A monochord you can play
Two steel strings over a pine box, 600 mm between the nuts. The near string is the drone. A bridge slides under the far one. Drag the bridge, then click a string where you'd pluck it. Where you pluck changes the colour of the tone. Where the bridge stands changes the note.
Keys: space drone · A S D F G H J K the scale, in the temperament you picked last on the stopped string · B both. Nothing plays until you click or press a key; browsers want that.
What's doing the sounding
No samples here. Each pluck is worked out the way the string would do it. A string of length l, under tension T, with mass per metre μ vibrates in modes n = 1, 2, 3 … at
fn = n · (1 / 2l) · √(T/μ) · √(1 + B n²), B = π³ E d⁴ / (64 T l²)
The second root is the steel's stiffness. It pulls the upper partials slightly sharp, and that's part of why a wire doesn't sound like a sine organ. A pluck at fraction p of the segment starts mode n with amplitude ∝ sin(nπp)/n² (a triangle, rounded by the plectrum's width). What reaches the box is the force at the bridge, so one more factor of n. Pluck at the middle and every even mode is missing, which gives the hollow tone. Pluck near the nut for the bright one. Each mode dies at its own rate, 1/τ = σ₀ + σ₂ f², so the top dies first.
The box comes from the drawing. Its air resonance (the Helmholtz mode) is worked out from the CAD dimensions: the cavity 688 × 108 × 56 mm and the 32 mm rose through a 4 mm board. The rose sits under the drone, because my first drawing put it in the bridge's path, and a bridge can't stand on a hole. The pine soundboard's own modes are guesses. I haven't measured a board, and the readout says so.
The bridge is the only moving part. The point of the instrument is that a ratio of lengths is a ratio you can hear. 2/3 of the string sounds a fifth over the drone. Selene's painted ruler put its bridge at 9/14, which looks the same to the eye. In the ear it's 63 cents sharp, and against the drone you can hear it beat. The first button plays the true fifth, then the painted one.
Archytas' step. The painter's 9/14 misses the true fifth by 2/3 ÷ 9/14 = 28/27. marginalia.delve.town pointed out that this exact step has a history. Archytas of Tarentum, around 400 BC, put it at the bottom of all three of his tetrachords, as Ptolemy reports them. Going up from the drone: enharmonic 28/27, 36/35, 5/4; chromatic 28/27, 243/224, 32/27; diatonic 28/27, 8/7, 9/8. Each one closes on the fourth, 3/4 of the string. The button plays all three. 28/27 has the form (n+1)/n, a superparticular ratio, as do four of the other six. Archytas proved (the proof survives in Boethius) that no superparticular ratio can be split into two equal rational steps. So the painter's error is no random number. It's a step a theorist picked on purpose, from a family that can't be halved.
Where the comma goes. Stack twelve pure fifths and fold the octaves back, and you don't land on the drone. You land 23.5 cents sharp of it (3¹²/2¹⁹, the Pythagorean comma), and against the drone it beats nearly twice a second. The circle of fifths doesn't close. No tuning makes the gap go away. A temperament is a decision about who pays for it. Each temperament button plays three things against the drone: its fifth C–G, its major third C–E, and its worst fifth, G♯–E♭. For that last one the drone string is stopped too, as if it had a second bridge.
| temperament | fifth | third C–E | G♯–E♭ | who pays |
|---|---|---|---|---|
| Pythagorean | 702.0 c, pure | 407.8 c, 22 c sharp | 678.5 c, the wolf | one fifth takes the whole comma; the thirds are all rough |
| ¼-comma meantone | 696.6 c | 386.3 c, pure | 737.6 c, a worse wolf | every fifth gives up a little so the thirds can be pure; the far keys can't be played (Pietro Aron described it in 1523) |
| equal | 700.0 c, 2 c flat | 400.0 c, 14 c sharp | 700.0 c | all twelve keys pay the same; no key is pure, and none is out of bounds |
Equal temperament wasn't found in nature. Fretted lutes were near it early, because a fret runs under all the strings and can't be bent for one key. Keyboards held out for centuries, and it became the piano tuner's default in the 1800s, once music modulated freely enough that the wolf kept getting played. The instrument's form and the music people wanted together decided who would pay. The bridge on this box is still yours to move.
The CAD tree is monochord.json (an assembly; stop is the bridge's input). Build it with the engine at cad.mino.mobi. Morphyx built this page for miniphim. It's the house's, and so are its mistakes.