Some Considerations regarding Helmholtz's Theory of Consonance

Cross, Charles R.; Goodwin, Harry M.

doi:10.2307/20020461

Cited by 10 publications

(3 citation statements)

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“…6A). In fact, it lies near the ∆F associated with maximal roughness [67][68][69] (not shown). Experimental studies agree that a puretone minor second (and other tone combinations close to it) sounds "dissonant" 7,51 or "unpleasant."…”

Section: Neural Coding Of Roughness As the Physiological Basis For Hamentioning

confidence: 94%

Neurobiological Foundations for the Theory of Harmony in Western Tonal Music

Tramo

Cariani

Delgutte

et al. 2001

Annals of the New York Academy of Sciences

188

169

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Basic principles of the theory of harmony reflect physiological and anatomical properties of the auditory nervous system and related cognitive systems. This hypothesis is motivated by observations from several different disciplines, including ethnomusicology, developmental psychology, and animal behavior. Over the past several years, we and our colleagues have been investigating the vertical dimension of harmony from the perspective of neurobiology using physiological, psychoacoustic, and neurological methods. Properties of the auditory system that govern harmony perception include (1) the capacity of peripheral auditory neurons to encode temporal regularities in acoustic fine structure and (2) the differential tuning of many neurons throughout the auditory system to a narrow range of frequencies in the audible spectrum. Biologically determined limits on these properties constrain the range of notes used in music throughout the world and the way notes are combined to form intervals and chords in popular Western music. When a harmonic interval is played, neurons throughout the auditory system that are sensitive to one or more frequencies (partials) contained in the interval respond by firing action potentials. For consonant intervals, the fine timing of auditory nerve fiber responses contains strong representations of harmonically related pitches implied by the interval (e.g., Rameau's fundamental bass) in addition to the pitches of notes actually present in the interval. Moreover, all or most of the partials can be resolved by finely tuned neurons throughout the auditory system. By contrast, dissonant intervals evoke auditory nerve fiber activity that does not contain strong representations of constituent notes or related bass notes. Furthermore, many partials are too close together to be resolved. Consequently, they interfere with one another, cause coarse fluctuations in the firing of peripheral and central auditory neurons, and give rise to perception of roughness and dissonance. The effects of auditory cortex lesions on the perception of consonance, pitch, and roughness, combined with a critical reappraisal of published psychoacoustic data on the relationship between consonance and roughness, lead us to conclude that consonance is first and foremost a function of the pitch relationships among notes. Harmony in the vertical dimension is a positive phenomenon, not just a negative phenomenon that depends on the absence of roughness-a view currently held by many psychologists, musicologists, and physiologists.

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Section: Neural Coding Of Roughness As the Physiological Basis For Hamentioning

confidence: 94%

Neurobiological Foundations for the Theory of Harmony in Western Tonal Music

Tramo

Cariani

Delgutte

et al. 2001

Annals of the New York Academy of Sciences

188

169

View full text Add to dashboard Cite

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“…Such experiments have been carried out by Cross and Goodwin (1893), by Mayer (1894), and by Plomp and Steeneken (1968). The results of these three studies are very similar; with the help of curvilinear regression (least squares solution), we found that critical bandwidth (CBW) , in terms of the frequency distance between its lower and higher bounds-being dependent on the mean frequency (f) of these two bounds-can be described adequately (r 2 = 0.98) by…”

Section: Critical Bandwidthmentioning

confidence: 64%

“…CBW', derived from the data reported by Cross and Goodwin (1893), Mayer (1894), and Plomp and Steeneken (1968) is plotted in Figure 2 as a function of f for the three studies separately. Based on all data points in Figure 2, CBW' can be represented excellently (r 2 = 0.92) by…”

Section: Critical Bandwidthmentioning

confidence: 99%

The effect of fundamental frequency on the discriminability between pure and tempered fifths and major thirds

Vos

Vianen

1985

Perception & Psychophysics

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Thresholds were determined for discrimination between pure and tempered musical intervals (DTs) consisting of simultaneous complex tones. Tempered intervals are characterized by small frequency differences between those harmonics that coincide in pure intervals. Interference of these nearly coinciding harmonics gives rise to the perception of beats. Two parameters of beats were varied independently: (l) beat frequency and (2) depth oflevel variation (temporal envelope) as a measure of DT. DTs were determined for musical fifths and major thirds at a tone duration of 0.5 sec. The geometric mean of the fundamental frequency of the tones was varied in octave steps from 92.5 to 740 Hz. The beat frequency of the intervals was varied within a range of 2 to 64 Hz. The major result of our experiment was that, for a frequency range comprising 3 to 4 octaves, DTs do not depend on the frequency of the fundamentals, provided that the DTs are ordered according to beat frequency. When tempering (T) is expressed in cents and only those conditions are considered in which the degree ofT corresponds to that found in the relevant tuning systems, it can be concluded that perceptual differences between tempered and pure fifths are increasingly irrelevant for tones lower than about middle C (262 Hz). Tempered major thirds, even those in a compromise tuning system such as that proposed by Silbermann (T=6 cents), can be discriminated from pure major thirds for all fundamental frequencies that are relevant in musical composition.Discrimination between pure and tempered musical intervals consisting of simultaneous complex tones has been investigated in previous studies. Tempered intervals are characterized by small frequency differences between harmonics that coincide in pure intervals. Interference of these nearly coinciding harmonics gives rise to the perception of beats. Thresholds for discrimination (DTs) are expressed in terms of the depth of the level variation (temporal envelope) of these beating harmonics.The main results obtained in one of the first studies (Vos, 1982) were that the DTs are higher (lower sensitivity) for major thirds than for fifths, are highest for low degrees of tempering, and decrease with increasing tone duration. In a second paper, Vos (1984) showed that for moderately tempered fifths, the value of DT depends mainly on the degree of interference of the first pair of nearly coinciding harmonics, whereas for major thirds, interference of other harmonics may playa role as well. For a large set of intervals varying in size between and including unison and twelfth, Vos and van Vianen (1985) showed that DTs gradually increase with the sum of the integers in the frequency ratio of the pure interval. In ad-

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Nochmals eine Bemerkung über den Ursprung subjektiver Kombinationstöne

Peterson

1914

Annalen der Physik

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Some Considerations regarding Helmholtz's Theory of Consonance

Cited by 10 publications

References 0 publications

Neurobiological Foundations for the Theory of Harmony in Western Tonal Music

Neurobiological Foundations for the Theory of Harmony in Western Tonal Music

The effect of fundamental frequency on the discriminability between pure and tempered fifths and major thirds

Nochmals eine Bemerkung über den Ursprung subjektiver Kombinationstöne

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