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I know this is a little subjective in regards to what roughness is. So, imagine the sound of a sine wave, you could say this is the purest, or cleanest sound there is.

Now contrast that to a sawtooth wave, it's quite rough, though if you add in some AM or/and FM, then the roughness of a sawtooth wave (or any wave), will increase.

Now, what's interesting is that a sawtooth wave has more harmonics then the sine wave, this is obvious, however, consider that it's rougher than a square wave also, and when you trasition from a sine to a square to a saw, you add in more harmonics, and the overall roughness increases. Indeed, FM introduce an unholy amount of sidebands, and roughness seems to increase rapidly.

However, now consider the sounds of white noise, which is an equal distribution of frequencies low to high. This isn't rough at all, and so this shows that the frequency distribution of a sound is only correlated with roughness, but is by no means a direct cause.

What is, however, the direct cause of the roughness of a sound? What is the one factor from which roughness arises from?

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    This is really a wide-open question, 'specially since "roughness" is qualitative. Sawtooth and square waves both feature extremely high frequency components, so perhaps that's what you're after. Alternatively, massive clipping/overdrive can lead to a roughness of a different sort. White noise, BTW, has uncorrelated phase vs. frequency, while a well-controlled sawtooth has very specific phase vs. frequency behavior so as to keep the waveform's shape. – Carl Witthoft Aug 30 '15 at 12:43
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    I wouldn't agree that white noise “isn't rough at all”. Just, it's usually quieter! – leftaroundabout Aug 31 '15 at 0:27
  • Roughness must be in the ear of the listener, because I think white noise is the "roughest" possible sound there is. – Todd Wilcox Sep 15 '15 at 14:46
  • @Todd Wilcox, it's interesting that some people use white noise machines to sooth themselves to sleep. Maybe white noise isn't so rough? I don't know the science of wave forms, but maybe it's more about unequal or random distribution of frequencies. – Michael Curtis Jan 20 '16 at 16:14
  • Perhaps the 'envelope' needs to be considered too. To me a harsh tone that is faded in rather than sudden seems less harsh. Also, maybe if the harmonic change over the duration of a tone occurs in a random/uneven way it might make the sound harsh... or at least annoying. – Michael Curtis Jan 20 '16 at 16:18
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The term auditory roughness was first introduced by Hermann von Helmholtz in 1885. In general, to my understanding, a high level of auditory roughness is defined as possessing:

  • High ratio partials (harmonics far away from the fundamental).

  • Amplitude fluctuations of the spectrum (rate being very important).

  • Pitch instability.

From wikipedia:

According to psychophysical theory, the roughness of a complex sound (a sound comprising many partials or pure tone components) depends on the distance between the partials measured in critical bandwidths. Any simultaneous pair of partials of about the same amplitude that is less than a critical bandwidth apart produces roughness associated with the inability of the basilar membrane to separate them clearly.

From here:

Psycho-physiologically, the roughness sensation can be linked to the inability of the auditory frequency-analysis mechanism to resolve inputs whose frequency difference is smaller than the critical bandwidth and to the resulting instability or periodic "tickling" (Campbell and Greated 1994: 61) of the mechanical system (basilar membrane) that resonates in response to such inputs.

And from here:

The auditory sensation of roughness can be described as an auditory attribute that represents a rapid sequence of brief auditory events. With respect to their subjective characteristics, these events as such are poorly defined; what matters is just that they occur in rapid succession. Rapid means that, to be perceived as roughness, the time interval between successive events must be less than about 30 ms. When the time interval is greater than about 30 ms, the events are perceived as individual events, e.g., as peaks of an oscillating loudness-time pattern.

When the auditory events are physically evoked by a periodic train of sound impulses, the magnitude of roughness (briefly just termed roughness) essentially depends on (1) the shape of the individual impulses; and (2) on the pulse frequency. For a fixed pulse frequency, roughness increases when the impulses are made shorter in duration. For a given impulse duration, roughness decreases when pulse frequency is increased (e.g., Miller & Taylor 1948a, Fastl 1976d, 1977f).

So, roughness indeed is an auditory percept that represents a sound's time structure, though in a "time-integrated" manner. Time structure is aurally represented without concsious apprehension of the individual events.

There's much more to say about it, but my experience in this is very limited. Perhaps someone with more experience can expand on this subject.

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Plenty of things: low sampling rate, low bit-depth, introduction of "noisy" elements such as you mentioned. I think the general principle of "roughness" you describe would fall under the subject of "inharmonicity."

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