fractalmusic - Preference, Complexity, and Fractal Melodies...

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M ICHAEL W. B EAUVOIS Loughborough, Leics, United Kingdom PARTICIPANTS RATED THE PERCEIVED complexity and melodiousness of fractal (1/f ß ) tone sequences with spe- cific ß values. Plotting the mean complexity and melod- icity ratings against each other and against ß indicated that: (1) a melody’s spectral power density slope ( ß ) can be used as an objective measure of its perceived com- plexity; (2) ß 1.50 for optimally preferred melodies; (3) perceived complexity is determined by the distribu- tion of pitch intervals such that optimally preferred melodies have a preponderance of small intervals com- pared to large ones; (4) the poor quadratic fits found in music-preference studies are due to the use of an inde- pendent-subjects design or the stimuli not covering the whole possible range of complexity; (5) ß < 2 for most music stimuli; and (6) ecologically valid melodies only exist over the ß range 0.67 to 2.35, with melodies whose ß values lie outside this range not being perceived as musical due to their extreme complexity or simplicity. Finally, converging experimental and neurophysiologi- cal evidence is discussed that suggests that these results are a consequence of the auditory system being opti- mally tuned to the statistical properties of speech. Received December 1, 2004, accepted June 8, 2006. Key words: complexity, fractal, melody ,melodiousness, preference I THAS BEEN CLAIMED THAT the power spectra of the pitch and loudness of successive notes in pieces of music exhibit a 1/f ß spectral density, where ß = 1 (Voss & Clarke, 1975, 1978). Here, the power spectra exhibit a power-law decrease as a function of frequency that, when plotted on a log-log graph, is a straight line with a slope of – ß (see Figure 1). Similar results were reported for the pitch intervals between the successive notes of melodies (Hsu & Hsu, 1990, 1991), and these findings have been widely reported in the literature as evidence that music is fractal in nature (e.g., Gardner, 1978; Schroeder, 1991). However,Voss & Clarke’s (1975, 1978) studies have been criticised by Nettheim (1992) on the basis that their data-acquisition method was flawed, and Hsu & Hsu’s (1991) study has been criticised by Henderson-Sellers & Cooper (1993) on the basis of the small range of pitch intervals examined and the fitting method used. Furthermore, more recent studies of 1/f ß noise in music that have examined the frequency spectra of melodies have typically found that 1 < ß < 2. For exam- ple, Nettheim (1992) found that for melodies by Bach, Beethoven, Chopin, Mozart, and Schubert, ß ranged from 1.19 to 1.88 ( M = 1.57), and for the isochronous pitch sequences derived from these melodies, ß ranged from 0.93 to 1.56 ( M = 1.33). Similar results were found by Yadegari (1992) for the melodies of the prel- udes and fugues of Bach’s Well-Tempered Clavier, Part I ( M = 1.47), and by Brillinger & Irizarry (1998) for melodies taken from examples of Baroque, Classi- cal, Romantic, Spanish guitar, Jazz, and Mambo music ( M = 1.32, and 1.41 when Jazz and Mambo were excluded from the sample).
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fractalmusic - Preference, Complexity, and Fractal Melodies...

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