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Add together some sinusoids with irrational or "nearly" irrational frequency ratios, and you get something pseudoperiodic. If there aren't too many, you can hear that it has only pitched components, even if you can't isolate them by ear.

If there are enough of them playing together, it will sound like steady noise.

The same goes with random transients. If they're separated in time well enough, you can hear them as distinct sounds. If they become very dense it sounds like steady noise.

Is there a model that predicts when a long-playing signal sounds like steady noise, and when it sounds merely rough or dissonant?

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    $\begingroup$ i have no idea why there are so many close votes. $\endgroup$ – robert bristow-johnson Oct 10 '18 at 8:06
  • $\begingroup$ Yes there are. These kinds of questions are answered by the field of "Psychoacoustics". But the way your question is posed, it makes it sound too broad. So, for instance, if your sound clip is less than 10ms it will be perceived as a click rather than the beginning of some sound. That is, the brain does not have enough time to "lock and recall" what the sound is. Between 0-100ms there are a lot of things happening in terms of timing and how it affects perception of sound. Do you think you can you make the question a bit more specific? $\endgroup$ – A_A Oct 15 '18 at 15:07
  • $\begingroup$ @A_A - I've edited the question. Maybe it's better now. $\endgroup$ – MackTuesday Oct 16 '18 at 1:05
  • $\begingroup$ Thanks for letting me know...This is still a bit vague: There are answers from psychoacoustics (perception) but also from musical theory. Which one are you after? Is this an audio application or musical application. It is strange to be making a distinction between music and sound (!) but I am trying to separate any kind of sound recording from recordings of specifically music. $\endgroup$ – A_A Oct 16 '18 at 8:18
  • $\begingroup$ @A_A - This question is purely about audition. It isn't limited to music. Does that answer your question? $\endgroup$ – MackTuesday Oct 17 '18 at 0:57
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As with any perceptual model, the answer is not exactly cut and dry. In my research, I’ve found that many of the hearing models I’ve encountered are based on band pass filter banks. These banks are numerous and tightly spaced. The more adjacent banks that are stimulated, the ‘noiseyer’ the signal would sound.

There are examples of inharmonic signal analysis that take noise into account, such as: https://pdfs.semanticscholar.org/b5bf/98425051f47b0685e75da986ee166e41d6ec.pdf. I’ve never tried to implement such a structure, but as I understand it, you analyze the peaks in the output of a frequency transform to create a ‘deterministic’ signal. You then subtract the deterministic signal from the original to get the ‘residual’ noise. You would then perform further analysis on the residual. I assume that this isn’t exactly what you were looking for in terms of a model, but I figured it may be informative.

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  • $\begingroup$ Yes. I'm asking about the part where you decide how nondeterministic a signal really needs to be for one to perceive it as steady noise. A nonstationary, stochastic signal will usually sound like noise, but it won't necessarily sound like steady noise. On the other side of the coin, you can add a bunch of sinusoids together and it will sound deterministic, unless the sinusoids are too dense. $\endgroup$ – MackTuesday Oct 17 '18 at 1:05
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...is there a mathematical model that predicts when a signal sounds like steady noise, and when it sounds merely rough or dissonant?

Roughness is a quantifiable concept in Psychoacoustics. It is measured in aspers and the most commonly used method of measuring it is due to Aures (see also references at the second article I am linking above) which attempts to generalise Roughness as defined over a single tone to multiple bands.

But, Roughness characterises loudness fluctuations and I am not sure how much it may cover all the examples you are mentioning (?)

Another thing you might want to try is a direct quantification of the spectrum with metrics such as Shannon's Entropy as derived by the amplitude spectrum or the entropy of the phase spectrum. In other words, you would be assessing the "coherence" of the sound. This does not involve perception, but it would be sensitive to whether the waveform starts sounding like noise and as a quantity, it could be used in regression in case you are trying to use it to drive some sort of decision about the composition of the generated spectrum.

A significant unknown here is the number of components the sound is made up from. If you have 3 tones at random intervals between them, you get a different type of "dissonant" than if you had 81 tones. The potential for noise-like waveforms is higher in the second case. So there might be a limit of "density" beyond which the perceptual and structural metrics do not deviate much and you might be able to assess the structure of the sound with the simpler metrics I mention above.

Hope this helps.

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