1
Conservatorio Conservatorio de Las Rosas A. C.
Tesina que presenta Leonardo Miguel Solorio Reyes en el área de Composición
Modulación Métrica: de sus posibilidades proporcionales proporcionales a sus características perceptuales perceptuales
Asesor
Juan Sebastián Lach Lau
Morelia, Michoacán, Junio, 2016
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1
Introducción
Capítulo I. Concepto de Modulación Métrica
1.1.
Concepto
3
1.2.
Orígenes
3
1.3.
Elementos Esenciales
6
1.3.A. 1.4.
Cálculo Alternativo de MM
Elementos secundarios, dos consideraciones sobre la MM
1.4.A.
Ecuación del tempo
1.4.B.
Sobre las indicaciones italianas de tempo
8 9 9 13
Capítulo II. La Modulación en la Partitura, Recuento Recu ento Histórico y Análisis
1.1.
Recuento Histórico
15
1.2.
Análisis
20
1.2.A.
Primer Cuarteto de Cuerdas de Elliott Carter
1.2.A.a. 1.2.B.
Sobre la Estructura de la Obra
Segundo Cuarteto de Cuerdas de Elliott Carter
20 27 38
1.2.C. Reliquary, Movimiento I de Dum Transisset de de Brian Ferneyhough
31
1.2.D.
34
Estudio II de Estudios sobre la Modulación Métrica, de José Manuel López López
1.2.E.
Silbury Air de Harrison Birtwistle
41
Capítulo III. Coincidencias y Disidencias
MM y Modulación Tonal
47
Capítulo IV. Modulación Métrica, Constitución y Expansión
1.1.
Constitución
53
1.2.
Expansión, Redes de Tempo
58
1.3.
Los Márgenes Perceptuales de la Modulación Métrica
66
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1.3.D.
Intervalo de Indiferencia
71
1.3.E.
Subjective Rhythmization
73
1.3.F.
Las Regiones
75
1.3.G.
El Presente Sicológico
76
1.3.H.
Percepción de Modulación de Tempo
79
4. Sobre una obra personal con MM, Post Sicut (2016)
83
Conclusiones
85
Bibliografía
87
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Introducción
1
El drum and bass y sus ritmos rápidos e irregulares que mediante secuenciadores crean texturas de complejidad superior a la realizable por cualquier intérprete humano, crearon en mí el deseo de componer una obra para percusiones acústicas que incorporara esta estética. El primer acercamiento primer acercamiento para hacerlo vino a través tra vés de la psicoacústica, ps icoacústica, en especial de un estudio1 sobre la ilusión auditiva de un tempo que parece acelerar infinitamente. El estudio me introdujo a un extenso abanico de artículos y obras enfocadas al aspecto temporal de la música, es decir al aspecto que relaciona los eventos sonoros con el tiempo cronométrico, aquel que resulta de retirar el timbre, el espacio, la dinámica y las alturas de la música. Respecto a este elemento elemento de la música, la primera mitad mitad del siglo XX fue fue particularmente interesante. El ritmicón de Henry Cowell o los estudios de la percepción del sonido y su relación con el tiempo de Paul Fraisse son ejemplos de una multitud de eventos que revolucionaron la teoría y la práctica musical. En los años cincuenta, Elliott Elliott Carter encontró una noción musical que entramaba e ntramaba los cambios de pulso, de compás y de subdivisión, consecuentemente el tempo fue entendido como una parte activa de la música y por consiguiente del compás y su contenido. Esta nueva perspectiva abandonó la concepción unidimensional del ritmo y se adentró en un sinfín de tempi de relaciones proporcionales. Ensayos como "… How time passes…" de Karlheinz Stockhausen abrieron un campo de experimentación temporal que apuntaba hacia la emancipación del ritmo. Estas nuevas investigaciones sugerían que los umbrales perceptuales del continuo de frecuencias sonoras por debajo de 20 Hz contenían universos musicales únicos y poco explorados. Sin embargo, como otros aspectos musicales que el siglo XX nos legó inconclusos, la temporalidad
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2
éstas una sensibilidad musical informada. El punto de partida es la composición influenciada por las teorías perceptuales del tempo, con el objetivo de crear y manipular elementos musicales. El objetivo principal de este trabajo es enriquecer mis composiciones a través del entendimiento integral de la modulación métrica, y promover activamente una herramienta musical que proponga retos a la práctica compositiva general y provea de recursos para una nueva valoración del tempo y su rol activo en la música. Teniendo en cuenta estas consideraciones, el presente trabajo está estructurado en cuatro capítulos: el primero discierne las diferencias semánticas y prácticas de todo lo que ha h a sido nombrado “modulación “modu lación métrica”; en este mismo capítulo se demuestra el funcionar básico de la misma. El segundo compila y
clasifica un vasto número de obras con modulación métrica y posteriormente analiza cinco obras que presentan un u n alto a lto número nú mero de modulaciones métricas. m étricas. El tercero tercer o propone pr opone un análisis comparativo c omparativo entre modulación tonal y modulación métrica que muestra los conceptos comunes que ambas comparten y las direcciones individuales en las que cada una se desarrolla. El último capítulo plantea una técnica que proporciona un cúmulo de tempi proporcionales al que cualquier tempo puede acceder y describe d escribe las particularidades psicoacústicas del ámbito de frecuencias sonoras pertinentes a la modulación métrica.
