Other famous examples are Philip Kirnbirger's The Ever Ready Composer of Polonaises and Minuets




Дата канвертавання24.04.2016
Памер61.44 Kb.
Other famous examples are Philip Kirnbirger's The Ever Ready Composer of Polonaises and Minuets (1757 1st edition; revised 2nd 1783) and Joseph Haydn's Philharmonic Joke (1790). http://imagine.xs4all.nl/bram/mozart/

Musikalisches Würfelspiel
(Musical Dice Game)

Musikalisches Würfelspiel (Juego de dados musical para componer minuetos) a menudo atribuido a Mozart. Podemos crear cientos de Minuetos al estilo Mozart. La idea es enganchar medidas musicales pre-escritas seleccionadas al azar hasta crear un minueto.

272 medidas musicales y una tabla de reglas para seleccionar medidas específicas a partir de los valores de los dados. El resultado es la selección aleatoria de 16 bar minuet y 16 bar trio.

El juego fue publicado por primera vez en 1793, después de la muerte de Mozart, po J.J. Hummel. A pesar de que nunca se encontró el manuscrito original de "Musikalisches Würfelspiel", ni Mozart hizo referencias conocidas al mismo, su autoría nunca ha sido cuestionada por los musicólogos.

Steven Goodwin creador de Mozart Dice 1998.

Dos dados son usados para determinar cada una de los 16 compases que componen un minueto. Para el primer compás se lanzan los dos dados con posibles resultados: 2, 3,..., 12, es decir un total de 11 posibilidades.
http://sunsite.univie.ac.at/Mozart/dice/generate_score.cgi

En su obra Musikalisches Würfelspiel (Juego de dados musical) K516f,  de 1787, Mozart compone 176 compases para los minuetos y 96 compases para los tríos. Cada pieza consta de 16 compases. Estos compases están sueltos, pero Mozart ofrece unas reglas basadas en el lanzamiento de dados que permite combinarlos de múltiples formas. ¿De cuántas?:



Minuetos: 1116 (casi 46 mil billones) formas no equiprobables correspondientes a dos dados. 

Tríos:   616 (casi 3 billones) equiprobables correspondientes al lanzamiento de un solo dado. 

Obra conjunta (minueto + trío): 6616  (más de 1029 ). Es decir, más que granos de arena hay en la Tierra. 
Se escoge al azar uno de los Hay 16x11=176 posibles compases de Minueto y 16x6=96 posibles compases Trio para escoger. El resultado de los dados se usa con una tabla de reglas para determinar que medida se escoge
Measure numbers are indicated in the horizontal axis.

-Dice roll indicated in the vertical axis (2-12 for the Minuet,

1-6 for the Trio).

-For example, in composing measure 1, if you rolled a 9, you would

play measure 119 of the 196 possible Minuet measures.

Minuet
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


2 96 22 141 41 105 122 11 30 70 121 26 9 112 49 109 14

3 32 6 128 63 146 46 134 81 117 39 126 56 174 18 116 83

4 69 95 158 13 153 55 110 24 66 139 15 132 73 58 145 79

5 40 17 113 85 161 2 159 100 90 176 7 34 67 160 52 170

6 148 74 163 45 80 97 36 107 25 143 64 125 76 136 1 93

7 104 157 27 167 154 68 118 91 138 71 150 29 101 162 23 151

8 152 60 171 53 99 133 21 127 16 155 57 175 43 168 89 172

9 119 84 114 50 140 86 169 94 120 88 48 166 51 115 72 111

10 98 142 42 156 75 129 62 123 65 77 19 82 137 38 149 8

11 3 87 165 61 135 47 147 33 102 4 31 164 144 59 173 78

12 54 130 10 103 28 37 106 5 35 20 108 92 12 124 44 131
----------------------------------------------------------------------------
Trio
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
1 72 6 59 25 81 41 89 13 36 5 46 79 30 95 19 66

