The Nature of Harmony told the story, Show Me the Number made the argument, and the Sound Laboratory built the bench. This note goes one step further into the books themselves. The old treatises are full of instruments that were diagrammed and then silenced — monochord divisions, tetrachord arcs, string tables with every pitch worked out to the integer — waiting five centuries for a reader who could hear them. Below, three of those diagrams are replicated and strung. Nothing is a recording; every sound is synthesized live from the numbers on the page.
I. The falsifier's own apparatus
The Sound Laboratory's first station weighs the smith's hammers and watches the legend break. What we did not notice until we read further into Vincenzo Galilei's Dialogo is that the falsifier left build instructions. On p. 145 he describes the exact bench: “if one takes two strings of the same length, thickness, and quality; if these are stretched to unison over a flat surface, and one of them is deprived of half its length by means of a bridge, fret, or even the fingers of the hand, one will hear (as has been said before) the Diapason consonance every time they are struck together or one after the other” — and, a line later, “From these observations, the use of the Monochord easily took its origin” (verified verbatim). The presets below are his sentence, word for word. The weight pan is the half of the story he put on trial: a string answers weight only by its square root, so the legend's two-to-one lands on the tritone, and an octave costs fourfold.
Galilei’s two strings
His presets, in his words — deprive string II of:
…or leave the bridge alone and load the pan instead:
String II
220.0 Hz
string I stays 220.0 Hz
Interval
0 ¢
✓ a unison
The law
f ∝ (1/L)·√W
L 1.00 · W 1.00
II. Four numbers, three laws
Boethius gives Mercury's lyre four strings in the proportions 6 · 8 · 9 · 12 — the same numbers as the hammers. But the authorities could not agree which end was low. Galilei lays out the dispute on p. 144: Plutarch's camp “applied the number six to the first, eight to the second, nine to the third, and twelve to the fourth and last named” — six on the lowest string — while the Zarlino camp assigns the numbers “exactly the opposite” way (verified verbatim). His diagnosis, same page: Plutarch's numbers cannot be monochord divisions — they only make sense as weights, while the reversed order reads them as measures of length. The direction of a scale is the fossil of a physical hypothesis. And a hypothesis about sound is audible: each reading is a different law, so the same four numbers sing three different chords. That is how you know for sure — authority picks a direction; the bench picks the law.
The Lyre of Mercury — but which way up?
sounding low → high: 12 · 9 · 8 · 6 — bigger number, deeper voice.
Two honest footnotes. The struck-hammer reading is the one a browser cannot fully settle: our tone follows the cube-root scaling of similar solid bodies, but in a real smithy the anvil sings its own note almost regardless of the hammer — and Galilei himself guessed that direction wrong, writing that the twelve-pound hammer “made the high sound.” To know that for sure you need iron, not a web page. And the legend never agreed with itself about its own data: in Boethius's telling there are five hammers and Pythagoras rejects one as dissonant — Galilei replays that arithmetic — while Gaffurius's woodcut draws six.
III. The page that begs to be played
In our fifteenth-century Boethius manuscript, the whole two-octave system of ancient music is drawn as nested arcs in red ink — string names, interval spans, and the monochord number of every string, in ladders for the chromatic and enharmonic genera. Below, the same ladder is replicated and strung. The drawn length of each string is true to its number on the 9,216 ruler, and the diatonic division reproduces the manuscript tradition's canonical integers exactly: 9216, 8192, 7776…down to 2304, two octaves up. Switch the genus and the brass strings — the movable notes — re-divide, just as the facing diagrams on the page do; the chromatic pulls Lichanos up to an F♯, and the enharmonic lands on genuine quarter-tones, notes that fall between the frets of every modern instrument.
The Greater Perfect System, strung
The middle column gives each string's nearest modern note with its deviation in cents. Even the diatonic sits a few cents off a modern tuner — pure 9:8 tones against equal temperament — while the enharmonic's quarter-tones land ~50¢ between any modern frets. The dieses here split the semitone equally; Boethius divides the ruler arithmetically, a hair's difference the ear forgives.
Every diagram is a score
A monochord division is not an illustration about music; it is music, stored in the only medium its author had. The three instruments on this page took an afternoon to string because the hard work was done centuries ago — the numbers are on the pages, worked out to the integer. The library holds many more: division tables in Zarlino and Salinas, interval ladders in Gaffurius, circle diagrams in Descartes, tuning grids from Ming China. We intend to keep stringing them. If there is a page you want to hear, tell us with the suggest-an-edit link below.