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3
Capítulo I Modulación Métrica
1. Concepto
La corta existencia temporal de la modulación métrica (de aquí en adelante referida como MM) permite entenderla como un término que aún evoluciona. Ha generado una multiplicidad de definiciones sin haber logrado todavía un consenso final. De la musicología al death metal se se ha nombrado MM a una variedad de conceptos diferentes. En este trabajo entenderemos a la MM como el cambio proporcional de tempo a través de una duración común. Las siguientes páginas explicarán con detalle lo que en este momento puede parecer como una formulación enigmática. 2. Orígenes
El término ‘modulación métrica’ fue inventado en 1951, en el análisis que Richard Franko Goldman2
realizó de Cello Sonata de Elliott Carter. Es el nombre que le dio al cambio de tempo y de compás.
Ejemplo 1: Cello Sonata, III, c. 6 y 7
Es una denominación derivada de un solo evento auditivo: el cambio de velocidad o del paso,3 que era la principal preocupación de Carter.4 Sin embargo la independencia de compás y tempo llevó a la
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4
de un pulso,7 cambios en la agrupación de acentos8 y agruparlas bajo el mismo nombre. Es posible que varios conceptos rítmicos hayan sido nombrados como MM porque comparten un modo de operación rítmica no tradicional; son ajenos a la acentuación habitual del compás o ajenos a la periodicidad permanente del pulso. En un artículo publicado en 1989 François Nicolas cuenta una anécdota: (...)hace ya cierto tiempo, Ellliott Carter fue interrogado en la radio sobre el significado del término “modulación métrica” frecuentemente utilizado en relación a su música. Fue
cuestionado en ese punto por un crítico musical -dotado de la competencia musical requerida- en presencia de un joven compositor del conservatorio -formado -f ormado por algunos maestros eruditos de esa institución-. Ninguno de los tres protagonistas supo (o quiso) proponer ese día una formulación simple del término, cada uno de ellos dio su opinión que se trataba de un fenómeno muy complejo para ser definido en pocas palabras(...). 9
El estado teórico de donde surgió la MM carecía de la fortaleza académica de otros campos musicales contemporáneos como el espectralismo, serialismo o la indeterminación. Los primeros intentos concisos de formular una definición clara se darían hasta principios del siglo XXI. Fernando Benadon,
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Los siguientes ejemplos muestran la equivalencia entre cambio de compás y modulación de tempo, en términos de Benadon. El ejemplo 2a muestra una MM, el 2b un cambio de compás. Ambos ejemplos son auditivamente idénticos y sólo se distinguen en el papel.
Ejemplo 2b. Cambio de compás Ejemplo 2a. Modulación Métrica
Después de la publicación del análisis de Cello Sonata de Franko Goldman, dijo que Carter "ha preferido el término modulación de tempo ya que es el tempo el que cambia, no la métrica".11 El argumento que Carter y Benadon comparten es claro e irrefutable: la modulación de tempo indica que el tempo cambia, la MM se refiere al cambio de métrica (o sea la distribución jerárquica de pulsos). A pesar de ello en el presente trabajo nos referiremos a la MM como el cambio proporcional de tempo
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• La tercera razón la otorga Jonathan Bernard al publicar en 1988, “sin embargo porque el término se ha vuelto común y porque Carter mismo lo utiliza, aquí sigo la práctica general”14 haciendo referencia al término “metric “metric modulation”. modulation”. Esta publicación es posterior a la de Schiff quien afirma que Carter prefería el término “Tempo “Tempo Modulation”. Modulation”.