2 56 82 42 74 14 7 26 71 76 20 64 84 8 35 47 88

3 75 39 54 1 65 43 15 80 9 34 93 48 69 58 90 21

4 40 73 16 68 29 55 2 61 22 67 49 77 57 87 33 10

5 83 3 28 53 37 17 44 70 63 85 32 96 12 23 50 91

6 18 45 62 38 4 27 52 94 11 92 24 86 51 60 78 31

The Minuet is in D Major and the Trio is G major. The Trio follows immediately after the Minuet Usamos dos dados para seleccionar cada uno de los 16 compases del minueto. Como hay 11 posibilidades para cada compás tenemos 1116 posibilidades. Los tres últimos compases 616. En total 1116 *616 = 1.3 *1029.

Mozart - Musical Game in C K. 516f*

By Hideo Noguchi (Kobe/Japan)


During the latter half of the 18th century and the earlier half of the 19th century some copies of musical games called "Musikalische Würfelspiele" were published in many cities all over Europe. All the publishers stated that an infinite number of compositions could be written by any amateur, even if he were not familiar with the techniques or rules of composing. The tools recommended to select the musical figures were either one to three dice pieces or tops with six or nine faces in assistance with numeral tables. Some of the methods required the players to select the bars directly from the staves, where no tables were given. The musical styles were, except for a few cases, dances (such as minuets, contredanses or waltzes) or marches. The compositions were limited either to a single melody without accompaniment, to two lines for clavier, or to three lines for trio scoring[1].

Several kinds of "Musikalische Würfelspiele" were spuriously published under the name of Mozart[2] and are designated as K. Anh. 294d/Anh. C 30.01 in the Köchel 6th edition of 1964. We can find an autographed genuine musical game by Mozart in the Bibliothèque Nationale, Paris (Collection Malherbe) with signature Ms. 253, which is designated as K. Anh. 294d/516f but which was not recorded in Mozart's "Verzeichnüß aller meiner Werke". A facsimile of the autograph was published for the first time in 1986 in Japan[3].

This one leaf (see Figure 1 and Figure 2) includes in piano reduction, on the upper two staves of what we may call the first side or page, a six-bar incipit from the third movement of Mozart's g-minor String Quintet K. 516[4], on the lower six staves of this side and on the first seven staves of the reverse side are the scribbled one-voice figures of the Musical Game in C K. 516f. Following the first two-bar motif on both pages, Mozart prepared many groups of two-bar melodies (with one exception) which may be selected at random, and the last staves on the two pages are an example of one solution written by Mozart himself. But since there are no instructions of how to select from the alphabet the small or capital letters, or from the numbers either 1 or 2 (which correspond to the two bars of music), a complete explanation for this Musical Game bas not been reported.

The following themes will be considered in the present article: (1) the meaning of the alphabet letters; (2) the method of the sample solution by Mozart; (3) the meaning of the numbers 1 and 2; (4) the date of completion of the work; (5) for whom the work was written; and (6) how to play this musical game.

Nissen wrote " Von Mozart und seine Handschrift" on page 1, while the Köchel sixth edition excerpted the incipit from page 2. Which is actually the first page? We will in any case begin with the page we designated above as side 1 (see Figure 3). Here we find 24 small alphabetical letters ("j" and "x" do not appear) for each of the two bars after the first two bars; these are then followed by 14 capital letters (here the vowels and "C", "H","J", "R", "V", "X" and "Y" do not appear). Since Mozart could easily have used all 26 letters of the alphabet[5], it is likely that he purposely deleted some letters. For example, "Q" seems originally to have been another letter which might have been an "S"; therefore "Q" was necessary for Mozart's purpose. The reason that "j" ("J") is deleted can be explained by the fact that "j" ("J") is the consonantal form of the letter "i" ("I"). But it can not be readily explained why "x" ("X") or almost half of the capital letters were not necessary for Mozart's purpose. The following scheme showing the deletions may be a hint to the answer to this question :

_B_D_ FG__ KLMN_ PQ_ ST__ W_ Z.

