(Pitch) (self similarity) (tone) 2. (Melody) Aristotle. (Rhythm) (fractal theory)[1] Voss Hsu. (meter) (Harmony) (chord) (triad) (Multifractal)

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1 (self smlarty) Arstotle (fractal theory)[] VossHsu (Multfractal) Hölder.. (Ptch) (tone). () 3. () (meter) 4. (Harmony) (chord) (trad) 4:5:6 ::5 : :3 3:4 4:5 5:6 8:9 5:6 :

2 7 5 f / f f / f = Voss Charke 5 / = /3. 7 / = / Voss Charke 6 / =.44 S( f ) (Loudness) 77 ( ) f Hz khz Mandelbrot (Hz-kHz) S( f ) f Voss (scalng nose) Clarke [] (whte nose) f S( f ) S( f ) f 9 Hsu Hsu[6] Bach Mozart S ( f ) f β β ( ) F F D D D 3 (Brownan nose) β Hsu Hsu[7] (x, y),4,8 Voss Charke (BachMozart Beethoven) f [8] [-3]f Sh[9] Hurst f f f Bgerelle Iost[] (Varance Method) 8 f Sh Madson[] Hurst [4-5] Hurst f 3. Fourer Mandelbrot f Hurst (long range correlaton)

3 ( ) d f ( α ) = [( ) D ] ( ) D (5) d D (4)(5)) f (α α () (Generalzed Dmensons)[] N (r) (r) (mass densty) f (α ) p (moment) µ ( r, ) = p ( r) / p ( r) (6) D log p( r) = lm () r log r p ( r) = N ( r) N r / (sub-cover) µ ( r, ) log p ( r) D α ( ) = lm (8) r D log r < D ( ) > D 4. () (multfractal spectrum)[3-4] (local scalng exponent) Hölder exponent α log p ( r) α = lm () r logr α α m+ n ( α ) dα α α + dα 6 n(α ) 6 r power-law D Legendre [( ) ] d α = D (4) d D [5-6] p µ ( r, ) log µ ( r, ) f ( α ) = lm r log r (7) ( )m f ( α ) n( α ) ~ r (3) 8 4 f (α ) ( )4-= α f (α ) α () [3-4] f (α) α 5. (I) Gossec

4 ()(II) Sant-Sanes ()(III) Bach Gounod (Ave Mara) Hölder f (α) Hölder ( ) r f (α) N (r) p log r log p ( r) Hölder Hölder Hölder x y Hölder (Devl s ( Gaotte Starcase) 4 ) r 9/ 6. r p (r) (6) µ (7)(8) r µ log µ µ log p r µ log µ µ log p log r µ log p log r µ log µ f (α) α () f () f (α) α f (α ) (7)(8) Mandelbrot, B. B., The Fractal Geometry of Nature, Freeman, New York, 983 Voss, R. F. and Clarke, J., / f Nose n Musc

5 and Speech, Nature, Vol. 58, pp , 975 3Voss R. F. and Clarke J., f Nose n Musc: Musc from / f Nose, J. Acoust. Soc. Am., Vol. 63, No., pp. 58-6, 978 4Campbell, P., Reples: Is There Such a Thng as Fractal Musc?, Nature, Vol. 35, pp. 766, 987 5Voss, R. F., Fractals n Nature: From Characterzaton to Smulaton, In The Scence of Fractal Image, ed. Petgen, H-O, and Saupe, D., Sprnger, New York, pp. -7, 988 6Hsu, K. J. and Hsu, A. J., Fractal Geometry of Musc, Proc. Natl. Acad. Sc. U. S. A., Vol. 87, pp , 99 7Hsu, K. J. and Hsu, A. J., Self-Smlarty of the / f Nose Called Musc, Proc. Natl. Acad. Sc. U. S. A., Vol. 88, pp , 99 8Hsu, K. J., Fractal Geometry of Musc: From Brd Songs to Bach, In Applcatons of Fractals and Chaos, ed. Crlly, A. J., Earnshaw, R. A., and Jones, H., Sprnger-Verlag, New York, pp. -39, 993 9Sh, Y., Correlatons of Ptches n Musc, Fractals, Vol. 4, pp , 996 Bgerelle, M. and Iost, A., Fractal Dmenson and Classfcaton of Musc, Chaos soltons fractals, Vol., pp. 79-9, Madson, G., Propertes of Expressve Varablty Patterns n Musc Performances, J. New Mus. Res., Vol. 9, pp , Grassberger, P., Generalzed Dmensons of Strange Attractors, Phys. Lett., Sect. A, Vol. 97, pp. 7-3, 983 3Frsch, U. and Pars, G., On the Sngularty Structure of Fully Developed Turbulence, In Turbulence and Predctablty n Geophyscal Flud Dynamcs, ed. Ghl, M., Benz, R., and Pars, G., North-Holland, Amsterdam, pp , 985 4Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procacca, I. and Shraman, B. I., Fractal Measures and ther Sngulartes: the Characterzaton of Strange Sets, Phys. Rev., A, Vol. 33, pp. 4-5, 986 5Chhabra, A., Meneveau, C., Jensen, R. V. and Sreenvasan, K. R., Drect Determnaton of the f (α) Sngularty Spectrum and ts Applcaton to Fully Developed Turbulence, Phys. Rev., A, Vol. 4, pp , 989 6Chhabra, A. and Jensen, R. V., Drect Determnaton of the f (α ) Sngularty Spectrum, Phys. Rev. Lett., Vol. 6, pp , 989 Multfractal Analyses of Musc Seuences Zh-Yuan Su, Tzuyn Wu Natonal Center for Hgh-Performance Computng Department of Mechancal Engneerng, Natonal Tawan Unversty Abstract Many spatal patterns and physcal phenomena of Nature are ether rregular or fragmented to such an extreme degree that shapes are not easly descrbed by tradtonal Eucldean geometry. Nevertheless, they often possess a remarkable smplfyng nvarance under changes of magnfcaton (fractal). The fractal geometry has helped provde both a descrpton and a mathematcal model for many of seemngly complex forms found n Nature. If musc s created to mtate the heavenly sound and harmony of our great Nature, then wthout excepton t would also show the ubuty of fractal character n ts structure. By utlzng fractal theory, prevous studes have shown that musc exhbts long-range correlatons over decades of notes smlar to those found n many naturally occurrng fluctuatng phenomena. Ths underlyng structure mght explan why musc sounds pleasng and how musc mtates the harmony of Nature. However, tradtonal fractal analyses cannot expose local propertes and hence the subtle dfferences n structures of musc seuences. In ths study, we adopt a multfractal methodology to analyze both the melody and rhythm seuences constructed from musc of dfferent styles. Dstrbutons

6 of local Hölder exponent and multfractal spectra ( α f (α ) curves) of varous musc scores are calculated. By nspectng the famlar nverted, downward-openng parabola shape of the spectra, dfferences n musc styles (melody and rhythm) can be uantfed and measured. Present analyss thus provdes a new way to categorze mu sc and to construct fractal musc out of naturally evolvng seuences. Keywords: musc, fractal, multfractal, Hölder Exponent f () Ave Mara..9.8 ( ).7 f () Ave Mara Hölder exponent..9.8 Ave Mara Ave Mara Hölder exponent f () ( ) f () Ave Mara