Pienso que estos problemas terminológicos validan preguntarse por los papeles que el tempo y el tiempo juegan juegan en la práctica musical actual. Las conexiones entre el estado de la la teoría rítmica y la MM apuntan a una tendencia estética que oculta al tiempo detrás de otros aspectos sónicos. El énfasis de este interés es en ver al ritmo no solamente como lo que constituye la coordinación musical, sino que también es posible imaginarlo como poseedor de una estructura propia. Estudiar el ritmo es estudiar toda la música. El ritmo organiza y es a su vez organizado por todos los elementos que crean y dan forma a los procesos musicales. 15
Wallace Berry en 1987 decía: Todos los procesos-elemento son rítmicos. En un sentido importante, el estudio del ritmo es por lo
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Ejemplo 3. Figuras y su proporción a
partir de una redonda
La primera pregunta puede ser entendida con el siguiente mecanismo:
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Este primer enfoque avisora el segundo enfoque, que transita mas fácilmente por la totalidad de tempi que la MM permite. Esta formula de calcular o pensar la MM será útil para explorar con mas facilidad el ir del tempo a la figura, que será explorado en el capítulo IV. Mientras, este procedimiento es suficiente para comprender el funcionamiento de la MM. 3.A Cálculo alternativo de MM
Weisberg, ávido intérprete y director de la obra de Carter, sugiere dos maneras sencillas y efectivas para deducir las modulaciones:17
Un ejemplo de modulación, mediante el large way, es que a través de una negra de tresillo encontremos un valor más largo a través de la siguiente pregunta: ¿cuál es la figura a la que se
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metronómico de la negra (100 x 3 = 300) ya que hay tres corcheas de tresillo dentro de una negra. Si una corchea de tresillo es igual a 300 bpm, entonces una negra de tresillo que es dos veces más lenta es igual a 150 bpm.
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10
Ejemplo 6 . El aprendiz de brujo, parte de flauta, compases 12-15.
Sin embargo, la aceleración del tempo de 90 a 168 engendra un carácter musical de tiempo más denso dentro de un tiempo dilatado; la notación contribuye a hacer más claro este cambio de velocidad temporal. El número del lado derecho de la indicación de tempo enuncia la frecuencia del pulso elegido. Es el
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11
un pulso a 72 bpm resulta indiscernible (para el oído no entrenado27) de 73 bpm hasta 76.2 aproximadamente.28 François Nicolas, hace una aclaración muy precisa precisa al respecto: (…) no conviene confun dir métrica y compás.
La métrica, tiempo ha, es el objeto de comentarios
en el campo de la poética contemporánea... contemporánea... propongo simplificar las cosas así: el compás, co mpás, primer nivel de agrupamiento de los pulsos, discierne entidades articuladas articuladas que no se hacen perceptibles como aquellas perceptibles en una métrica, ese nivel más vasto de agrupamiento que a la vez establece un periodo (envolviendo más compases) y al mismo tiempo articula ar ticula interiormente cada uno de los compases (diferenciando sus pulsos de acuerdo a la polaridad fuerte/débil). Así la métrica subsume al compás de suerte que éste se convierte a la vez en elemento (uno de
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12
6a). De igual forma si la indicación de bpm cambia y la figura musical permanece tendremos también una modulación (ejemplo 6b).
Ejemplo 6a. MM que afecta a la
figura musical
Ejemplo 6b. MM que afecta al
tiempo físico
En los ejemplos 6a y 6b, la modulación reduce el tempo original a la mitad -al doble de lento- lo cual es
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13
Sobre las indicaciones italianas de tempo
A pesar de que las unidades de bpm son notadas en la gran mayoría de las MM, en realidad son prescindibles para indicar un cambio de tempo. Probablemente el primer ejemplo histórico de MM (realizada sin notación de bpm) sea Sub Arturo plebs. Compuesta por John Aleyn en 1358, este motete de disminución isorrítmica, 30 permuta la
duración de una línea melódica de 9 a 6 y finalmente a 4 unidades rítmicas.31 Esta música anterior a la tonalidad contiene también compases simultáneos de 3/4, 6/8 y 2/4 y ninguna indicación de tempo. Al prescindir de la estructura tonal, esta música modal encontró en el tempus perfectum e imperfectum32 y en la notación mensurada en general, un alto nivel de organización de eventos sonoros
; característica inherente a la música europea.
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La principal intención de este trabajo es crear un enfoque práctico de la expansión de la MM
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Capítulo II La Modulación en la Partitura Recuento Histórico y Análisis
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Alter Face (2004) • Benjamin, George
Antara (1985-7), p. 72. Fantasy on Iambic Rhythm (1985), compás 101.
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Lachrymae (1976), p. 12. Suite for cello solo (1964) Winter Words, 3. Wagtail and Baby (1953). Gloriana, 5. first Lute Song (1977), pg. 85, 86, 87.
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Body Mandale (2006), p. 46. Concerto Percussion (1997), p. 60. • Hurtado, José Luis
L´Arditto (2006), pp. 8, 10.
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54
La siguiente tabla muestra las modulaciones que la tabla anterior contiene, mediante las figuras señaladas estas figuras permiten.
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55
Ejemplo 3a. MM 3/5
Ejemplo 3b. MM 4/7
Ejemplo 3c. MM 8/9
El total de de tempi distintos generados en la tabla es el siguiente: 1. 6.6 bpm 2. 7.5 3. 8.6 4. 10 5. 12 6. 13.2 7. 15 8. 17.14 9. 20 10. 22.5 11. 24 12. 25.71 13. 26.4 14. 30 15. 33.28
16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
33.6 36 37.5 40 41.85 45 46.2 48 50 51.42 52.5 52.8 27.5 60
28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.