We must be satisfied here to agree that Mozart decided the selection of the letters by some unknown rules. From the traces that "a" through "P" are not lapped over the notes in the lower staff but "Q", "T" and "Z" are partly lapped over in Figure 1, the following sequence would be suggested; (1) Mozart filled in the music from the third to the seventh staff, (2) wrote down the alphabet "a" to "Z" with care not to be lapped over the notes in the each lower staff, and (3) made the sample composition in the eighth staff for he might not mind that "Q" ,"T" and "Z" were already rushed into the staff. The selected paired bars in the eighth staff represent those musical figures denoted by the letters "fanciS" (reading from left to right). To consider the meaning of this word, the following words will be examined: (1) FANCI[E] S (Eng.), (2) FANCI [ULLE]S[CA] (It.), (3) F[R]ANCIS, and (4) F[R]ANCIS[CA]. The reason an English word "fancy" is considered is that Mozart had begun to learn English sometime before March 1787[6], and it suggests that the method of composition of the musical game is secret and "pure fancy"[7]. "Fanciullesca" is used as in "con fanciullesca ingenuità" which suggests that the musical game should be played "with the innocence of a child". But it seems likely that a name, either Francis or Francisca, who was a possible acquaintance of Mozart, was intended.

The music on page 1 is rather poor in the configuration, i.e., we find neither rests, chords, cadences nor double bars in any of the two-bar groups. It is likely that Mozart intended not to complete a piece of music but only to check the possibility of "ars combinatoria" on page 1, because rests and cadences are indispensable to a music. One might suggest the possibility that the music on page 1 was not composed by Mozart himself but is a copy of another's dice game, which he made in order to continue with something more complex. But this possibility may be very slight because if the version before us should be considered the copy, rests, cadences and double bars should have been transfered by Mozart from the original.

The music on page 2 (see Figure 2) is more manifold in melodies and rhythms and also has more slurs, as well as rests, chords and cadences. Each two-bar group is generally clearly separated by a double bar. Here the numbers 1 and 2, with a couple of corrections by Mozart, are assigned to each two-bar group. We also find a correction in the solution written out by Mozart. He might have made the corrections in accordance with some rules. But on what basis can we select either figure 1 or 2? We must remember that the name of an acquaintance would lead the music toward the solution. Let us therefore supply an alphabet for the musical figures on page 2: "a" for the first paired bars 1 and 2, "b" for the next pair, etc., with "j" and "x" deleted as on page 1. All of the bars require just 24 alphabetical letters as shown in Figure 4. Mozart's solution in the seventh staff is read as follows:

f1 r2 al n1 c2 i2 s2 c1 a2.

Thus it is proven that the name intended was Francisca.

Why was Francisca "Fancis" on page 2? Is "Fancis" a nickname for Francisca? Surely not, and I would propose that (1) "r" was carelessly left out by Mozart, (2) the letters "c" and "a" were left out because they had already been used, and a second selection of the same letters was probably prohibited by the rules, (3) capital letters might have been permitted to replace small letters where a letter recurred, but "C" and "A" were initially, and perhaps only accidentally, omitted on the first page, and accordingly (4) the selection on page 1 might then have been abandoned and the music on page 2 might be created as a revised new version of the musical game.

The problem of recurring letters was improved on page 2, where Mozart set up the numbers 1 and 2 so that each could be represented by one letter of the alphabet. It might have been Mozart's intention to select the last two paired bars (= "z1", "z2") as the C major ending. The rules of selection might be applied as follows:

Rule 1.


Write a name, appending "z" at the end.

Example 1.

franciscaz

Rule 2.


Rewrite the letters in alphabetical order.

Example 2.

aaccfinrsz

Rule 3.


If a letter recurs, place it on second appearance at the end of the main group, and separated by the mark +.

Example 3.

acfinrsz + ac

Rule 4.


