{"id":3785,"date":"2022-09-15T15:22:21","date_gmt":"2022-09-15T13:22:21","guid":{"rendered":"https:\/\/erlebnisland-mathematik.de\/?page_id=3785"},"modified":"2023-01-26T11:52:02","modified_gmt":"2023-01-26T10:52:02","slug":"advanced-text-musical-dice-game","status":"publish","type":"page","link":"https:\/\/erlebnisland-mathematik.de\/en\/advanced-text-musical-dice-game\/","title":{"rendered":"Advanced text Musical Dice Game"},"content":{"rendered":"

[vc_row drowwidth=”sidebar-biest-default sidebar-biest”][vc_column][vc_column_text]<\/p>\n

Musical Dice Game<\/h1>\n

It was the (nowadays almost unknown) composer and musicologist Johann Philipp Kirnberger<\/em> (1721–1783) who made musical dice games fashionable as a popular pastime in 1757 with his publication “Der allezeit fertige Polonoisen- und Menuettenkomponist<\/em>“. Previously, Johann Sebastian Bach<\/em>‘s second son, Carl Philipp Emanuel Bach<\/em> (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<\/em>“.<\/p>\n

The best-known musical dice game of this kind is attributed to Wolfgang Amadeus Mozart<\/em> (1756–1791). His “Anleitung so viel Walzer oder Schleifer mit zwei W\u00fcrfel zu componiren so viel man will ohne musikalisch zu seyn noch etwas von der Composition zu verstehen<\/em>” was published in 1793 after his death by Johann Julius Hummel<\/em> (Berlin-Amsterdam).<\/p>\n

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.<\/p>\n

By means of the experiment “Musical Dice Game<\/em>” 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<\/em>” piece of music (waltz), can be realised. Acoustically too!<\/p>\n

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

Table 1<\/p>\n

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

Table 2<\/p>\n

[\/vc_column_text][vc_column_text]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<\/em>” 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.<\/p>\n

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.[\/vc_column_text][vc_column_text]<\/p>\n

And now … the mathematics:<\/h3>\n

[\/vc_column_text][vc_column_text]Of course, one wonders about the number of different pieces consisting of 16 bars in this musical experiment. Combinatorics<\/em> (systematic “counting<\/em>“) provides the answer: It is exactly <\/p>\n

  <\/span>   <\/span>\"\[11^ {16} = 45,949,729,863,572,161,\]\"<\/p>\n

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 Gie\u00dfen<\/em>:<\/p>\n

“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.”<\/em><\/p>\n

Conclusion: Every visitor who carries out the experiment with the two dice will almost certainly “compose<\/em>” a new Mozart waltz![\/vc_column_text][vc_column_text]<\/p>\n

Literature<\/h3>\n

[\/vc_column_text][vc_column_text][1] Beutelspacher A. u.a., Mathematik zum Anfassen<\/em>, Mathematikum Gie\u00dfen, 2005.<\/p>\n

[2] Mozart, W.A.: Mathematisches W\u00fcrfelspiel<\/em>, Hrsg. K.-H. Taubert, Schott Musik International, Mainz, 1956.<\/p>\n

[3] Kirnberger, J.Ph.: Der allezeit fertige Polonoisen- und Menuettencomponist<\/em>, publizert bei Christian Friedrich Winter, Berlin, 1757.<\/p>\n

[4] Reuter, Ch.: Musikalische W\u00fcrfelspiele<\/em>, 1 CD-ROM … von Mozart, Haydn und anderen gro\u00dfen Komponisten, Schott Musik International, Mainz, 2001.[\/vc_column_text][\/vc_column][\/vc_row]<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

[vc_row drowwidth=”sidebar-biest-default sidebar-biest”][vc_column][vc_column_text] 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 … Advanced text Musical Dice Game<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"folder":[],"class_list":["post-3785","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/3785","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/comments?post=3785"}],"version-history":[{"count":16,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/3785\/revisions"}],"predecessor-version":[{"id":4011,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/3785\/revisions\/4011"}],"wp:attachment":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/media?parent=3785"}],"wp:term":[{"taxonomy":"folder","embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/folder?post=3785"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}