67.5 68.57 70 72 75 77.14 80 84 90 96 100 105 108 120 135
43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54.
140 150 160 180 210 240 270 300 360 420 480 540
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56
En el ejemplo 1 se muestran modulaciones que son prácticamente iguales, como 4:5 y 7:9 que en el tempo ejemplificado de 60 tienen un bpm de 75 y 77.14 respectivamente, lo cual representa una variación del tempo tempo de 2.8%, que como se verá más adelante, resultan perceptualmente indistinguibles. Sin embargo, existirán razones para elegir un tempo sobre otro, aún con con diferencias tan sutiles. El contexto puede ayudar a determinar la fracción adecuada, dependiendo del tipo de subdivisión de la que se parta y a la que se llegue. Otra razón para elegir un cambio de tempo sobre otro con características semejantes podrá ser el efecto acumulativo que ejerzan a lo largo de 10 compases, por ejemplo si se modula a través de 7:9 en compás de 4/4 a partir de negra 60, después de diez compases habrán transcurrido 12 segundos y 14 segundos en 4:5; variación que sin ser especialmente significativa en términos musicales, tiene la capacidad de seguir acumulándose para crear accelerandos mesurados con velocidades distintas. Como ya hemos visto, la duración común entre dos tempi es un factor fundamental para la MM. Cada tempo ofrece un vasto abanico de duraciones, desde las más comunes como negras y corcheas hasta subdivisiones irregulares dentro de irregulares. El uso de nueve subdivisiones produce 54 tempi proporcionales al tempo original. En teoría cada uno de los 54 nuevos tempi conducen a 54 tempi alternativos, con un total de 54² tempi. Si iteramos
el proceso, cada nuevo tempo brinda 54 tempi, que resulta en 54³. Una secuencia de modulaciones como estas, tiene una pauta de crecimiento exponencial, segmentado por el número de iteraciones del proceso.
bpm
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57
estructura rítmica de la obra. Tal situación puede ser presentada así: q = 75 (800 ms.) y q = 90 (666.667 ms.) comparten al menos un pulso común, que es un tresillo de dieciseisavo en 75 bpm y un quintillo de dieciseisavo en 90 bpm. La imagen contiene cinco figuras musicales las cuales determinan el número de tempi posibles para modular. Mediante tuples anidados el número de subdivisiones es teóricamente ilimitado, al igual que tempi posibles a modular. Para el escucha la modulación sólo será reconocible una vez que la duración común entre ambos tempi funja bajo una subdivisión distinta a la original, o su agrupación sea diferente a la del compás anterior. Hasta el momento hemos revisado las modulaciones que se dan de un tempo a otro a través de todo el rango de
Ejemplo 5. Subdivisiones de dos tempi
subdivisiones simples, (corcheas, semicorcheas, tresillos, quintilos, sietillos, fusas y nonillos) por lo tanto, las fracciones que expresan la relación entre tempi se pueden representar con fracciones con numeradores y denominadores que transitan todas las posibilidades del 1 al 9 como 1:9, 3:4, 6:7, 9:1 (ver tabla tal y tal otra). Estas relaciones pueden trascender a una infinidad de nuevos tempi con relaciones proporcionales más complejas, pensadas en el siguiente concepto.
2. Expansión, Redes de Tempo
Como recién vimos un tempo contiene un amplio número de subdivisiones. Sin embargo la cantidad de tempi a donde se puede transitar es limitada, un rejilla discontinua.58
Es posible expandir el campo de duraciones accesibles a través de una idea propuesta por Fernando Benadon,59 que es en la concatenación de dos modulaciones métricas para permitir el acceso a un número exponencial de posibles duraciones que él llama Tempo Networks. En su breve pero tremendamente conciso artículo describe a los Tempo Networks como una herramienta capaz de
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58
(0.88) y 9/8 (1.125). Utilizando las mismas subdivisiones es imposible acceder a tempi como 0.9 o 1.1 del tempo inicial. La concatenación de modulaciones permite internarse en áreas temporales inaccesibles para la modulación simple, que llegan a ofrecer hasta 25 gradaciones en donde la otra ofrecía 2, por lo que el tempo original puede acceder a puntos como 1.01 de su velocidad inicial. Los Tempo Networks pueden ser utilizados de dos d os maneras: • Como dos modulaciones métricas sucesivas • Como una modulación compuesta de dos modulaciones simultáneas.
Ambas maneras tienen el mismo fin, su uso dependerá del contexto. Un ejemplo de estas redes de tempo: Dos modulaciones sucesivas:
Ejemplo 6
Dos modulaciones simultáneas:
Ejemplo 7 .