Assign the number 1 or 2 to each letter one after the other; where the number 1 is assigned to the main group, the number 2 must be assigned to the group after the + sign, and vice versa, in order not to duplicate numbers and therefore musical figures.

Example 4.

al c2 f1 i2 n1 r2 s1 z2 + a2 c1

Rule 5.


Return to the original spelling.

Example 5.

f1 r2 al n1 c2 i2 s1 c1 a2 z2

A rigid application of the above rules would give a slightly different solution than that offered by Mozart, namely in the assigning of numbers to the letter "s". This suggests that one must temper the rules for better results, though it seems difficult to determine in this case which solution on purely musical grounds is correct, s1 or s2.

The two bars of g1 seem very similar to Zerlina's "Vedrai, carino" shown in Figure 5. It is likely that g1 is the development of the preceding bars(f2), therefore this work might have been written independently of and before "Don Giovanni" (K. 527).

If on page 1 of our manuscript the Clavierauszug of K. 516 is not a sketch for the composition of the Quintet, as the Neue Mozart-Ausgabe points out[8], Francisca might have been a piano pupil of Mozart. - Was this the announcement that she was promised to be provided with a clavier edition of the Quintet? Was this the corrected version for a completed clavier edition of the Quintet? We can only say that K. 516f might have been written between 16 May 1787 (date of K. 516) and 28 October 1787 (date of K. 527 in Mozart's "Verzeichnüß"), based on the above assumption. If the name Francisca was taken after St. Joanna Francisca de Chantal (1572-1641), the date of her feast on 21 August might be worth considering.

It is not difficult to identify our Francisca from among Mozart's acquaintances in the year 1787. Franc. Cajetan a Ployer, who wrote a Latin poem in Mozart's album on 28 June 1787, was once thought to be Francisca Cajetana Ployer[9], but the signature has proven to be that of Franz Kajetan Ployer[10]. The only possible female is therefore Francisca[11] von Jacquin (1769-1853), daughter of the famous botanist Professor Nicolaus Josef Baron von Jacquin (1727-1817). To Francisca's brother Gottfried von Jacquin (1767-1792) Mozart wrote letters at least four times in 1787. He and she both were furthermore pupils of Mozart[12], who in turn wrote many works for the household music-making of the Jacquin family, namely:

1) Six vocal ensembles K. 439, 438, 436, 437, 346/439a, 549 and the "Bandel" trio K. 441 were written between 1783 and 1788 during the intimate friendship of the Mozarts and the Jacquins in Vienna. This might be the reason why none of these works were published in Mozart's life time and why copies of these works were not known outside the circle of friends connected with Gottfried[13].

2) The piano trio K. 498 and the four-hand piano sonata K. 497 might have been written for Francisca as pianist in August 1786. The trio was surely played in Jacquin's house with Mozart as violist and Anton Stadler as clarinettist[14].

3) The flute quartet K. 298 was written around the last quarter of the year 1786 (or perhaps somewhat later). A theme with variations in the first movement, similar to Franz Anton Hoffmeister's song "An die Natur", could have been suggested by Gottfried[15].

4) The aria K. 513 was written on 23 March 1787 as the friendly gift for Gottfried, who possessed a well-trained bass voice. But the range requirement for the voice part is rather moderate, from A to es'[16].

5) The double-canon K. 228/515b with Mozart's English compliment, "Vienna the 24 April 1787 / don't never forget your true and faithfully - WAM" was found in the album (1787) of Joseph Franz von Jacquin[17].

6) The song K. 520 was written on 26 May 1787 in the Vienna Landstrasse, in Gottfried's room, and the song K. 530, also for Gottfried, on 6 November 1787 in Prague. Both were published under Gottfried's name, probably with Mozart's permission, on 26 March 1791[18].

7) The four-hand piano sonata K. 521 was written on 29 May 1787 and was sent to Gottfried with a letter advising Francisca to tackle it at once, because it was rather difficult[19].

