Calculation of 1/f Fluctuation from Sound Signal and Comfort Evaluation

Abstract

Providing a comfortable sound for users is an important factor for high-value products. Therefore, many studies have investigated pleasant sound levels for developing and manufacturing new products. Notably, sounds containing 1/f fluctuations provide a relaxing effect in humans. There are many studies on the influence of sound signals, including 1/f fluctuations; however, the verification of fluctuations, including sound signals, has not been performed. In this study on fluctuation, the discrete Fourier transform was used to directly calculate the time of the sound signal. We evaluated the duration of music and the 1/f fluctuation via the discrete Fourier transform using the time history of the music data. Furthermore, we investigated the relaxation effect of music containing a 1/f fluctuation. We determined a person’s comfort according to the difference in the calculated fluctuation coefficient by subjectively evaluating the comfort felt by people when listening to music with two different fluctuation coefficients, and we examined the improvement in the fluctuation coefficient and human comfort.

1. Introduction

Noise is an undesired sound that makes people uncomfortable. Additionally, it interferes with listening to voices and music and hinders or impairs our lives [1]. Therefore, sounds that are unpleasant and disturbing to the listener are considered noise, even if they are sounds that have frequency, amplitude, or timbre [2]. It has been reported that noise causes serious problems in humans, such as causing sleep disorders [3], hindering daily life, affecting work efficiency [4], and causing hearing loss [5]. Noise, among various other types of pollution, is closely related to our daily lives, and it has several sources. Therefore, it is important to reduce the sound pressure level of noise using various noise-reduction technologies to suppress noise and establish a quiet environment.

Noise control changes the acoustic state of the controlled object to the desired acoustic state via a chosen method. There are two main methods used for noise control: passive and active noise control. Passive noise control is the method of absorbing sound in a noise-transmission path to reduce the sound pressure level of noise using fiber-based materials, such as glass or rock wool, foamed metals, and ceramic granules [6]. In this method, the sound energy input to the soundproof material is converted to heat energy. Then, the reflected and transmitted energies decrease. To maximize the sound absorption coefficient of the soundproof material, it is necessary to make the thickness of the sound-absorbing material one-quarter or more of the wavelength of the noise. Therefore, when the frequency of the noise decreases, it is necessary to make the soundproof material thicker [7].

Active noise control is a method of controlling noise using the interference of sound waves from a secondary sound source [8]. This method was proposed in the 1980s and has been used in various applications with the development of digital signal processing technology [9]. Notably, active control enables wideband noise control from low-frequency to high-frequency by exploiting sound wave interference.

Moreover, automobiles and electrical appliances need to provide comfortable sounds for users, and this has recently become an important factor for high-value products [10]. In previous sound design research, Toi et al. [11] improved the sound quality of the camera shutter sound. Additionally, Furuya et al. [12] improved the quality of the hit sound and wind noise of golf clubs. These improvements provide differentiation from other companies and high value aimed at “noise reduction” and “pleasant sounds” to increase comfort.

In a study on comfortable sound, participants listened to a sound signal, and subjective evaluations via a questionnaire and an evaluation via human biological signal measurements were performed [13,14]. In a study by Asakura [15], the impression and comfort caused by white noise and the murmuring sound of a river were evaluated using brain waves and a questionnaire given to the experiment’s participants. Pope [16] evaluated the comfort of listening to a performance of an orchestra; the impression of the sound and the music output from a DVD were evaluated using a questionnaire. We considered that the evaluation of comfortable sound differs depending on the experimental environment and the preferences of the study participants who perform the comfort evaluation after listening to a sound signal. Sound evaluation is also time-consuming and costly because it is continuous and requires a large amount of data. To systematically provide a comfortable sound signal, comfort should be quantitatively evaluated. However, an effective method has not yet been established.