Benadon infiere que el uso natural de Tempo Networks es la consecución de dos o más modulaciones. En realidad las redes de modulaciones se pueden concatenar ocultando las costuras, es decir, sin establecer puntos de estabilidad que permitan al escucha reconocer la modulación, como el ejemplo siguiente:
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59
Ejemplo 8.
En el segundo compás el pulso se mantiene en la subdivisión generada en el primer compás. Hasta ese punto es posible pensar pe nsar que el tempo ha permanecido perma necido inalterado (desde el punto pu nto de vista del escucha). Sin embargo la nueva subdivisión a tresillos de negra indica un tuple anidado o una modulación, que es el pivote de una segunda modulación, que en este caso particular da pie a una tercera. La atípica agrupación de 5 corcheas en el compás de 10/8 tiene por fin no alterar la distribución métrica(métrica en el sentido de jerarquía de pulsos), que bien podría ser notada de manera tradicional y remediar la acentuación con ligaduras de fraseo. Como se vio en el ejemplo 1, nueve figuras musicales distintas permiten 54 modulaciones – 72 72 si contamos las fracciones no simplificadas que musicalmente tienen sentido, como 2:4 con 1:2 – . A través de las redes de tempo, con el mismo número de figuras podemos acceder a cerca de 6300 modulaciones (contando todas las modulaciones, incluyendo fracciones no simplificadas). El total de tempi a los que se puede modular bajo este concepto, utilizando las nueve figuras ya utilizadas es de 483 que permiten teóricamente crear pulsos desde 12 ms. (5000 bpm) hasta 81 s. (0.74 bpm) a partir de un pulso de 1 s ó 60 bpm.
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60
Representado de forma musical la expresión (1/1 x 2/1):
Ejemplo 9
La negra es el valor 1 y a partir de él se derivan los valores de las demás figuras. Del lado izquierdo de la ecuación hay una negra, entonces hay 1 unidad (numerador) con valor de 1(denominador). Del lado derecho de la ecuación tenemos una corchea, de la cual caben dos en una negra. Por lo tanto tendremos una unidad (denominador) con valor de 2 (numerador), que podemos representar de la siguiente manera:
Temp o Inicial
Temp o Fina Fina l
Nú m e ro d e unidades
Valor de la un ida d
Valor de la un ida d
Nú m e ro d e unidades
1
1
2
1
1
1 1/1
2/1 Ejemplo 10. Figuras rítmicas y su representación numérica
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61
Tempo Inicial nicial
A
Nú m e ro d e unidades
Valor d e la unidad
Valor d e la unidad
Nú m e ro d e unidades
2
3
2
7
2
2 B
7
2
4
3 3
7
1
4
4
D
7
2
4 C
Tempo Fina Fina l
3 3
3
1
7 7
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62
Las cuatro multiplicaciones tienen un sólo producto, 4/21, lo que significa que podemos transitar a este tempo de cuatro formas musicales distintas. En aras de la claridad a continuación expongo todas las relaciones de tempo que las nueve figuras utilizadas contienen, excluyendo productos redundantes. Las figuras precisas y modulaciones que estos productos representan pueden ser deducidos dedu cidos a través del método mostrado. 81/1 (81)
81/7 (11.571)
54/35 (1.5428)
16/27 (0.59)
72/1 (72)
45/4 (11.25)
49/32 (1.53125)
7/12 (0.5833)
64/1 (64)
56/5 (11.2)
32/21 (1.5238)
9/16 (0.5625)
63/1 (63)
54/5 (10.8)
40/27 (1.4814)
14/25 (0.56)
56/1 (56)
32/3 (10.66667)
72/49 (1.46938)
27/49 (0.5510)
54/1 (54)
21/2 (10.5)
35/24 (1.4583)
35/64 (0.5468)
49/1 (49)
72/7 (10.2857)
81/56 (1.446)
15/28 (0.5357)
48/1 (48)
81/8 (10.125)
36/25 (1.44)
8/15 (0.533)
45/1 (45)
10/1 (10)
10/7 (1.428)
21/40 (0.525)
42/1 (42)
49/5 (9.8)
64/45 (1.4222)
25/48 (0.5208)
81/2 (40.5)