8) The aria K. 621a was possibly composed for Gottfried in 1787 (or 1791?) in Prague, if in fact it is the same aria described in Mozart's letters of 15 October and 4 November 1787 to Gottfried from Prague. Constanze Mozart, by the way, attributed the work to Gottfried, not to Mozart[20].

In the facsimile we find a horizontal crease between the sixth staff and the seventh and a vertical crease on the left hand side of page 2 (see Figure 2). It has also been reported that the lower edge of the paper was cut off, leaving only eight staves[21]. How many staves were originally prepared on this paper? I turned my attention to the irregular pattern in the profile created by the left ends of the staves. The first line of the first staff, for example, projects further to the left than the other lines of that staff, while the first line of the fourth staff is set further to the right etc.[22]. The paper therefore shows a striking similarity with that used for the "Twelve Duos for two Horns K. 487/496a" (see Figure 6) and is thus suggested to be the same type of the twelve-staff paper[23]. For further confirmation the paper has a watermark (three moons, of a particular size and shape) that is able to be identified as that of a particular paper-type which was always used by Mozart with twelve staves[24]. It is likely that because the paper was folded exactly in two, the crease remained between the sixth staff and the seventh, which means the cutting away would have been done after the folding. The sequence would be as follows:

1) A clavier version of music from K. 516 was written on page 1.

2) A preliminary version of Musical Game K. 516f was written on page 1 but abandoned.

3) An improved version of Musical Game K. 516f was written on page 2.

4) The paper was folded once horizontally, in the middle, and (two or) three times vertically.

5) The lower four staves were cut away.

Why was the paper folded? Why were the lower four staves cut off[25]?

It is still impossible to answer these questions.

Mozart's second sample composition on the bottom of page 2 was possibly not intended for voice, in as much as a G clef is used instead of a C clef or an F clef. The structure does not seem to allow canons. It might have been recommended to play a piano accompaniment with the left hand "ad libitum". For our own purposes we may add that if a given name includes "x" in the spelling, "chs" may be substituted in its stead; if a single letter recurs more than twice in a name - "Nannerl", for instance, we may use more + signs in rule 4. But we must be careful because it seems that these are perhaps not of Mozart's intention.

So much for one possible method of playing Mozart's Musical Game K. 516f. We might also add that Mozart had a great inclination to play with names. We can find in his letter to Gottfried on 15 January 1787 a name-list with the nicknames he and his party invented on the journey from Vienna to Prague (see Table 1)[26]. Mozart might have remembered the above name play when he wrote K. 516f; or rather, these names and nicknames (which, of course, do not in themselves make musical sense at all) might have given Mozart the idea of writing the Musical Game K. 516f. The interval of more than four months between the journey and the game requires no special explanation, since the canon "Lieber Freistädtler, lieber Gaulimauli" K. 509a, which includes one of the above nicknames, was written even later, between 4 July and 1 October 1787[27].

A few trials of the musical game were played to test the conclusions of this essay and the results are shown in Figure 7.

Should I estimate here the music itself of Musical Game K. 516f? Since there is no limit to the number of names and the number of letters one may choose, the number of compositions will be infinite. I recommend that all Mozart lovers in the world play this musical game using their own names and then estimate their/Mozart's music by the composition which results.

***


With special thanks to the Bibliothèque Nationale, Paris, and the Archiv der Gesellschaft der Musikfreunde, Vienna, for the courtesy of the photographs.

* I would like to thank Dr. Wolfgang Plath (Augsburg) for consenting to read a draft of this article and for offering helpful comments. I would also like to thank the editor for shaping it into a more readable form.

