6.3 Beats Induced by Dual Missing Fundamentals

2.Within this periodicity range, even if a complex tone with a missing fundamental has only two harmonics (n=2,3), the pitch corresponds to the delay time of the first major peak in the NACF.

6.3 Beats Induced by Dual Missing Fundamentals

It has been observed that a monaural beat can be induced by two complex tones that have slightly different missing fundamentals. This is true even when the envelopebeat component is realized by random-phase components. The beat stimulus is constructed by mixing two complex tones, A and B, that have missing fundamental frequencies at F0a and F0b respectively. When we listen to these complex tones A and B together, a beat is heard that corresponds to the frequency difference between the fundamentals ( f = F0b–F0a). When all components are in phase, the composite tone (A + B) has a waveform repetition and envelope periodicity corresponding to f. When the components are in random phase, however, the envelope periodicity f disappears. Experimental results show that, in both cases, beats of f ≤ 4 Hz were perceived clearly for stimuli with missing fundamentals up to 256 Hz. These results show that beats that are independent of the envelope component can be detected. These phenomena can be explained in terms of the delay time of the maximum peak extracted from the ACF of the sound signal.

An experiment on monaural beats induced by two complex tones with missing fundamentals was conducted (Shimokura and Ando, 2004). Each stimulus signal consisted of two complex tones, A and B, mixed together. Let F0a and F0b be the fundamental frequencies of A and B, respectively, each consisting of upper harmonics (n ≥ 8). φa and φb are phases of complex frequency components. Amplitudes of all components were equal. Fundamental frequencies F0a of the first tone A were either 32, 64, 128, 256, or 512 Hz, while the fundamentals F0b of the second tone B differed from the first by 2, 4, 8, or 16 Hz, respectively, so that F0b = F0a + f. The lowest component of A was always fixed at 1,024 Hz. For example, when F0a = 128 Hz and f = 2 Hz, then the components of A consisted of three harmonics n = 8–10 of 128 Hz: 1,024, 1,152, and 1,280 Hz, while the components of B (F0b = 130 Hz) were harmonics n = 10–12 of 1,300, 1,430, and 1,560 Hz. The total peak sound pressure level measured at the center position of the center of the head was fixed at 74 dB SPL.

Figure 6.11a shows the waveforms of the two complex tones with F0a = 128 Hz and f = 2 Hz. When components are in-phase, the envelope has a periodicity corresponding to f. When phases are random, however, the envelope regularity disappears as shown in Fig. 6.11b. Figure 6.12 shows the results of the ACF analysis of the two complex tones. As is well known, the ACF is identical for in-phase and out-of-phase signals. The maximum peak, τ 1 = 0.5 s, corresponds to 2 Hz. In the ACF for τ < 10 ms shown in Fig. 6.12b, two initial fundamental frequencies (128 and 130 Hz) are indicated by arrows.

Fig. 6.11 Waveforms of two complex tones. (a) In-phase condition. (b) Random condition

Three 23to 24-year-old subjects participated in the beat matching test. Each subject was seated in the listening room, and the same sound signal was fed to the two ears via headphone (Sennheiser, HE60). First, subjects were presented with the two combined complex tones and were asked to listen for a single beat in the sound signal. Then, subjects were presented a train of pulse tones generated by an oscillator and asked to adjust the pulse rate to match the beat perceived for the combined complex tones. This process was repeated until subjects could match an identical beat.

Fig. 6.12 NACF analyzed for both conditions. (a) τ ≥ 2.0 s. (b) τ≤ 0.01 s

Beat-matching tests for the set of five stimuli were presented in random sequence, with each composite complex tone being presented a total of 10 times.

Figure 6.13 shows the probability of the subjects correctly matching the perceived beat frequency by adjusting the pulse rate f within the one-third octave that was generated separately. This is shown as a function of the fundamental frequency F0a and as a parameter of the beat frequency f. What we find remarkable is that when f = 2 to 4 Hz, the probabilities for the frequency range of F0a = 32 to 256 Hz almost always exceeded 80% for both in-phase and out-of-phase conditions. When f = 8 to 16 Hz, the probabilities all decreased below 65%. For in-phase conditions, beat-matching probabilities were smaller than those for out-of- phase conditions, only except for one condition, F0a = 512 Hz (p < 0.025). This beat that is perceived is independent of the envelope of the waveform; consequently it was distinguished from an envelope beat.

As discussed previously, the pitch-matching test of the single complex tone shows that the listeners hear a pitch at the fundamental frequency, which can be described by the delay time of the first peak in the ACF below 1200 Hz (Inoue et al., 2001). However, the beat phenomenon induced by the dual missing fundamentals was observed in the range of 32 Hz < F0a < 256 Hz. The periodicity-limiting mechanism of this fundamental frequency range for F0a is unknown.

These experiments demonstrated that:

1.Fundamental frequencies of multiple complex tones can induce an additional secondary fundamental frequency that is perceived as a beat. The perceived beat rate corresponds to the delay time of the maximum peak of the ACF of the whole signal.

2.The perceived beat was independent of the existence of regularities and fluctuations in the waveform envelope of the two-tone stimulus. This rule holds for fundamental frequencies below 256 Hz.