48/5 (9.6)
45/32 (1.40625)
14/27 (0.518)
40/1 (40)
28/3 (9.33)
7/5 (1.4)
18/35 (0.514)
36/1 (36)
64/7 (9.142)
48/35 (1.37142)
25/49 (0.5102)
35/1 (35)
9/1 (9)
49/36 (1.3611)
32/63 (0.5079)
32/1 (32)
35/4 (8.75)
27/20 (1.35)
40/81 (0.493)
63/2 (31.5)
42/5 (8.4)
21/16 (1.3125)
24/49 (0.489)
30/1 (30)
25/3 (8.333)
64/49 (1.306)
35/72 (0.4861)
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63 20/1 (20)
48/7 (6.857)
32/27 (1.1851)
35/81 (0.432)
56/3 (18.666)
27/4 (6.75)
7/6 (1.1667)
7/16 (0.4375)
18/1 (18)
20/3 (6.667)
8/7 (1.14)
27/64 (0.4218)
35/8 (4.375)
45/7 (6.4285)
9/8 (1.125)
20/49 (0.4081)
30/7 (4.2857)
32/5 (6.4)
28/25 (1.12)
25/63 (0.3968)
64/15 (4.26667)
63/10 (6.3)
10/9 (1.111)
32/81 (0.395)
21/5 (4.2)
25/4 (6.25)
54/49 (1.1020)
25/64 (0.3906)
25/6 (4.1667)
56/9 (6.22)
35/32 (1.093)
7/18 (0.3888)
49/12 (4.08333)
49/8 (6.125)
49/45 (1.08889)
8/21 (0.3809)
81/20 (4.05)
6/1 (6)
27/25 (1.08)
10/27 (03703)
63/16 (3.9375)
35/6 (5.833)
15/14 (1.07)
18/49 (0.3673)
35/9 (3.8889)
81/14 (5.7857)
16/15 (1.066)
9/25 (0.36)
27/7 (3.85714)
40/7 (5.7142)
21/20 (1.05)
16/45 (0.3555)
15/4 (3.75)
45/8 (5.628)
25/24 (1.0416)
7/20 (0.35)
56/15 (3.7333)
28/5 (5.6)
28/27 (1.03)
25/72 (0.3472)
18/5 (3.6)
49/9 (5.444)
49/48 (1.020833)
28/81 (0.3456)
25/7 (3.5714)
27/5 (5.4)
36/35 (1.02)
12/35 (0.34)
32/9 (3.5556)
16/3 (5.333)
64/63 (1.01587)
21/64 (0.328125)
7/2 (3.5)
21/4 (5.25)
63/64 (0.9843)
16/49 (0.3265)
24/7 (3.428571)
36/7 (5.142)
48/49 (0.979591)
9/28 (0.3214)
27/8 (3.375)
81/16 (5.0625)
35/36 (0.972)
8/25 (0.32)
10/3 (3.33)
49/10 (4.9)
24/25 (0.96)
20/63 (0.317)
49/15 (3.2667)
24/5 (4.8)
27/28 (0.9642)
14/45 (0.311)
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64 81/28 (2.89285)
35/16 (2.1875)
8/9 (0.88)
16/63 (0.2539)
72/25 (2.88)
54/25 (2.16)
7/8 (0.875)
20/81 (0.2469)
20/7 (2.857)
15/7 (2.1428)
6/7 (0.857)
12/49 (0.2448)
45/16 (2.8125)
32/15 (2.133)
27/32 (0.84375)
6/25 (0.24)
14/5 (2.8)
21/10 (2.1)
21/25 (0.84)
15/64 (0.2343)
25/9 (2.77778)
25/12 (2.0833)
40/49 (0.816)
7/30 (0.233)
49/18 (2.722)
56/27 (2.074)
45/56 (0.80357)
9/40 (0.225)
27/10 (2.7)
72/35 (2.05714)
64/81 (0.7901)
8/35 (0.22)
8/3 (2.66)
49/24 (2.041)
25/32 (0.78125)
7/32 (0.218)
21/8 (2.625)
81/40 (2.025)
7/9 (0.777)
10/49 (0.2049)
18/7 (2.571)
63/32 (1.96875)
27/35 (0.7714)
16/81 (0.1975)
64/25 (2.56)
49/25 (1.96)
49/64 (0.7656)
7/36 (0.1944)
81/32 (2.53125)
35/18 (1.944)
20/27 (0.7407)
9/49 (0.1836)
63/25 (2.52)
27/14 (1.928)
36/49 (0.7346)
8/45 (0.177)
49/20 (2.45)
48/25 (1.92)
35/48 (0.7291)
7/40 (0.175)
12/5 (2.4)
40/21 (1.904)
18/25 (0.72)
14/81 (0.172)
64/27 (2.3703)
15/8 (1.875)
32/45 (0.711)
6/35 (0.17)
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65 12/1 (12)
25/16 (1.5625)
49/81 (0.6049)
35/3 (11.667)
14/9 (1.556)
25/42 (0.5952)
El total de tempi en esta red específica es de 483, a través de cerca de 6300 maneras distintas de hacerlo. Hemos utilizado un rango de 9 subdivisiones distintas; si utilizáramos 13 subdivisiones, tendríamos acceso a un aproximado 28,561 maneras de modular. Hasta el momento hemos revisado el alcance de una red de tempi de dos niveles. Benadon expone la posibilidad de expandir este concepto c oncepto a través de la repetición y suma del proceso. Una U na red de tempi de tres niveles, con las mismas nueve subdivisiones que hemos utilizado permite 531, 441 posibilidades de modulación, por lo que plantear rítmicamente intervalos como el ragisma (4375:4374) (4375:4374)60 es posible con una modulación de cuatro niveles. Desde la práctica musical el alcance de esta herramienta es ilimitado; abre la puerta a un reino en el cual se puede transitar a cualquier tempo de forma precisa.