Therefore, research is being conducted on factors and components that give people a feeling of comfort and relaxation by analyzing sound signals. It has been proven that work efficiency and comfort are affected by the tempo of sound signals [17,18]. These studies have quantitatively evaluated the components contained in the sound signal to then evaluate comfort. We propose that it is possible to regularly select sound signals and create a feeling of comfort by establishing a method to analyze the relevant factors. Furthermore, this paper found that a sound signal including 1/f fluctuation also made the experiment’s participants comfortable. In this study, we focused on the 1/f fluctuation included in sound signals to analyze factors that give people a feeling of more comfort.

In the 1/f fluctuation, the power spectral density is inversely proportional to frequency f. The 1/f fluctuation included in music has a relaxing effect on humans [19]. In various research fields, the psychological influence of sound signals, including the 1/f fluctuation, has been recently investigated [20,21,22,23,24,25,26,27,28,29]. Sugai et al. [30] reported that music has strong 1/f fluctuation characteristics. Additionally, music was exhibited by electroencephalographic α and β waves via the measurement of participants’ brain waves while they listened to the music, and the 1/f fluctuation was visualized by the frequency analysis of the music. Moreover, Watanabe et al. [31] calculated the change in the fluctuation coefficient for the time history of the instantaneous frequency of the sound signal. Thus, there is extensive research on the 1/f fluctuation in various fields. However, there is no existing description of the calculation method used for the 1/f fluctuation in a sound signal, and there is insufficient sound signal information in studies on sound signals that include the 1/f fluctuation. Moreover, the method to detect 1/f fluctuations in sound signals has not been established. According to Musya [32], there is a 1/f fluctuation in the change of acoustic power and instantaneous frequency. However, the calculation method used to obtain these two changes from the sound signal was not clarified. Although there is extensive research on the 1/f fluctuation, such as the calculation method and the evaluation of its relaxation effect on humans, as described above, there are no studies on the most suitable calculation method for the fluctuation.

This study investigated fundamental considerations of the established calculation methods used to determine the frequency and amplitude fluctuation coefficient via a discrete Fourier transform using time changes to clarify the 1/f fluctuation in sound signals. In this paper, “fluctuation” indicates the fluctuation from a physical quantity’s average value or increase/decrease from the average value. Additionally, the fluctuation of the sound signal means the fluctuation of “pitch of sound = frequency” and “volume of sound = amplitude”, which are elements of sound. We considered a subjective evaluation of comfort when a human listened to a sound signal including two different 1/f fluctuation coefficients to determine the comfort of the human according to the difference in the calculated fluctuation coefficient. We also conducted experiments to improve human comfort on the basis of the fluctuation coefficient.

2. Calculation Method for Fluctuation Coefficient of the Sound Signal

2.1. Fluctuation Coefficient of the Whole Sound Signal

We analyzed the fluctuation included in a sound signal using a .wav file, which is the time history of the sound signal. Figure 1a shows the sample of the sound signal .wav file. The spectrum exhibited the discrete Fourier transform, which is defined as follows:

$$F\left(k\mathsf{\Delta }f\right)=\frac{N}{T_0}{{\displaystyle \sum }}_{n=0}^{N-1}x\left(n\mathsf{\Delta }t\right)e^{-j\frac{2\pi }{N}kn}$$

(1)

$$f_s=\frac{1}{\mathsf{\Delta }t}$$

$$k=0,\pm 1,\pm 2,\cdots \pm \frac{N}{2}$$

where T₀ is the music running time [s], Δt is the sampling time [s], N is the sampling number of wav data, x is the sampled music data, Δf is the frequency resolution, and F is the spectral intensity for each frequency range. In this experiment, for all cases, the sampling frequency f_s of the music was 44.1 kHz.

The least squares method was used to calculate the slope of the spectrum. The relationship between frequency f and spectrum F is given as follows:

$$F=af+b$$

(2)

Slope a and intercept b are respectively shown as:

$$a=\frac{{{\displaystyle \sum }}_{i=n_1}^{n_2}\left\{\left(f_i-\overline{f}\right)\left(F_i-\overline{F}\right)\right\}}{{{\displaystyle \sum }}_{i=n_1}^{n_2}{\left(f_i-\overline{f}\right)}^2}$$

(3)

$$b=\overline{F}-a\overline{f}$$

(4)

where f_i is the discretized frequency, and F_i is the spectrum at f_i. In addition, n₁ and n₂ are integers to satisfy fn₁ = 0.05 Hz and f_n2 = 0.5 Hz, and $\overline{f}$ and $\overline{F}$ are the averages of frequencies and spectra in the sections. In this study, we defined the fluctuation coefficient of the whole sound signal λ_m, which was calculated using the above method.

