Pink noise is a type of noise whose power spectral density (PSD) follows the curve 1/f^a (where a = 1, corresponding to a -3db/octave slope), and whose energy is equal across the frequency spectrum. Pink noise appears in many phenomena and systems including tides, electronic devices, pulsars, and EEG recordings of human brains. Thus, it has become a subject of interest among scientists and retarded hippies alike.
The article claims that the frequency spectrum of Hoshimachi Suisei's voice follows a pink noise curve on average. The author does not clearly outline his methodology, choosing only to say that he used the least squares method on the data, and that he used the entire human hearing range from 20 Hz to 20 kHz[1].
We can use the Welch averaging method to estimate PSD. Welch's method involves splitting the signal into overlapping segments and windowing them, after which a PSD estimate, or periodogram, is calculated by taking the squared magnitudes of the discrete Fourier transform on each windowed segment. This method reduces signal noise in exchange for reducing frequency resolution, producing a smoother signal. I used 2048 segments with 50% overlap, hamming windows, 4096 discrete Fourier transforms. All audio files were recorded or downloaded with the minimum standard audio sampling rate of 44.1 kHz. I opted to use the range from 300 Hz to 20 kHz as the data below 300 Hz is not necessary for this analysis.
There are two ways to fit the data gathered from the PSD estimate to the 1/f curve: linearize the data and function and use linear regression, or use nonlinear least squares regression. I never took stats, so I relied on Matlab for both of these.
y=1/x^a can be rewritten as y=x^-a, a power function. It can then be linearized as log(y) = -a*log(x). Performing linear regression with pink noise as the input consistently yields a = 1 with less than 1% error, proving that this method works. However, on the two Suisei acapella recordings, we get a ≈ 5. This is not consistent with pink noise. A Risu acapella gives a ≈ 6, and a well-recorded raw vocal gives a ≈ 5.4. Looking at the graphs, this method seems to ignore data at high frequencies.
Using nonlinear least squares regression, we can fit the data directly to 1/x^a. Fitting real pink noise to this curve consistently yields a = 1 with less than 5% error. The Suisei acapellas give a = 0.865 and a = 0.855. The Risu acapella and raw vocal give a = 0.856 and a = 0.857 respectively.
Averaging the values between the two methods does not give us anything close to a = 1. Using a larger bandwidth lowers a values from 0.85 to 0.7 for the nonlinear method, and makes no significant change for the linear method. Alternatively, we can look at the power per octave of each audio file and see that they barely resemble pink noise, except for over short, insignificant frequency ranges.
Alternatively, use your ears.
[1] J. Retard "星街すいせいの歌声スペクトル解析および1/fゆらぎの確認."
https://suichyan-ha-kyoumokawaii.amebaownd. com/posts/17805948
