Fibonacci 数列
关于数列，音乐上很早就有人使用了：
Bartok's的 《Music for Strings ,Percussion and Celestra》
Bartok 是一近代作曲家，其风格异于古典之作曲方式。他尝试以数学性的作曲
方式，作为处理音乐细部的新方法。他同时关心以纯粹科学的方法来达到目标，
并在意这些方法对于作曲家所产生的刺激效应，他相信在不断尝试后，终究可以
成为一新的音乐理论。
在这首译为"为弦乐、打击乐和钢片琴所做的音乐"中，Bartok以Fibonacci数列：1,2,3,5,8,13,21,34...，后面一个数字为前面两位数字和的数列，作为他操作的依据。整首曲子的张力，在于第一乐章的每个转换点均和这个数列相关，如弦乐在第34小节上提高音量；在第56小节时达到一个高潮；钢片琴于第77小节的末尾一拍介入。这些手法导致了适合于此数列的一种错置之多层次结合，而这错置被Bartok用来完成他的音乐。
1.What is Fibonacci series?
F(n+2) = F(n+1) + F(n) is the official definition of the Fibonacci sequence, as provided by the Fibonacci Association,
This sequence arises from the resolution of one of the problems of Abacci Liber:
If one places a rabbit couple in an enclosed place, how many rabbits would one obtain after a certain time if it is presupposed that they reproduce once per month, and that those born can reproduce at the age of a month ?
One obtains 1,1,2,3,5,8,13,21,34,55... after each month, and one can check that the relation F(n+2) = F(n+1) + F(n) is correct, for example for n=8, one has : 55=34+21.
Obviously, in reality the this is not the case; however, if there were perfect conditions, without the presence of harmful bacteria for example, one sees intuitively that this model could apply.
In fact, the Fibonacci Sequence contains much more than the 'natural proportions' that artists have traditionally seen. Indeed, very recently, Robert Devaney, eminent professor at the University of Boston, has discovered the appearance of Fibonacci numbers in the Mandelbrot Group.
In visual arts and music, the use of Fibonacci numbers is generally related to proportion, particularly linked to the golden section; examples of this can be found in the work of Michel -Ange, J.S Bach, Brahms, Scriabin, and many artists in the twentieth century.
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Bartok's的 《Music for Strings ,Percussion and Celestra》
Bartok 是一近代作曲家，其风格异于古典之作曲方式。他尝试以数学性的作曲
方式，作为处理音乐细部的新方法。他同时关心以纯粹科学的方法来达到目标，
并在意这些方法对于作曲家所产生的刺激效应，他相信在不断尝试后，终究可以
成为一新的音乐理论。
在这首译为"为弦乐、打击乐和钢片琴所做的音乐"中，Bartok以Fibonacci数列：1,2,3,5,8,13,21,34...，后面一个数字为前面两位数字和的数列，作为他操作的依据。整首曲子的张力，在于第一乐章的每个转换点均和这个数列相关，如弦乐在第34小节上提高音量；在第56小节时达到一个高潮；钢片琴于第77小节的末尾一拍介入。这些手法导致了适合于此数列的一种错置之多层次结合，而这错置被Bartok用来完成他的音乐。
1.What is Fibonacci series?
F(n+2) = F(n+1) + F(n) is the official definition of the Fibonacci sequence, as provided by the Fibonacci Association,
This sequence arises from the resolution of one of the problems of Abacci Liber:
If one places a rabbit couple in an enclosed place, how many rabbits would one obtain after a certain time if it is presupposed that they reproduce once per month, and that those born can reproduce at the age of a month ?
One obtains 1,1,2,3,5,8,13,21,34,55... after each month, and one can check that the relation F(n+2) = F(n+1) + F(n) is correct, for example for n=8, one has : 55=34+21.
Obviously, in reality the this is not the case; however, if there were perfect conditions, without the presence of harmful bacteria for example, one sees intuitively that this model could apply.
In fact, the Fibonacci Sequence contains much more than the 'natural proportions' that artists have traditionally seen. Indeed, very recently, Robert Devaney, eminent professor at the University of Boston, has discovered the appearance of Fibonacci numbers in the Mandelbrot Group.
In visual arts and music, the use of Fibonacci numbers is generally related to proportion, particularly linked to the golden section; examples of this can be found in the work of Michel -Ange, J.S Bach, Brahms, Scriabin, and many artists in the twentieth century.