The Golden Ratio and Fibonacci Numbers
James Tenney reconceived his piece "For Ann (Rising)", which consists of up to twelve computer-generated tones that glissando upwards (see Shepard tone), as having each tone start so each is the golden ratio (in between an equal-tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.
Ernő Lendvaï analyzes Béla Bartók's works as being based on two opposing systems: those of the golden ratio and the acoustic scale. In Bartok's Music for Strings, Percussion, and Celesta, the xylophone progression at the beginning of the 3rd movement occurs at the intervals 1:2:3:5:8:5:3:2:1. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose Croix.
The golden ratio is also apparent in the organization of the sections in the music of Debussy's Image, "Reflections in Water", in which the sequence of keys is marked out by the intervals 34, 21, 13, and 8 (a descending Fibonacci sequence), and the main climax sits at the φ position. "Prelude to the Afternoon of a Faun" also reaches a climax point, marked by the entrance of the antique cymbal, at the φ position.
Many of the important musical events in Krzysztof Penderecki's "Threnody for the Victims of Hiroshima" occur at φ positions.
Boards of Canada have discussed using the golden ratio and Fibonacci numbers in their work. Song titles such as "Music is Math", and numerous songs featuring samples of people counting or discussing numbers, also illustrate the influence of mathematics on their music.
Read more about this topic: Mathematics Of Music
Famous quotes containing the words golden, ratio and/or numbers:
“He hangs in shades the orange bright,
Like golden lamps in a green night,
And does in the pomegranates close
Jewels more rich than Ormus shows;
He makes the figs our mouths to meet,
And throws the melons at our feet;
But apples plants of such a price
No tree could ever bear them twice.”
—Andrew Marvell (1621–1678)
“A magazine or a newspaper is a shop. Each is an experiment and represents a new focus, a new ratio between commerce and intellect.”
—John Jay Chapman (1862–1933)
“Old age equalizes—we are aware that what is happening to us has happened to untold numbers from the beginning of time. When we are young we act as if we were the first young people in the world.”
—Eric Hoffer (1902–1983)