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1 Listen to the next generation of Beethoven’s compositions: Digital music generator using fractal and genetic crossover concepts FractalsGeneMatrixResultsConclusions

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2 What Are Fractals ? FractalsGeneMatrixConclusionsResults Fractals have the property of similarity within diversified changes. They are similar in any section and direction.

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3 Sierpinski Gasket FractalsGeneMatrixConclusionsResults

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4 Sierpinski Carpet FractalsGeneMatrixConclusionsResults

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5 Sierpinski Gasket FractalsGeneMatrixConclusionsResults

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6 Sierpinski GasketSierpinski CarpetSierpinski Pentagon Sierpinski HexagonCantor SquareBox Fractal FractalsGeneMatrixConclusionsResults

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7 Fractal Music - Melody FractalsGeneMatrixConclusionsResults

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8 The Genetic Crossover Method was proposed by John H. Holland, a professor from the University of Michigan, in 1975 [6]. It was based on the concept of “Survival of the fittest”, he created an environment that allows chromosomes to crossover and produce genes of the best possible recombination. What are Genetic Crossover Concepts (GCC) ? FractalsGeneMatrixConclusionsResults

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9 Fractal Music - Beats FractalsGeneMatrixConclusionsResults

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10 Fractal Music - Beats FractalsGeneMatrixConclusionsResults

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11 This study selected parent music that is “pleasant to ears” such that “the music similarity with parent music” is used as the basis of judgment, offspring generation music similar with the characteristics of parent music, will be generated (“music similarity” here refers to the relative note length position) to eliminate “different” offspring. It is thought that more pleasant music will then be composed. Imitated Parent Judgment (IPJ) FractalsGeneMatrixConclusionsResults

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12 Definitions of “Imitated Parent Judgment Function” A is the assembly of note pitch sequence of initial notes ; A = { a 1, a 2, a 3...... a n } B is the assembly of note pitch sequence of next generation; B= { b 1, b 2, b 3...... b n } Flag i refers to the determinant Boolean constant of Rule i ; V t is the total vote V i is the vote provided by each rule MAX( A ) is the maximum value of the sequence; min(A) is the minimum value of the sequence; and Mid(A) is the average of the sequence. FractalsGeneMatrixConclusionsResults

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13 The position of the maximum note length of the immediate offspring generation corresponds to that of the parent note. If a M = MAX( A ) And b M = MAX( B ) Then Flag 1 = True V t =V t +V 1 FractalsGeneMatrixConclusionsResults

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14 The position of the minimum note length of the immediate offspring generation corresponds to that of the parent note. If a m = min( A ) And b m = min( B ) Then Flag 2 = True V t =V t +V 2 FractalsGeneMatrixConclusionsResults

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15 The position of the second maximum note length of the immediate offspring generation corresponds to that of the parent note. If a M = MAX( A ), a S = MAX( A – { a M }) And b M = MAX( B ), b S = MAX( B – {b M }) Then Flag 3 = True V t =V t +V 3 FractalsGeneMatrixConclusionsResults

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16 The position of the medium note length of the immediate offspring generation corresponds to that of the parent note. If a k = Mid( A ) And b k = Mid( B ) Then Flag 4 = True V t =V t +V 4 FractalsGeneMatrixConclusionsResults

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17 The sum of the relative deviance value of the immediate offspring generation note and parent note is smaller than the first critical value. If ∑ | a i - b i | < T 1 Then Flag 5 = True V t =V t +V 5 Where T 1 is the first threshold value. FractalsGeneMatrixConclusionsResults

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18 The sum of deviance value of relative fluctuation of the immediate offspring generation note and the parent note is smaller than the second critical value. Let c t = a t - a t -1 d t = b t - b t-1 ( 2 t n) If ∑ | c i - d i | < T 2 Then Flag 6 = True V t =V t +V 6 Where T 2 is the second threshold value. FractalsGeneMatrixConclusionsResults

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19 Production procedure for the generation of fractal music FractalsGeneMatrixConclusionsResults

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20 Screen Window of fractals and program used for the creation of digital music FractalsGeneMatrixConclusionsResults

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21 Test listening result of digital music of this study ResultsNot GoodGood Melody143.9%56.1% Melody 227.2%72.8 % Melody 336.7%63.3 % FractalsGeneMatrixConclusionsResults

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22 Ratio of Poor Response to Melodies Generated by Different Methods MethodPoor Response Ratio Random Selection92.2% Fractal Iterated Function Systems and Genetic Algorithm Method 61.8% Fractal Iterated Function Systems, Genetic Algorithm Method, and Imitated Parent Method 14.6% FractalsGeneMatrixConclusionsResults

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23 Think of it! If we enter a piece of music, for example Mozart’s Concerto No. 24 into a type of function system, will it become possible for us to generate the output of Concerto No. 25 or might we end up having another music style such as that of Beethoven? Although this may sound weird, it is undeniably, a very interesting thought. Conclusions FractalsGeneMatrixConclusionsResults

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24 In the future, it is quite possible that with the assistance of computer aided interface, composers can more easily finish their digital music. Furthermore, it is also the hope of this study that we are able to determine the direct relationship between “great music” and mathematics. Conclusions FractalsGeneMatrixConclusionsResults

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25 Future Improvement FractalsGeneMatrixConclusionsResults A. Processing for any MIDI format; B. Automatic generation of Track 2 and Track 3 for vocal harmony and music accompaniment; C. Determinant mechanism extends “imitated parent. D. The possibility to add “mutation” in genetic crossover. E. Adding “crossover between beats” in crossover method. F. Connecting to external input equipment such as note input from keyboard and human voice input from microphone.

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26 1.Rob Eastanway (Translated by Tsai Cheng-chi), How Long is a Piece of String: More Hidden Mathematics of Everyday Life, Trafalgar Square Publishing Ltd, 2004 2.Stewart, Ian. Four Encounters with Sierpinski's Gasket, The Mathematical Intelligencer, 17, No. 1, 1995, p.52-64. 3.Wu Wen-chen, Fractal, http://alumni.nctu.edu.tw/~sinner/ complex/ fractals/index.html 4.Chao Peng-chen, Theory and Application of Genetic Algorithm, CHWA Publishing, 2001 5.Goldberg, D, E., Genetic Function systems in Search, Optimization, and Machine Learning, Addison-Wesley, New York, 1989 6.Lin Chern-sheng, Digital Signal-Image and Sound Processing, CHWA Publishing, 1997 References

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