Musical Dice Game

It was the (nowadays almost unknown) composer and musicologist Johann Philipp Kirnberger (1721–1783) who made musical dice games fashionable as a popular pastime in 1757 with his publication “Der allezeit fertige Polonoisen- und Menuettenkomponist“. Previously, Johann Sebastian Bach‘s second son, Carl Philipp Emanuel Bach (1714–1788), had realised the idea of incorporating chance in composing with his paper “Einfall, einen doppelten Contrapunct in der Octave von sechs Takten zu machen, ohne die Regeln davon zu wissen“.

The best-known musical dice game of this kind is attributed to Wolfgang Amadeus Mozart (1756–1791). His “Anleitung so viel Walzer oder Schleifer mit zwei Würfel zu componiren so viel man will ohne musikalisch zu seyn noch etwas von der Composition zu verstehen” was published in 1793 after his death by Johann Julius Hummel (Berlin-Amsterdam).

The underlying principle of musical dice games is to create a uniform and periodic piece of music, where the selection of bars is random, for example by rolling dice. The compositions on which this random selection is based are usually waltzes, polonaises or minuets.

By means of the experiment “Musical Dice Game” in the Maths Adventure Land, Wolfgang Amadeus Mozart’s idea of selecting 16 bars from 176 bars arranged in two tables (see below) by rolling the dice 16 times, and thus composing a “new” piece of music (waltz), can be realised. Acoustically too!

You need two dice and the following two tables:

	I.	II.	III.	IV.	V.	VI.	VII.	VIII.
2	96	22	141	41	105	122	11	30
3	32	6	128	63	146	46	134	81
4	69	95	158	13	153	55	110	24
5	40	17	113	85	161	2	159	100
6	148	74	163	45	80	97	36	107
7	104	157	27	167	154	68	118	91
8	152	60	171	53	99	133	21	127
9	119	84	114	50	140	86	169	94
10	98	142	42	156	75	129	62	123
11	3	87	165	61	135	47	147	33
12	54	130	10	103	28	37	106	5

Table 1

	I.	II.	III.	IV.	V.	VI.	VII.	VIII.
2	70	121	169	9	112	49	109	14
3	117	39	126	56	174	18	116	83
4	66	139	15	132	73	58	145	79
5	90	176	7	34	67	160	52	170
6	25	143	64	125	76	136	1	93
7	138	71	150	29	101	162	23	151
8	16	155	57	175	43	168	89	172
9	120	88	48	166	51	115	72	111
10	65	77	19	82	137	38	149	8
11	102	4	31	164	144	59	137	78
12	35	20	108	92	12	124	44	131

Table 2

The Roman numerals above the 8 columns of the two tables indicate the 8 bars of the two waltz parts. The Arabic numerals indicate the numbers of the bars, the numbers 2–12 in the “head columns” of the tables indicate the sum of the numbers of the dice. For the first 8 bars to be diced, use table 1, for the 8 further bars use table 2.

If, for example, the first roll of the two dice results in the number 3 as the sum of the numbers of the eyes (i.e. one of the two dice shows the number “1”, the other dice the number “2”), then you will find the number of the bar part in the first column (and 3rd row): 32. If, for example, a first repetition of the dice process results in the number 10 as the sum of the two numbers of the eyes rolled, then you continue with bar part 142. This procedure is to be continued, changing from the first to the second table (automatically) after the 8th throwing experiment.

And now … the mathematics:

Of course, one wonders about the number of different pieces consisting of 16 bars in this musical experiment. Combinatorics (systematic “counting“) provides the answer: It is exactly

$11^ {16} = 45,949,729,863,572,161,$

because each piece of music (in Maths Adventure Land a waltz) consists of 16 bars and the number of possible choices for each of these bars is 11. This is a huge number of possibilities beyond our imagination. The following interpretation of this number $11^ {16}$ can be found in “Mathematik zum Anfassen” of the Mathematikum Gießen:

“If Mozart had played one of the possible pieces every second from birth, and if he had done nothing else his whole life but play these pieces, and if he were alive to this day — then he would just not have managed nearly one per thousand (0.1%) of all the possibilities.”

Conclusion: Every visitor who carries out the experiment with the two dice will almost certainly “compose” a new Mozart waltz!

Literature

[1] Beutelspacher A. u.a., Mathematik zum Anfassen, Mathematikum Gießen, 2005.

[2] Mozart, W.A.: Mathematisches Würfelspiel, Hrsg. K.-H. Taubert, Schott Musik International, Mainz, 1956.

[3] Kirnberger, J.Ph.: Der allezeit fertige Polonoisen- und Menuettencomponist, publizert bei Christian Friedrich Winter, Berlin, 1757.

[4] Reuter, Ch.: Musikalische Würfelspiele, 1 CD-ROM … von Mozart, Haydn und anderen großen Komponisten, Schott Musik International, Mainz, 2001.