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66
88 bpm a 89 bpm? ¿Existe una similaridad cognitiva entre proporciones de tempo y proporciones de altura? ¿Qué “propiedades de los objetos matemáticos se derivan en cualidades sensoriales” 63?
Lo que mejor ha respondido a estas preguntas son las ciencias cognitivas y perceptivas, que por más de un siglo han generado un alud de información útil para la presente investigación. La selección y discusión de estudios al respecto no pretende negar el complejo fenómeno (en todo caso confirma lo lejos que las ciencias se encuentran de dar una explicación completa respecto a temas elementales en música) de la percepción del tempo, sino discutir con elementos empíricos empíricos (si bien no ideales) ideales) todas las características que el tempo expresa en quien lo escucha, produce e imagina. Al principio de esta investigación, yo mismo mantenía mis reticencias respecto a incorporar esta clase de información a mi proceso compositivo; muchos muc hos estudios ricos en datos y muestras apenas rasgan la superficie su perficie de temas que como músicos comprendemos o creemos comprender intuitivamente. Me parecía el intento de algunos científicos por entender algo que para nosotros parece instintivo. Sin embargo, tomados como un cúmulo de información, que en ocasiones puede ser contradictorio, toma un cariz mucho más útil y atractivo de lo originalmente pensado. Estas investigaciones son particularmente importantes para las teorías de tempo musical. Sugieren de manera imponente que hay limites perceptivos y cognitivos sobre el rango de tempo aplicado a la música y, más importantemente, que existen diferencias entre nuestro entendimiento e interpretación de
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67
3. A. Elementos a Explorar
En estos estudios el tempo tiene el mismo sentido que el tempo musical, sin embargo, el concepto de modulación de tempo responde a cualquier modificación de tempo, sea continua (accelerandos, ritardandos) o discontinua. La principal búsqueda de éste compendio de estudios es conocer los aspectos relacionados con el tempo y su percepción así como conocer la manera en que el contexto afecta la percepción del cambio de tempo. Este espectro de búsqueda implica fenómenos como: • Tempi preferidos, los tempi caracterizados como los más cómodos, sean producidos o escuchados. • El rango en el que somos más propensos a escuchar un pulso o la “ritmización subjetiva”64
decir el periodo que tendemos a juzgar como ni largo ni corto. • El intervalo de indiferencia, es decir el • Nuestra sensibilidad a los cambios de tempo, en distintos tempi iniciales y en distintos contextos.
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68
perceptuales es un arma de dos filos. Por un lado nos provee con la maravillosa evidencia de fenómenos insospechados por la intuición más fina y por el otro terminamos con un cúmulo de información que parece imposible de emplear con éxito.
3. B. Terminología
Antes de adentrarnos en estos estudios y sus posibles conclusiones es necesario entender a que se refiere su terminología. • Un Event Onset marca el punto en el tiempo en el que un solo evento sonoro (un sonido)
comienza. Usualmente, en estos estudios, la longitud de las duraciones es pequeña por lo que generalmente resulta práctico medirlas en milisegundos (ms). • Inter-Onset Interval (abreviado como IOI) se refiere al intervalo que ocurre entre los Onsets de
dos eventos auditivos sucesivos.
