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The oldest theory of musical consonance is that perceptions of consonance and dissonance are governed by ratios of whole numbers. Pythagoras is thought to have first articulated the principle that intervals of small integer ratios (cf. Fig. 3B; Fig. 4C) are pleasing because they are mathematically pure (Burns 1999). He used this principle to explain the musical scale that was in use in the West at the time, and Pythagoras and his successors proposed small-integer-ratio systems for tuning musical instruments, such as Just Intonation (JI). Modern Western equal temperament (ET), divides the octave into 12 intervals that are precisely equal on a log scale. ET approximates JI, and transposition in ET is perfect, because the frequency ratio of each interval is invariant. Apart from octaves however, the intervals are not small integer ratios, they are irrational.The fact that intervals based on irrational ratios are approximately as consonant as nearby small integer ratios is generally considered prima facie evidence against the theory that musical consonance derives from the mathematical purity of small integer ratios. [emphasis mine]
Tones with small integer ratio relationships (1:1, 5:4 and 3:2 – a Neurodynamics of Music 24 tonic triad) produced a stable memory in the neural oscillator network (cf. Fig. 2). Although a leading tone (8:15 ratio with the tonic frequency) could be stabilized through external stimulation, when the external stimulus was removed, the leading tone frequency lost stability as those oscillators that had responded at the leading tone frequency began to resonate at the tonic frequency. In other words, the tonic frequency functioned as an attractor of nearby oscillators. Thus, nonlinear resonance predicts both memory stability of small integer ratios and tonal attraction among sequentially presented frequencies (Large in press).
The tuning systems of the world’s largest musical cultures, Western, Chinese, Indian, and Arab-Persian, are based on small integer ratio relationships (Burns 1999)2. However, each tuning system is different, and this has led to the notion that frequency relationships do not matter in high level music cognition; rather, auditory transduction of musical notes results in abstract symbols, as in language (see, e.g., Patel 2007). If this were true, stability and attraction relationships would also have to be learned presumably based solely on the frequency-of-occurrence statistics of tonal music (for a current overview, see Krumhansl and Cuddy, this volume).
However, stability and attraction relationships are not learned per se, but are intrinsic to neural dynamics given a particular set of frequency relationships.
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