Figure 1b shows a result of the spectral slope obtained by the above equations. The fluctuation coefficient of the whole sound signal λ_m was −0.873.

2.2. Fluctuation Coefficient of the Frequency

The instantaneous frequency of the sound signal was calculated using the zero-cross method for the time history of the sound signal data. The zero-cross method calculates the frequency from the number of times the amplitude value of the sound signal intersects the zero point at a particular time. In this study, the time interval between the zero-cross instances of the sound signal was 25 ms to detect frequencies of more than 20 Hz, which is the absolute threshold of human hearing. Figure 2 shows the result of cutting out the sound signal shown in Figure 1a from 21.5 to 25 ms. The x plots are zero-cross points in the figure. The average frequency in each interval was obtained using the following equation:

$$f_{zc}=\frac{1}{T_{zc}}\frac{n_{zc}}{2},$$

(5)

where f_zc is the average frequency [Hz], T_zc is the sampling period, and n_zc is the number of zero-crosses at a particular time. Figure 3a shows the time history of the average frequency of sound signal every 25 ms. The spectrum was calculated using the discrete Fourier transform, as defined in Equations (1)–(4). The time history of the average frequency in each interval was used to analyze the fluctuation as the frequency changes value. The fluctuation coefficient λ was defined by calculating the spectral slope from 0.05 Hz to 0.5 Hz. In this study, we defined the fluctuation coefficient of the frequency as λ_f, which was calculated using the above method.

Figure 3b shows the result of the spectral slope obtained by the equations. The frequency fluctuation coefficient λ_f was −2.148.

2.3. Fluctuation Coefficient of the Amplitude

Changes in the amplitude value of the sound signal were calculated using the root mean square (RMS) of the time history of the sound signal data. The calculation interval of the amplitude value was 25 ms, which was the same as that of the frequency fluctuation. Figure 4a shows the time history of RMS of sound signal every 25 ms. The spectrum was calculated using the discrete Fourier transform, as defined in Equations (1)–(4). The time history of the RMS was used to analyze the fluctuation, which provides the change in amplitude. The fluctuation coefficient λ was defined by calculating the spectral slope from 0.05 Hz to 0.5 Hz. In this study, we defined the fluctuation coefficient of the amplitude λ_a, which was calculated using the above method.

Figure 4b shows a result of the spectral slope obtained by the equations. The amplitude fluctuation coefficient λ_a was −1.134.

2.4. Fluctuation Coefficient Calculation

Figure 5 shows the relationship between the frequency and amplitude of the fluctuation coefficient for 1500 pieces of music. In this figure, the vertical and horizontal axes represent the amplitude and frequency of the fluctuation coefficient, respectively. The results were plotted for each fluctuation coefficient in one song. The red line indicates a fluctuation coefficient of −1. From the result, the average frequency fluctuation coefficient was −0.793, and the average amplitude fluctuation coefficient was −1.083. Furthermore, the range of the frequency fluctuation coefficient was −3.159 to 0.802, and the range of the amplitude fluctuation coefficient was −2.279 to 0.165. Moreover, the standard deviation of the frequency fluctuation coefficient was 0.502, and the standard deviation of the amplitude fluctuation coefficient was 0.369. Therefore, there was no correlation between the frequency and amplitude fluctuation coefficients. In the 1500 pieces of music, the frequency coefficient −0.99926 had the closest value to −1. However, there was no music for which all fluctuation coefficients, including the sound signal, frequency, and amplitude, became almost −1. Additionally, the range of the frequency fluctuation was wider than that of the amplitude fluctuation. We considered that the fluctuation coefficient for frequency changes according to various factors, such as the musical instrument, the singing voice, and the melody of the music. However, the amplitude fluctuation coefficient was detected for the same type of sounds, such as the sound pressure of the volume of the voice and the musical instrument. Therefore, the range of the amplitude fluctuation was narrower than that of the frequency fluctuation.