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69
(primer Onset anómalo) es definido como el primer onset • El FAO es el First Anomalous Onset (primer que ocurre más temprano o más tarde de lo que es esperado de un patrón de pulsos establecido previamente. • Tempo aquí es definido como la tasa a la que pulsos sucesivos ocurren. Aquí la medida
usualmente es bpm. Una herramienta que resultará práctica en las siguientes páginas es la relación entre milisegundos y bpm. Si conocemos la velocidad de un pulso en milisegundos es posible calcular su bpm de la siguiente
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70
estudios el Tempo Change es entendido como cualquier tipo de cambio en el tempo, sea gradual (accelerandos, ritardandos) o con cambios bruscos relacionados a la expresividad o estilo.65 Antes de proceder es necesario mencionar que a través de estos estudios puede ser imposible llegar a una conclusión clara y firme respecto a nuestro entendimiento perceptivo de las modulaciones de tempo. Podría pensarse que estos estudios carecen de una validez absoluta en el mundo real de la música, debido a la gran cantidad de variables contextuales que ocurren en una situación musical real. Sólo se circunscriben a tener una validez parcial. Sin embargo su estudio y conocimiento es pertinente y en él, a pesar de no siempre llevarnos a donde deseamos, existe información necesaria para la práctica como compositor c ompositor que me fue inesperada. Por nombrar un caso, cas o, hay estudios estud ios66 que enfatizan el fenómeno de la percepción de acentos y duraciones y cómo se entrelazan para modificar la sensación de altura. Idea que para nosotros como músicos resulta irreal y hasta risible. Hemos aprendido a
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71
respecto concurren en que el área circundante a 700 ms es un umbral importante de la percepción y en él se encuentra el promedio del tempo preferido, también llamado “Tempo Espontáneo˝ o “Ritmo Natural” por p or no ser un tempo planeado, como encontró Fraisse F raisse69 que investigó este mismo fenómeno
pidiendo a los sujetos de su estudio estud io marcar un pulso a una u na velocidad cómoda, un método común c omún para la exploración de la percepción relacionada con el tempo, con resultados similares. A pesar de que la mayoría de la evidencia al respecto es concordante en términos generales y que es posible decir que esa región reg ión de la percepción es real, rea l, los intervalos de indiferencia son únicos para cada persona y cambiantes dependiendo de la situación y el e l contexto. Ya en 1979 Brown70 publica un estudio en el que señala que las muestras individuales de tempo preferido pueden llegar a ser tan cortas como 100 ms. Sin embargo agrega una idea muy valiosa asociada con la música: “el área de 750 ms. en la sicología del tempo y del movimiento no puede ser
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72
crear una visión dual: si escribo una estructura de tempi a partir de ideas musicales o a partir de la fisiología. Cada una de estas estructuras provee de un marco específico de referencia pero es posible que no difieran mucho entre ellas, ya que estos datos sobre la percepción proveen de detalles sutiles que pueden ser asimilados por la música. El ciclo Scardanelli del mismo Holliger, contiene un movimiento en el cual cada cantante debe de cantar con un tempo independiente del resto a partir de su propio pulso, tomado de su muñeca por ellas mismas. Desconozco las consideraciones consideraciones de Holliger sobre el pulso de este movimiento pero es claro que entendía al tempo como parte activa y presente en su obra, que es uno de los objetivos de esta investigación. También ha sido probado que estados de ánimo, funciones corporales conscientes o inconscientes
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73
los ritmos y pulsos tienen a partir de este nuevo tempo. ¿Son corcheas de este tempo o tresillos del anterior? Establecer un punto de referencia permite juzgar las relaciones temporales. Sin embargo es posible que existan subterfugios en la interpretación compositiva de estos datos, ya que a partir de la renuencia que los sujetos expresaban a cambiar de tempo podemos deducir que existe una capacidad para precisar planos temporales discernibles entre ellos.Aquí es necesario enfatizar la importante diferencia que existe entre establecer un tempo en la partitura y establecerlo en quien lo escucha. 3. E. Subjective Rhythmization
Dos estudios presentados por Handel75 relacionados con la percepción del tempo dieron más evidencia coincidente con el hecho de tener un tempo preferido entre 500 y 600 ms. En estos estudios los sujetos
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74
Como fue el caso del Intervalo de Indiferencia, la Ritmización Subjetiva también tiene una larga historia de estudios que corroboran y especifican de manera detallada los pormenores de este fenómeno. Un estudio realizado en 1990 por Monahan y Hirsch encontró que podemos discernir discernir un intervalo con mucha precisión en zonas periodos tan cortos como 100 ms.77 También notaron que
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75
discernimiento de tempo sobre just noticeable difference. Sus resultados cuentan que sin importar la formación musical es posible discriminar diferencias en el tempo de hasta 2.5% cuando las muestras son iguales a 240 ms. o más grandes. Cuando son menores el grado de imprecisión es mucho más amplio.
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76
tocaran 4 notas seguidas al tempo de su elección. En promedio fueron tocadas 1.8 veces más rápido que el grupo de 2 notas. El grupo de 6 notas fue tocado 2.2 veces más rápido que el de 2, lo que sugirió que los sujetos estaban intentando unificar el grupo de notas cuando el número de elementos aumentaba. A través de estudios posteriores se ha encontrado que existe una interacción entre el número de
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77
tiempo es deliberadamente comprimido emerge una sensación de presión, de “ muy poco tiempo” tiempo” como
un factor importante que condiciona la recepción - algo que indica al escucha al categorizar la música como “rápida” .87
Que es el mismo concepto que Cooper y Meyer anuncian,
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El fenómeno del presente sicológico fue explorado en 1956 cuando Fraisse busca establecer los límites de este concepto a través de una serie de experimentos91 y encuentra que 5 o 6 s son el límite máximo para que comprendamos un evento musical como unidad. Gracias a sus resultados logra descartar las
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