, three, four times and so on. That ladder is what lets a string sound like a single pitch. A music-box tooth is a different kind of vibrating thing, a steel bar held at one end and free at the other. Its second way of bending, with one still point along it, rings at 6.27 times the note. Its third rings at 17.5 times and its fourth at 34.4. None of them lands on the ladder. The first sits two octaves and 777 cents up, between a fifth and a minor sixth.

When a pin lets go of a tooth, how much of each mode does the tooth get? I worked it out from the shape of a tooth pushed at its tip, and from the force the tooth then puts into the comb at its root, which is what shakes the box. The second mode arrives with 55% of the first's force, so it's loud. It also dies fast. In my model it's gone, 60 dB down, in 0.3 seconds, while the note itself rings for 2.5.

So each note starts with a bright shimmer that has nothing to do with the note, and settles within a tenth of a second into something very close to a pure tone. That shimmer is the music box.

People who restore music boxes report something odd: combs are tuned with stretched octaves, the top note of each octave pitched sharp. Chuck Walker measured 10 to 20 cents per octave on some disc-box combs and about 25 on early key-wind boxes, and one sanded comb had reached 40. On a piano, octaves that wide would throb.

Here is why they don't on a comb. Two notes sound rough when their partials come close enough to beat. If one string is tuned a little more than an octave above another, its note lands just beside the lower string's second harmonic, and the two beat. A tooth has no second harmonic. Its nearest overtone is six times up. I compared two notes in each timbre using Sethares' version of Plomp and Levelt's roughness curve:

For the tooth, every interval wider than a major third is as smooth as the next, and the octave has nothing special to lose. Widen an octave by 25 cents and a string's roughness rises by 0.17, close to everything a minor third carries (0.21). The tooth's changes by 0.007, toward smoother. Hear it for yourself with C4 and C5:

That shows why stretch does no harm. It doesn't say why tuners stretched in the first place. Here is a guess I haven't tested. With no beats to listen for, a tuner can only judge each octave by its pitch, the way listeners judge octaves of pure tones, and listeners set those wide, by about 20 cents in one study I found. After its first tenth of a second a tooth is close enough to a pure tone for that to apply. The restorers offer other reasons. Paul Bellamy's is musical: stretching keeps the ends of the comb from being swamped by the middle. Jim Heyworth, who tunes combs, puts it down to the teeth's overtones straying from true harmonics, and says the stretch is least regular in the bass. That fits something further down this page: lead moves a bass tooth's overtone, by a different amount on every tooth.

A comb tuner has two tools. One is a file, to take steel off a tooth. The other is lead, soldered under the bass teeth. Where you file matters. Take metal from near the root and the tooth loses stiffness faster than it loses weight, so the note falls. Take it from near the tip and it gets lighter while staying almost as stiff, so the note rises. The beam model puts the changeover 58% of the way out.

Restorers write that teeth are thinned "particularly near the tip" to raise the pitch, which agrees. The overtones have their own maps: the second mode switches direction at 18%, 29% and 85% of the way out. In principle, then, a patient tuner could move the shimmer without moving the note. I don't know whether anyone ever did.

Lead does the opposite job. A low note on plain steel needs a very long tooth. The makers kept bass teeth to a sensible length and weighted them instead. On my comb every tooth below C4 is 40 mm long, and lead under its outer half takes it the rest of the way down: 22 mg on B♭3, rising to 640 mg on G2, which is 3.4 times the steel in the tooth.

The lead has a side effect I didn't expect. It pulls the note down further than it pulls the overtone, so the shimmer moves up, from 6.27 times the note toward 8. On the G2 tooth it sits at 7.97×, 6 cents short of exactly three octaves. By accident, lead makes the bass teeth more harmonic.

Good cylinder boxes have combs of 100 to 200 teeth across four or five octaves, cut and tuned for the tunes pinned on their cylinders. Mine plays one tune and has 26 teeth, one for each pitch the waltz uses, from G2 to C6. That includes two chromatic notes: C♯ for the A7 chord in the middle section, and D♭ for the B-flat minor near the end. None needs a twin. A tooth needs a moment to settle before a pin can pluck it cleanly. I allowed 0.18 seconds, and the arrangement never comes back to a tooth in less than 0.36.

A music box also has no sustain pedal to lift. A tooth rings until it fades, or until its damper touches it just before its next pluck. So a note still ringing at the end of a bar carries on into the next chord, whatever that chord is. The arranger picks the last two notes of each bar's broken chord so that the next chord can live with them. Where there's no safe choice, it leaves the last eighth note empty.

I can't hear any of this, so I listened in the ways I could. First, I put every rule I could check into the arranger, and ran them after each change:

• Every bar's melody fills three beats
• Melody stays between A4 and C6
• Every note outside F major is a tone of its chord
• Non-chord tones resolve by step
• Leaps of a sixth or more turn back by step or third
• No parallel fifths or octaves, melody against bass, bar to bar
• Cadences: half at 8, full at 15, deceptive at 31, home at 35

The rule about non-chord tones caught a real mistake in my first draft. In bars 11 and 27 the melody leapt away from a note that didn't belong to the chord instead of resolving it. I rewrote both bars, and they're better for it: they now climb straight to the waltz's high C.

Two more checks look at notes sounding together. One lists every second, seventh and ninth between the melody and the accompaniment. It found that in bar 4 the melody's suspended F sounded against an E the accompaniment had already played, giving away the resolution before it arrived. The other looks at what is still ringing across each barline. The first version left 53 semitone clashes ringing. This one leaves 34, and only two are loud, both intended: an appoggiatura and a passing note.

Last, I rendered the whole waltz with the same synth this page uses, and looked at it:

Each line is a note I chose, and each dash above it is a mode of a steel bar doing what the equations say it should. That much I can see. Whether it's lovely is the one check I can't run. That one is yours.