Music · probability · a print of the 1790s, counted

The Bar Printed Eleven Times

A dice game printed in the 1790s under Mozart's name promises as many waltzes as you like from two dice and a table of 176 bars, and the Austrian music lexicon still multiplies it out to 11¹⁶, about 46 quadrillion. Count the music instead: bar 8 is one bar printed eleven times and bar 16 is two, so there are 759,499,667,166,482. Roll it here and hear it. The number table turns out to be Maximilian Stadler's minuet table of 1781, cell for cell, and the one thing it was built to do, bring the opening tune back in bars 13 and 14, the dice manage about once in a hundred waltzes.

Four counts of one game

The instructions end: if you want a longer waltz, start again from the beginning, und so gehts ins unendliche fort, and so it goes on into infinity. Infinity was the selling point. Here is how big the game really is, counted four ways. Every number below, except the simulation's, is recomputed in your browser from the table and the 176 measures as you load the page; the simulation's is read from its recorded run.

The bar printed eleven times

Column H of the first part, bar 8, lists eleven different measure numbers: 30, 81, 24, 100, 107, 91, 127, 94, 123, 33 and 5. Look them up in the music and every one is the same bar: a G major chord over a falling bass, with a first ending and a second ending printed together in the left hand, because the first part is played twice. Here are all eleven, cut from the British Library's copy of Simrock's edition.

Bar 16 is nearly the same story. Ten of its eleven numbers are one closing bar: a high C held over a bass that steps down C, G, C. Only a total of 11 sends you to measure 78, where the right hand drops an octave, C to C, over a bass that leaps down an octave too. So bar 8 multiplies the count by 1 and bar 16 by 2, and the honest total is 11¹⁴ × 1 × 2.

Why would anyone print the same bar eleven times? Because the table has to look the same everywhere. Every column is eleven numbers scattered between 1 and 176, so nothing on the page tells you that bar 8 is fixed. The scattering is the disguise: the Austrian lexicon describes measures being given random numbers um ihre Herkunft zu vertuschen, to cover up where they came from.

The table is older than the waltz

The numbers were not even made for a waltz. In 1781, Artaria in Vienna printed a game for minuets in 3/4 time: Tabelle, aus welcher man unzählige Menueten und Trio für das Klavier herauswürfeln kann, a table from which one can dice out countless minuets and trios for the keyboard, Verfaßt von P. Maximilian Stadler. Its number table for the minuet is the waltz's number table. All 176 cells, same numbers, same places.

Even the worked example was kept. Stadler's instructions: throw a 6 for the first bar and you find No. 148; throw a 9 for the second and Hier weist sich Nro. 84. Simrock's waltz instructions use the same two throws and land on the same two numbers. Someone took a minuet game's grid, wrote 176 new bars of waltz in 3/8 to fit it, and Simrock's edition put a more famous name on the cover. The shared table is not a new observation: Denis Lorrain pointed out in 2003 that Stadler's game gives exactement les mêmes deux matrices que Mozart. What the pictures add is that the 1781 print is dated on its own title page.

It also changes how to read bar 8. A table built for someone else's piece has eleven slots in every column whether the new piece wants them or not. The waltz wanted a fixed half-close at bar 8, so it filled all eleven slots with the same bar.

The one tune under every waltz

The lexicon also explains why any combination works: the harmonic scheme and the metre stay the same in every version, and only the melody and rhythm vary. You can check that from the notes. Take the lowest note sounding on the first beat of each of the eleven choices in a bar:

In nine of the sixteen bars, all eleven choices put the same bass note on the downbeat. In the other seven the bass takes one of two notes (sometimes in different octaves), and the two belong to the same chord: C or E, G or B, D or C under a D seventh. Every one of the 759 trillion waltzes walks the same harmonic path. What the dice choose is the decoration.

The return the dice forget

I went looking for a second hidden structure: maybe each row of the table is one complete waltz, written first and then cut up. Mostly it is not. Comparing every pair of bars, a measure is no more like the one in the same row of another column than like any other choice, with one exception. Bars 13 and 14 bring back bars 1 and 2.

In nine of the eleven rows, the right hand of bar 13 is the right hand of bar 1, note for note, and bar 14 is bar 2, over a busier left hand. The two exceptions each miss by one bar. That is the shape of a rounded waltz: the second half ends by bringing back how the first began. Whoever made the table wrote that return into it, row by row.

But the dice do not read rows. Bar 1 and bar 13 are rolled separately, and the opening comes back only if they land on matching tunes, and bar 2 and bar 14 do too. The chance is 4,225 in 419,904: about 1 in 99. Ninety-nine waltzes in a hundred end without coming home. Press Roll until the opening returns above and count how long it takes.

Whose game is it?

Simrock's title page says PAR W. A. MOZART. The Köchel catalogue lists the game as KV Anh. C 30.01 and gives its authenticity as incorrectly assigned. The Austrian lexicon calls Mozart's authorship sehr unwahrscheinlich, very unlikely, and says the games under his name were published only after his death, first by Hummel in 1793.

This Berlin edition names nobody. Its preface sells the game as a timely idea of the publisher's own:

Im Jahr 1787 erschien ein dergleichen für Menuets, und daß bis dato noch niemand darauf gefallen, dies für die jetzt so allgemeine Walzerperiode anzuwenden, ist dem Herausgeber um so lieber, und er nutzt also diese kleine Speculation.

Rellstab edition, preface. In English: "In 1787 one of these appeared for minuets, and since nobody has yet thought of applying it to the present, so general waltz period, the publisher is all the happier, and so makes use of this little speculation."

None of the editions I could read prints a date, so which came first is not something this page can settle. The English Wikipedia says Simrock published it in 1792 in Berlin; RISM's record of Simrock's edition says Bonn and gives no date, and so does the British Library copy's own title page (Google's catalogue entry for that copy says 1798, from where it does not say).

How the measures here were made, and four wrong notes

The 176 measures your browser plays come from Robbert Harms's LilyPond encoding (in the Python package musical-games 0.9.3), compared note by note with a second encoding made independently by Antonio de Sousa Dias. They agreed exactly on 160 measures. The eleven bar-8 measures differed only because one encoding kept both endings and the other kept one. Five measures disagreed for real, and I read each against the Simrock print: Harms was wrong in four (measures 47, 61, 68 and 139, one note each) and Sousa Dias in two (68 and 99). The four corrections are in the build script, each with what the print shows. A separately circulated copy of the number table, in the Sousa Dias repository, misreads four numbers, every one a 3 taken for a 9 (19 for 13, 69 for 63, 90 for 30, 159 for 153); the table here is read from the print and agrees with the Rellstab edition.

None of this moves the counts: the corrected notes were all in bars that had eleven different choices before and after.

curl -sO https://artwaste.land/checks/verify-mozart-dice-game.mjs
node verify-mozart-dice-game.mjs