The Mechanical Invariance Factor in Musical Acoustics and Perception (Revisited)

Akpan Jimmy Essien

doi:10.11648/j.ajmp.20180701.11

Abstract

Mechanical, acoustical, and neurophysiological investigations in music, acoustics, and auditory perception repose on the Pythagorean string ratio theory of musical pitch intervals (6th century B.C). Recently, the mechanical validity of the string ratio theory and its psychological import have been challenged and denied on grounds of invariance. In this regard, Essien (2014) demonstrated experimentally that, contrary to established tradition in physics of sound, the tension of a string is not constant when string length is modified even though the balanced-force exerted on the string is held constant. The data revealed the existence of two sources of force in a vibrating string: (1) The oppositely-directed force applied externally to the string (labelled Fex); (2) The force that is the intrinsic property of the string (labelled Fin). The latter is the missing parameter in Pythagorean auditory psychophysics. The omission lured researchers into acoustics and neurophysiology of pitch without an invariant physical correlate of pitch. Essien’s (2014) data showed that all transformations to string length or the balanced-force exerted on a string are various ways to modify the string’s resistance to deformation. Thus, the force in a string varies inversely with string length even though Fex is held constant. In the present paper, string length is shown to have very little or no effect at all on a string’s vibrational frequency and subjective pitch. Because psychoacoustic theories of hearing are founded on the string ratio theory, the data not only offer the missing psychological element that deprived the string ratio theory of a scientific status, but also refute both Ohm’s acoustical law (1843) and Helmholtz’s resonance theory (1877). The force in a string is portrayed as the mechanical parameter in control of pitch regardless of vibrational frequency or spectral structure. Implications for future research in musical acoustics and auditory perception are discussed.

Highlights

IntroductionAcoustic cues for perception “represent a narrow-minded way of thinking which leads us into a blind alley when faced with the problem of discovering the real nature of the auditory mechanism.” [1]
Acoustic cues for perception “represent a narrow-minded way of thinking which leads us into a blind alley when faced with the problem of discovering the real nature of the auditory mechanism.” [1].The Pythagorean string ratio theory is the pivot around which revolve all work in hearing [2; 3]
Because the full string and its sub-lengths are tuned to the same subjective pitch, the experiment operates on the platform of invariance, seeking to detect, from among the different mechanical configurations of the string, the parameter of the sound source, or of the physical manifestations of the signal, that remains inseparably tied to pitch; or as Fechner [21] expressed it (p.47), “as effectively as the length of the yard is tied to the material of the yard-stick.”

Summary

Introduction

Acoustic cues for perception “represent a narrow-minded way of thinking which leads us into a blind alley when faced with the problem of discovering the real nature of the auditory mechanism.” [1]. Hearing research cannot attain the status of a science in the absence of invariance in the formulation of its foundation [22] In this regard, Essien [23] pointed out the existence of a force that is the inherent property of a string (hereafter Fin) outside the oppositely-directed force that is applied externally to a string (hereafter Fex). The former was shown to vary inversely with string length, such that the force in the string, in reality, is not constant but varies with pitch even though Fex (called tension in traditional string mechanics) is held constant This missing mechanical property of a string deprived Pythagorean psychophysics of a mechanical invariant parameter in pitch production; it lured hearing research into premature acoustic studies of music in the absence of an invariant physical correlate of subjective pitch. Implications for future research in pitch production, musical acoustics, and perception are discussed

Mechanics of a String

Equipment and Method

Results

The Problem of Non-Invariance in Psychoacoustics

The Mechanical Invariance Factor

Summary and Conclusions