2.5. Verification of Fluctuation Calculation Method

In the previous section, we demonstrated that the fluctuation coefficient of the whole sound signal, the fluctuation coefficient of the frequency, and the fluctuation coefficient of the amplitude were calculated from a wav data by the proposed calculation method. However, the accuracy of each calculated fluctuation coefficient was not proven. Therefore, we calculated each fluctuation coefficient using pink noise to prove the calculation method’s accuracy. Pink noise is the noise at which sound pressure is inversely proportional to a frequency, and the strength of every octave is a constant noise. Therefore, the fluctuation coefficient of the whole pink noise will be close to −1 when calculated by the proposed calculation method in the previous section.

Figure 6 shows the calculated pink noise result of the spectral slope obtained by the above equations. The figure shows that (a) is the fluctuation coefficient of the whole sound signal, (b) is the fluctuation coefficient of the frequency, and (c) is the fluctuation coefficient of the amplitude. For result (a), the fluctuation coefficient of the whole sound signal of pink noise was limitlessly close to −1, as we expected before the calculation. On the other hand, for results (b) and (c), each fluctuation coefficient was far from −1. Pink noise does not change frequency and amplitude even after the play time of the sound signal. Therefore, the frequency and amplitude fluctuation coefficient were not near −1.

The fluctuation coefficient of the whole pink noise was close to −1 according to the proposed calculation method in this paper. Furthermore, using this algorithm to calculate the fluctuation coefficient of the frequency and amplitude yielded the same result as calculating the spectrum slope of the whole sound signal. We considered that the calculation accuracy of the fluctuation coefficient of the frequency and amplitude was secured. Therefore, the proposed method can calculate the fluctuation coefficient accurately.

3. Psychological Condition Estimation and Comfort Evaluation by the Difference in Fluctuation Coefficients for Humans

The previous section clarified that the fluctuation coefficient of the whole sound signal, frequency, and amplitude can be derived using the proposed calculation method. Therefore, we studied the relationship between the calculated fluctuation coefficient of music and human comfort. This study focused on frequency fluctuation and comfort as fundamental considerations. In this section, we investigated the comfort of music with different fluctuation coefficients and its influence on a human using the sound signal of the same melody.

3.1. Music Selection for the Experiment

We needed to select the most suitable music to consider the influence of the difference in fluctuation coefficients for human comfort. Therefore, we focused on classical music with many samples for the same melody and recordings by various performers. We selected unaccompanied classical music by playing one instrument that did not exist in various music parts, such as an orchestra and accompanying music, to clarify the factors that affect 1/f fluctuation. In addition, we selected music playing only the cello, which is a stringed instrument. Using this instrument, it is structurally difficult to generate three or more notes simultaneously, in contrast to using a piano. We selected Johann Sebastian Bach: Cello Suites in this study considering the experimental condition above. The selected Johann Sebastian Bach: Cello Suites does not have existing original music notes. Therefore, tempo, volume, bowing, and accidentals are entrusted to the player’s interpretation. Furthermore, we used the prelude of BWV1007, which is used as background music in movies, television shows, and commercials.

Figure 7 shows the fluctuation coefficients for 15 players who played the prelude of BWV1007 from the Johann Sebastian Bach: Cello Suites. The results showed that all fluctuation coefficients were different for the same music depending on the players. On the basis of this result, we selected music A, which had the frequency fluctuation farthest from −1, and music B, which had frequency fluctuations closest to −1. In this experiment, we investigated the comfort that music with different fluctuation coefficients gave to a human using the sound signal of the same melody. In addition, the λ_f values of music A and B were 0.0068 and −0.982.

