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Repeating Noise Motifs to Make Tones with Different Timbres

As mentioned in my analysis of the coasting flywheel recording, a waveform made by repeating a noise segment has (1) a fundamental frequency that depends on the repetition rate, and (2) a harmonic composition (timbre) that depends on the specific noise waveform repeated. On this page I report an experiment in which randomly chosen noise motifs of equal length are concatenated to generate tones that differ in timbre. These are compared with a tone made by repeating one cycle from the flywheel recording at the wheel's maximum speed, and also the recorded wheel deceleration.

Using the noise generator function in Audacity software, I specified 250 milliseconds (msec) of pink noise (equal power per octave) with a maximum peak amplitude of 50 percent full-scale (-6 dBFS). From each of five randomly picked times within this noise waveform, I copied 57-sample segments (1.29-msec at 44.1-KHz sampling frequency) into the clipboard. Each segment was in turn pasted repeatedly to the end of the growing sound-file until five regions were formed, each about 250 msec long (about 193 repeats) and representing a different noise motif. Why use 57-sample segment lengths? It was to allow direct comparison with the repeated noise motif from the maximum-speed portion of the flywheel recording (see Analysis of Flywheel Recording); that experiment occupies the next 500 msec of the sound-file (about 386 repeats). This is followed by a copy of the flywheel recording between maximum speed and stall, which plays in just over two seconds (time compression: 165-fold).

I applied 8.1 dB gain to the portions of the experimental sound-file derived from the flywheel recording, to give them an amplitude similar to the synthetic noise-derived parts. Thus, if you listen to the original flywheel recording (FlywheelX165.wav [3.5 sec]), be prepared for louder playback when you click on the experimental sound-file, which is: NoiseRepeatExp.wav (4 sec). Plots of the experimental file's waveform overview and spectrogram, made using Spectrogram 16 software, are shown below; time and frequency scales represent 165-fold compressed time with respect to the original flywheel recording. The different parts of the experimental file are identified above the waveform overview. For the periodic signals, the 1st (fundamental), 4th, 8th, 12th ... 28th harmonics are marked toward the left of the spectrogram (in the noise).

Spectrogram of tones synthesized by repeating five different noise motifs, compared to tone made by repeating flywheel cycle, followed by deceleration portion of flywheel recording.

The spectrogram of the repeating noise motifs displays a set of 28 harmonics spaced 773.7 Hz apart, which equals 44,100 samples per second divided by 57 samples per cycle. The pattern of amplitude differences between the harmonics (i.e, the spectrum) varies between the different synthetic tones; we hear this as differences in timbre (listen to sound-file: NoiseRepeatExp.wav [4 sec]). In this case, timbral diversity is due to stochastic (chance) differences between the waveforms of the randomly picked noise segments.

Interestingly, compared to the spectra of the five pink noise-derived tones, the tone made by repeating the max-speed wheel motif seems deficient in harmonics beyond the 19th. As noted in my Analysis of Flywheel Recording,  a 3rd-order low-pass filter with 50-Hz corner was applied to the signal. At 165X time compression, this corner frequency represents 8250 Hz (between 10th and 11th harmonics). The filter does not readily account for the observed pattern; moreover, the pattern maintains as the wheel decelerates and the spectrum progressively shifts to lower frequencies. The spectrum probably indicates something about the surface characteristics of the flywheel. Apparently, the amplitude of "noise" on the wheel's surface is not completely independent of distance scale (frequency range).

Comparing the recorded wheel deceleration with the tone made by repeating its max-speed cycle, note that some of the latter's apparent harmonics (especially the "24th" and "27th") are actually mostly aliases. (See Analysis of Flywheel Recording for information about aliasing.) At maximum wheel speed, the aliases just happen to coincide with actual harmonic frequencies because the sampling frequency is evenly divisible by the fundamental frequency (44,100 Hz / 773.7 Hz = 57.0 time-compressed; 268 Hz / 4.70 Hz = 57.0 as originally recorded). Thus the "24th harmonic" is really the alias of the 33rd, and the "27th" is the alias of the 30th. The 28th harmonic seems absent in the tone made by repeating the wheel's max-speed cycle, but is represented in the deceleration. This probably means that one complete cycle is not precisely 57 sample periods long at maximum wheel speed.

An Evolutionary Method for Tone Synthesis. This experiment gave me an idea. General categories of analog tone synthesis are "additive" (mixing harmonics at the desired amplitudes and phases) or "subtractive" (filtering different input waveshapes). Digital approaches include physical modeling and arbitrary waveform generation. You might call these conventional methods "intelligent design." In contrast, my idea is an "evolutionary" approach, inspired by Darwinian theory. Starting from a large random "population" of tones made by looping different specific noise waveforms, the best-sounding tones are "selected" (by listening to them? by algorithmic comparison to some spectral criterion?), but the majority are discarded (become "extinct"). A "surviving" tone, which one might call the "fittest," may be subjected to iterative rounds of "mutation" and selection. For "mutation," the repeating unit may be mixed with different noise motifs, and these products looped to make another large population of cadidate tones, each looking for a "niche" in human musical sensibility. Perhaps among the "hopeful monsters" are undiscovered useful sounds--happy accidents.

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