臺灣國際科展

西爾平斯基船帆與掛毯圖形應用於數位圖形與數位音樂創作

科展類別
臺灣國際科展
屆次
2006年
科別
電腦科學
學校名稱
國立臺中第一高級中學
指導老師
梁銘誠
作者
林自均
關鍵字
西爾平斯基、數位圖形、數位音樂、Sierpinski、Digital patterns、Digital music

摘要或動機

西爾平斯基船帆(Sierpinski Gasket)與西爾平斯基掛毯(Sierpinski Carpet)都屬於碎形(fractals)圖形的一種,可以利用迭代運算系統IFS(Iterated Function Systems)碼來產生,代入迭代運算方程式後,經由多次的運算,可以得到重覆的圖形。本研究中,我將提出一些作法,找出西爾平斯基船帆與掛毯圖形其遞迴關係式,進而討論出其數位圖形之規律性及所涵蓋的內容與性質,著重在推廣西爾平斯基船帆與掛毯圖形的概念,將一段音樂曲取出,把它們看成反覆隨機迭代點,利用程式經由多次的插值運算,計算出各段音符。最後加入基因演算法來解決音符長短的問題,把製造好的音符染色體放置到交配池中,以隨機的方式在交配池中選取其中之一個染色體進行交配的動作,此二音符染色體會交換彼此的基因,產生下一代新的代表音符長短之染色體,應用於數位音樂創作,而衍生的西爾平斯基船帆與掛毯圖形新穎應用與創新的結果,希望能提供數位音樂創作的多樣性,更進而可以找出「好聽的音樂」與數學的直接關聯性。‘Sierpinski Gasket’ and ‘Sierpinski Carpet’ are two graphics that belong to fractals. They can be produced by IFS (Iterated Function Systems). By iterative computation of many times, we can obtain the similar graphics. In my research, there are some methods to generate Sierpinski Gasket, Sierpinski Carpet, and the iterative algorithms. In addition, I would discuss the regularity and the content as well as the properties of those digital patterns. At last, the advanced application of Sierpinski Gasket and Sierpinski Carpet to digital music pieces was presented. The program took a note of several measure of music as the beginning point, and made the IFS calculations for each new note in each measure. But there was no difference in beats if you just make the IFS iteration. So I changed the beats with genetic crossover method. In this research, the expression of the DNA to each beat of note was adopted. The same way, it took a note as a beginning point. And the system obtained the new DNA from the old notes for new ones randomly. That would make a piece of brand new music. What I want to do in this research is improve the multiformity of music and find what the relationship is of ‘good music’ and mathematical algorithms.


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