(This paper is presented in Mitteilungen der ISM 38 (1990), Heft 1-4, p.89-101)

: Musical Game in C K.516f

- Selection of Measures (k516f.lzh, 54kB, Excel95 File)


- File Converter (mml2m47.lzh, 70kB, Windows95 Program in Japanese)

Parts; CH1:Piano


Sound Generator; Roland SC-88VL
Source; Facsimile of Autograph





Author: Hideo Noguchi
E-mail: Please refer to homepage.
URL: http://www.asahi-net.or.jp/~rb5h-ngc/e/k.516f.htm
© All rights reserved.
(Originally uploaded:1997/10/19, Last updated:1997/12/28)


Juego de dados musical de Mozart

por: Hernando Ortega / Federico O´Reilly [hernando@sigma.iimas.unam.mx / federico@sigma.iimas.unam.mx]

A Jorge Velazco.

Wolfgang Amadeus Mozart (1756-1791) compuso la obra Musikalisches Würfelspiel, singular creación artística en la que el ingenio del músico lo llevó a componer no una pieza para piano sino un generador de valses. Esto es, la obra no contiene una partitura para un pequeño vals de 16 compases sino que tiene un sistema que, apoyado en el azar, puede generar un número mucho muy grande de valses diferentes de 16 compases cada uno. Mozart escribió 176 compases numerados del 1 al 176 y los agrupó en 16 conjuntos de 11 compases cada uno. El procedimiento para generar un vals particular a partir de esta combinación de habilidad en la composición y el uso del azar consiste en que cada compás del 1 al 16 se selecciona con unos dados, del correspondiente conjunto de 11 compases.

Estos 16 conjuntos o columnas de números, que identifican cada uno de los 176 compases, son los siguientes:




En el encabezado, en números romanos aparece el número del compás e identificando cada una de las filas aparece un número entre 2 y 12 que corresponde a la suma de las caras de dos dados que deben ser lanzados para definir en cada compás, cuál es el elemento que deberá incluirse en la partitura.


La obra aparece publicada por primera vez en la Edición de J.J. Hummel, Berlín-Amsterdam 1793.Existen en muchos sitios referencias al “Juego de Dados Musical de Mozart”, en los cuales se enfatiza el número de posibles combinaciones en la elección de la partitura.

Existen, también, en la red de Internet varios sitios en los que se simula este Juego de Dados e inclusive se escucha el vals particular con la calidad sonora de un sintetizador y las restricciones de audio del equipo de cómputo con el que se conecta uno a la red.

Sin entrar al detalle más fino como lo es el que algunos compases son iguales aunque tengan distinto número que los identifica, en principio, el número de posibles partituras corresponde al número 1116 que se lee como el número 11 elevado a la potencia 16. Este número es tan grande que se estima que si se interpretaran continuamente y con un orden sistemático todas las partituras posibles y cada interpretación tardara 30 segundos entonces para agotar todas las posibilidades se excederían 728 millones de años, interpretando la obra de día y de noche y de manera continua.


Dicho lo anterior es importante mencionar que no todas las realizaciones para la suma de dos dados son igualmente probables. La distribución probabilística para la suma de las caras de dos dados lanzados al azar se deduce haciendo la observación de que la suma = 2 sólo cuando en ambas caras aparece el número 1, esto es: (1,1) y la suma = 3 cuando: (1, 2) o bien (2, 1), y así las demás, como la suma = 9 cuando: (3, 6) o (4, 5) o (5, 4) o (6, 3). Se observa que el número total de pares (i,j) es 36. Las referidas probabilidades de la suma son entonces:

Prob(2) = 1/36 = Prob(12)
Prob(3) = 2/36 = Prob(11)
Prob(4) = 3/36 = Prob(10)
Prob(5) = 4/36 = Prob(9)
Prob(6) = 5/36 =Prob(8)
Prob(7) = 6/36

Los 16 lanzamientos del par de dados se hacen de manera independiente y observar que las 16 sumas dieran como resultado, por ejemplo,


(2, 4, 11, 6, 7, 6, 11, 8, 3, 5, 4, 8, 2, 12, 10, 7),
tiene una probabilidad asociada. Se calcula su probabilidad de ocurrencia multiplicando las 16 probabilidades que le corresponden a cada uno de los números ejemplificados:
la del 2, la del 4, la del 11, etcétera. En este caso el resultado es:

Prob = (1 x 3 x 2 x 5 x 6 x 5 x 2 x 5 x 2 x 4 x 3 x 5 x 1 x 1 x 3 x 6).

