ASA Lay Language Papers
166th Acoustical Society of America Meeting

How Does Fractal Music ("Self-similarity" or Predictability) Sound to You If You Are Amusic?

Fang Liu –
Department of Linguistics and Modern Languages
The Chinese University of Hong Kong, Hong Kong

Sherri L. Livengood –
The Roxelyn and Richard Pepper Department of Communication Sciences & Disorders
Northwestern University, USA

Cunmei Jiang –
Music College
Shanghai Normal University, China

Alice H. D. Chan –
Division of Linguistics and Multilingual Studies
Nanyang Technological University, Singapore

Patrick C. M. Wong –
Department of Linguistics and Modern Languages
The Chinese University of Hong Kong, Hong Kong

Popular version of paper 1pMU9
Presented Monday afternoon, December 2, 2013
166th ASA Meeting, San Francisco

Music is ubiquitous in our environment. We hear music in shopping malls, restaurants, gyms, and almost everywhere we go. While listening to music is enjoyable for most of us, music may sound like noise for a small proportion of the general population (Stewart & Walsh, 2002). These individuals may not even recognize their national anthems, as depicted in the popular English nursery rhyme: "There was an old fellow of Sheen, whose musical sense was not keen. He said: "It's odd, I can never tell 'God save the weasel' from 'pop goes the queen'!" (Kalmus & Fry, 1980:369). Suffering from congenital amusia (or "tone/note/tune deafness"; Peretz et al., 2002), a neuro-developmental disorder of musical perception, this "old fellow of Sheen" was accompanied by other famous amusics such as Che Guevara and two U.S. Presidents, Ulysses S. Grant and Theodore Rosevelt according to anecdotal reports (Munte, 2002).

Although the first case of "note-deafness" was reported more than a century ago (Allen, 1878), only recently has the disorder been systematically studied (Ayotte et al., 2002). Recent research indicates that amusia has a strong genetic component (Drayna et al., 2001), affecting around 4% of the general population for speakers of both tone (e.g., Mandarin Chinese, in which lexical tones are used to distinguish words at the syllable level) and non-tonal (e.g., English) languages (Wong et al., 2012). The core deficit of congenital amusia is predominately defined by impaired perception of musical tonal relationships (Peretz et al., 2003), although amusics are also impaired in fine-grained psychophysical and linguistic pitch processing (Foxton et al., 2004; Jiang et al., 2010; Liu et al., 2010).

It remains unclear why individuals with congenital amusia cannot make sense of music. To address this question, we compared how individuals with and without congenital amusia responded to a structure, or organizational pattern that has been found throughout music of the world (Manaris et al., 2005). The structure is called a fractal. A fractal is a self-simulating pattern, with each part of the object being similar to the whole. For example, snowflakes have a fractal organization. A snowflake demonstrates how a visual shape repeats to create larger and larger versions of itself (e.g., the Koch snowflake in Fig. 1).

Fig. 1a

Fig. #1: An illustration of the Koch snowflake made of equilateral triangles that result in a snowflake pattern that repeats in multiple sizes, created using the R script at Detailed description and animation of the Koch snowflake can be seen at

We live in a world of fractals (Mandelbrot, 1982). Fractal patterns have been identified not only in music (Hsu & Hsu, 1991; Voss & Clarke, 1975), but also in nature (e.g., the shapes of leaf patterns, cloud formations, snowflakes, etc.), dynamic systems (e.g., the ups and downs of the stock market), and biological systems (e.g., the branching patterns of our neurons and cells, heartbeat, human gate patterns). A fractal pattern can be thought of in terms of how complex an organizational pattern appears. Fractal relationships can be created and measured using a power law formula, 1/f^ β (Voss & Clarke, 1978). When the β-value is zero, there is zero or no self-similarity, meaning that the parts of the object are highly erratic and unorganized, and as a result appear complex. When the β-value is greater than 2, self-similarity is high to the point that parts are almost duplicates of each other, and as a result the object appears relatively simple.

A broad range of work has shown that moderate levels of fractal complexity are preferred for visual objects and art (Koch et al., 2010; Redies et al., 2007), natural landscapes (Redies et al., 2007), and tone sequences (Beauvois, 2007; Livengood, 2013). The combined research suggests the central nervous system is sensitive to certain levels of fractal organization, and the perception of music may be a unique reflection of that sensitivity.

The present study tested the hypothesis that individuals with congenital amusia process the structure of music differently than normal listeners. We compared normal and amusics' responses to a gradient of fractal complexity applied to pitch interval relationships. We predicted that both groups would perceive the full gradient of complexity; however, when fractal organization was moderate, only the normal group's responses would reflect a music listening experience. We created 14 random fractal tone sequences using β values from 0.0 (most complex, least predictable) to 2.6 (least complex, most predictable) in increments of 0.2 (Fig. 2).

Fig. 1a

Fig. #2: Samples of fractal ss-values applied to random tone sequences: A) β = 0.0 (most unpredictable, most complex), B) β = 0.6, C) β = 1.4 (moderately complex), D) β = 2.0, E) β = 2.6 (most predictable, least complex). Listen to samples by clicking on the links.

We asked eighteen native speakers of Mandarin (a tone language with four lexical tones: High, Rising, Low, and Falling) with congenital amusia and 18 matched controls to rate random fractal tone sequences for perceptual (complexity, melodicity) and affective (interest, ease, mood) attributes. All participants also performed a recognition memory task (Fig. 3).

Fig. 1a

Fig. #3: Perceptual, affective, and cognitive measures obtained in the study. Perceptual and affective ratings were in 7-point Likert scales. Recognition memory was a yes/no identification task.

As predicted, both groups rated complexity based on the β-values, demonstrating that amusics perceived the gradient of pitch interval complexity (Fig. 4A). However, amusics' ratings deviated from controls in measures of melodicity (Fig. 4B), affect (Fig. 4D-F), and memory performance (Fig. 4C). For controls, moderately complex sequences (fractal β-values = 1.4-1.6) were rated the most melodious, drove the highest emotional responses, and were the easiest to remember, whereas amusics' ratings did not respond to this range, but rather followed a more linear trend across the five measures (Fig. 4B-F).

Fig. 1a

Fig. #4: Mean Z-score ratings of complexity (A), melodicity (B), mean % accuracy of recognition memory (C), and mean Z-score ratings of affective attributes (D: mood; E: interest; F: ease) of the melodies as a function of β-value pairs in the amusic and control groups: 1 (0.2-0.4), 2 (0.6-0.8), 3 (1.0-1.2), 4 (1.4-1.6), 5 (1.8-2.0), and 6 (2.2-2.4). The bigger the β-value, the less complex the melody. Error bars represent standard errors.

The results from this study suggest that congenital amusia may reflect a difference in how the central nervous system responds to organizational patterns in our environment. Individuals with congenital amusia did not have trouble perceiving the gradient of fractal complexity, but rather lacked heightened sensitivity to specific levels of fractal complexity.

So, how does fractal music sound to you if you are amusic? Our study indicates that two people can listen to and hear the exact same thing, yet have a very different experience (i.e., "Yes, I hear what you hear, I just don't think that's interesting."). Both the ability to process music and the tendency to be drawn to expert musicianship have been theorized to have a genetic component. By looking at not only perception, but also emotion and memory, this work may lead to understanding when biology plays a role in driving people to develop different areas of expertise.


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