3.2. Experimental Consideration of Subjective Evaluation by Listening to Music including Different 1/f Fluctuation Coefficients

Using the previous consideration of this study, we described the calculation method used for the fluctuation coefficient of the sound signal. Therefore, we experimented with the effects of sound signals with fluctuation coefficients close to −1 on humans. In the experiment, we selected music A, with the frequency fluctuation farthest from −1, and music B, with frequency fluctuations close to −1. In addition, we conducted a subjective comfort evaluation using a questionnaire after playing music A and B to clarify the effects of sound signals with a fluctuation coefficient close to −1 on humans. In this paper, we focused on the difference fluctuation coefficients of frequency, which represent the characteristics of the sound signal to assess the relationship between the calculated fluctuation coefficient value from music and human comfort [19]. We studied the fluctuations, including sound signals. Therefore, it was not the purpose of our study to clarify whether the slope of the spectrum of the sound signal was close to −1.

Figure 8 shows the experimental flow. First, the participants listened to music A and B. After listening to two pieces of music, the participant selected the most applicable option, as shown in Figure 9. In this experiment, the participant could listen to music A and B many times until the participant understood that music A and B were clearly different. We considered that this question could prove that music B, which had a coefficient of frequency fluctuation close to −1, made the participant feel comfortable.

Additionally, this questionnaire included “I felt that or didn’t understand that both A and B were comfortable.” By including this option, we considered that it was possible to classify a participant who did not identify a difference between A and B. In this experiment, the length of the music was 30 s. Owing to the short listening time, it was difficult to forget the impression of the previous music. Thus, the participant could smoothly and intuitively perform the test. In addition, 102 participants in their 20s and 70s participated in the experiments. All participants were recruited via the home page of our laboratory, and the experiments were conducted anonymously.

Figure 10 shows the results of the comfort evaluation for the 102 participants. According to the results, 41 participants reported that music A was comfortable, 52 reported that music B was comfortable, and 9 reported no difference between music A and B in terms of comfort or that they did not understand.

To summarize the above results, we proved that the value of the fluctuation coefficient obtained by the proposed method was close to −1 and that a sense of comfort was given to the listener. However, approximately 40% of the participants answered that they were comfortable with music that had a frequency fluctuation coefficient that was far from −1. Therefore, we considered that more people felt comfortable with music that had a fluctuation coefficient and amplitude fluctuation coefficient of the whole sound signal close to −1.

3.3. Experiment to Determine the Impression of the Sound Signal

In the previous section, we proved that more than half of the participants felt comfortable when they listened to music with a coefficient of frequency fluctuation close to −1. Therefore, we investigated the impression of music that felt comfortable in this section.

In the experiment, the participants listened to music A again after the experimental consideration of subjective evaluation by listening to music including different 1/f fluctuation coefficients, as shown in Figure 8. After listening to music A, the participants selected an impression from 38 evaluation items, as shown in Figure 11 [33]. Next, the participant listened to music B and selected an impression again.

3.4. Consideration for the Impression of the Sound Signal

Figure 12 shows the evaluation items selected by 10 or more participants according to the impression of the music that the participants felt more comfortable with. The vertical axis shows the evaluation items, and the horizontal axis denotes the number of people. The blue bar denotes the number of participants who selected the evaluation items as the impression of music A when they were comfortable listening to music A (note that the frequency coefficient of variation for A was far from −1). The red bar denotes the number of participants who selected the evaluation items as the impression of music B when they were comfortable listening to music B (note that the frequency coefficient of variation for B was close to −1). From the results, we observed that when people felt comfortable, they tended to choose Clear, Calm, Deep, Luxury, and Beautiful.

Figure 13 shows the adjectives typically chosen by participant groups according to the music that they chose. In this figure, (a) shows the adjectives chosen by the participant group who felt that music A was comfortable; (b) shows the adjectives chosen by the participant group who felt that music B was comfortable. Both results exhibited a difference of five or more in the selected evaluation items. According to (a), many people who felt comfortable with music A chose Clear, Not Muffled, Awaking, Strong, and Powerful. Therefore, we considered that these participants tended to be comfortable when they listened to music that gave an easy-to-understand impression and had a large sound intensity. Moreover, according to (b), many people who felt comfortable with music B chose Vague, Calm, and Sleepy. Therefore, we considered that these participants tended to report that they felt comfortable when they felt physically and mentally stable by listening to music B. Furthermore, there was a difference between the answers Do Not Care and Quiet. According to this result, we considered that a person who listened to music B and felt comfortable did not care about the sound signal itself, even if the person listened to the music at a particular sound pressure level and had a quiet environment.