De todas las más de 45,949 billones de posibles realizaciones (1116), muchas comparten el tener la misma probabilidad de ocurrir pero sólo una de ellas se distingue, desde el punto de vista probabilístico, en tener la probabilidad de ocurrencia más alta.
Ésta corresponde a la realización en donde para cada uno de los 16 compases, los dados suman 7 en todas las ocasiones. La probabilidad de dicha realización es (1/6)16.


Si como se dijo anteriormente, cada 30 segundos se interpreta una realización del Juego de Dados pero siguiendo al pie de la letra la selección aleatoria, la realización más probable que se ha mencionado ocurriría “en promedio” cada 44,728 años.


Haciendo un cálculo similar, una de las menos probables, por ejemplo (2, 2, 2, ..., 2), ocurriría “en promedio” cada 126,184 billones de años, en donde recordamos que un billón es un millón de millones (no así en otros idiomas). Por ello no pensamos que sea una exageración el que cada vez que se anuncia que se interpretará el Juego de Dados, se presume como Estreno Mundial. Se estima que el Big Bang (inicio del Universo como lo conocemos) ocurrió hace aproximadamente 13 a 15 mil millones de años y que la existencia de nuestro astro solar, que lleva media vida, durará todavía unos 5 mil millones de años. Esto es sin duda información para reflexionar. La obra El Juego de Dados, interpretada siguiendo la selección aleatoria descrita, para permanecer en nuestra cultura y agotar sus posibles realizaciones, evidentemente requerirá de la colonización de otros sistemas solares y que desde luego, no se les olvide llevarla.


Aún cuando los 176 compases fueron escritos para piano suelen hacerse arreglos para incorporar otros instrumentos. La Orquesta Sinfónica de Minería le proporcionó a este Instituto, el IIMAS de la UNAM, un arreglo de 176 compases para cuatro cuerdas con la idea de hacer un sistema computarizado. Dicho arreglo fue capturado con un programa llamado Finale y se crearon archivos en lo que es referido como un arreglo matricial de 11 x 16, en el cual en cada celda de la matriz podría identificarse el compás correspondiente para los cuatro instrumentos. Cada objeto o elemento de esa matriz queda asociado a su vez a un elemento gráfico y se desarrolló un programa que lleva a cabo una simulación aleatoria de las 16 tiradas de un par de dados y se identifica entonces la secuencia de objetos gráficos que forman la partitura decidida por el azar. Dicha combinación se imprime en forma de partitura y también por separado para cada uno de los cuatro instrumentos de cuerda.

Se ilustra la partitura de la realización más probable:


Todas las obras de Mozart han sido catalogadas por su número Köchel y esta obra en particular, Musikalisches Würfelspiel, es la K. 294 (Anh.C), así que ha sido propuesto que cada realización pudiera tener un número particular Köchel que la identifique. Es relativamente simple hacer una extensión con 16 “dígitos” utilizando un sistema de base 11, por ejemplo, los “dígitos” 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 y A. La correspondencia entre la suma de los dados y cada uno de los 16 compases representaría a la suma igual a 2 con “0”, la suma igual a 3 con “1” y finalmente la suma igual a 12 con “A”. Así la partitura más probable tendría el número Köchel, K. 294.5555555555555555.



Agradecemos la colaboración para la captura de la base de datos formada por los 176 compases a Federico O’Reilly Regueiro. Asimismo, agradecemos a Víctor Hugo Godoy Aguirre de la Dirección General de Servicios de Cómputo Académico por su apoyo en el modelado geométrico y animación del personaje que representa a Mozart.













Principio del formulario


База данных защищена авторским правом ©shkola.of.by 2016
звярнуцца да адміністрацыі

    Галоўная старонка