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

Abstract

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-

Sensory cortical response to uncertainty and low salience during recognition of affective cues in musical intervals

“…Numerous psychoacoustic models have been proposed to explain the consonance/dissonance percept [47–51,55,57]. According to these models, the most consonant intervals would be the ones that could be expressed with simple frequency ratios, which has been supported by psychological studies [75,78,79]. Intervals such as the unison (1:1), the octave (2:1), perfect fifth (3:2), and perfect fourth (4:3) are regarded as the most consonant.…”

Music Psychology: Tonal Structures in Perception and Memory

“…However, ratings of subjective purity (V os 1985) deviated from precise predictions of the model (discussed earlier) of Plomp & Levelt (1965) and Kameoka & Kuriyagawa (1969a,b), suggesting that modifications are needed. Vos & van Vianen (1985), Hall & Hess (1984), and Elliot et al (1987) found tolerance for mistuning decreased when going from less consonant to more consonant intervals, again implicating absence of beating as a cue for pure tuning. The results of the latter two studies also suggested an additional strategy of matching the intervals to abstract standards of interval sizes.…”

Rhythm and pitch in music cognition.

“…Tuning judgments are somewhat more precise for smaller intervals and more consonant intervals (Hall & Hess, 1984). Detection of mistuning can be attributed largely to the strength of beats produced by tempered intervals (Vos, 1982, 1984, 1986, 1988; Vos & van Vianen, 1984, 1985). However, experiments with dichotic stimuli (two tones to different ears that produce no beats) suggest that musicians can also use internal interval standards to make tuning judgments (Hall & Hess, 1984).…”