Figure 14 shows the results of the impression evaluation in which more than 10 participants identified a difference between music A and B. From this result, we observe that many people who feel comfortable with music A tend to answer Clear, Strong, and Powerful. Moreover, many people who feel comfortable with music B tend to answer Vague and Sleepy.

Considering the above results, we conducted a multiple regression analysis for the result of the impression of sound signal for all participants.

Table 1. The p-value for each 21 evaluation items.

Evaluation Items	p Value (p < 0.05)
Beautiful	0.381
Luxury	0.318
Deep	0.853
Calm	0.327
Clear	0.969
Heavy	0.478
Powerful	0.643
Strong	0.001
Awakening	0.850
Don’t muffle	0.099
Sharp	0.874
Cleary	0.218
Light	0.115
Quiet	0.160
Does not care	0.811
Comfort	0.697
Sleepy	0.360
Muffled	0.192
Sedated	0.223
Dull	0.426
Vague	0.718

shows the result of the p-value for each of the 21 evaluation items; 21 of the 38 items in this table are shown in Figure 12, Figure 13 and Figure 14. The p-values indicate the significant probability of the selection that affects music A and B. If the p-value was 0.05 or less, it meant that the evaluation items had an effect on the selection between music A and B.

According to the results, Strong had a p-value of 0.01, indicating a significant difference. Strong was an evaluation item that could cause a significant difference when used as a question to clarify the difference in comfort when a person listened to music with different frequency fluctuation coefficients.

4. Conclusions

This study focused on 1/f fluctuation, which relieves people by masking audio signals used to reduce discomfort due to noise. We calculated the fluctuation coefficient of the sound signal and the comfort of the music, including the fluctuations given to humans.

First, to calculate fluctuations in sound signals for masking, we used the zero-cross method and RMS to extract the frequency and amplitude components from the sound signal data. We clarified the method used to calculate the three fluctuation coefficients of the entire sound signal, such as the whole sound signal fluctuation, frequency fluctuation, and amplitude fluctuation, by discrete Fourier transform. Second, we conducted a subjective evaluation of the calculated fluctuation coefficient and human comfort. On the basis of the experimental results, we proved that the experiment’s participants felt relatively comfortable with music with a frequency fluctuation coefficient close to −1. Particularly for men, listening to music with a frequency fluctuation coefficient close to −1 improved comfort. Moreover, on the basis of the impressive results of the music, we observed that people who felt more comfortable with music that had a frequency fluctuation coefficient close to −1 tended to be comfortable when they feel Vague, Calm, and Sleepy. Therefore, it is considered that music that has a frequency fluctuation coefficient close to −1 has the effect of stabilizing the mind and calming physical activity, which makes people feel comfortable. However, the study participants who felt more comfortable with music that had a frequency fluctuation coefficient far from −1 chose Clear, Not Muffled, Awakening, Strong, and Powerful. Therefore, we considered that these participants tended to be comfortable when they listened to music that gave an easy-to-understand impression and had a high sound intensity.

In future research, we will examine each calculated fluctuation coefficient and its effect on humans from the subjectivity of the participants and biological information. Therefore, we will conduct the same experiment multiple times with traceable participants and consider the changes in that tendency. Moreover, we will perform the experiment with not only frequency fluctuations but also amplitude fluctuations and sound signals with fluctuation coefficients of both a frequency and an amplitude close to −1. We aim to create a sound system that provides a comfortable space by stabilizing the mind with the sound signal including fluctuation. Conversely, it provides an alarm sound when a human is drowsy or inactive in the brain. Our ultimate goal is to build a system that switches sound signals according to the situation.