In the context of synthesizers, "buzzing" quality is associated with unfiltered basic waveforms: sawtooth, square, triangle (to a lesser extent), pulse (notably so). A sawtooth wave is used, for example, as the bass sound in Gorillaz' "DARE".

More generally, in my personal experiments, "buzzing" is associated with the presence of discontinuities in the waveform (i.e., the Dirac delta and its antiderivatives). Any discontinuity is associated with sonic energy at all frequencies, at a highly localised point in time. (See the Fourier transform of Dirac delta (anti)derivatives here [1].) Higher antiderivatives of the Dirac delta have progressively less energy at higher frequencies; beyond the 2nd antiderivative buzzing is not really audible.

Aside – a pulse wave is a series of Dirac deltas; a sawtooth is the 1st-order antiderivative thereof; a square wave is a series of sign-alternating 1st-order Dirac delta antiderivatives; and a triangle wave is alternating 2nd-order Dirac delta antiderivatives. Hence – buzziness in these waveforms.

The human ear has a Q of about 15 (very approximate) – meaning its response at any frequency lasts for about 15 cycles of that frequency. So, when presented with a periodic discontinuity (e.g. sawtooth wave), the sonic content below about 15 times the base frequency will tend to cohere together into a tone, while the sonic content above 10 times the base frequency will tend to be perceived independently of frequency – as a buzzing. (See Bell, "A Resonance Approach to Cochlear Mechanics".)

So, if you want to increase the amount of buzzing in a waveform, you can add localized "packets" of high-frequency sonic energy up to a rate 1/15 that of the lowest frequency content of said packets. You can experiment with this in Audacity by generating sawtooth waves of various frequencies (between 25-250 Hz, where buzzing is easily audible) and low- and high-passing them appropriately to separate the "tonal" (low-frequency) content from the "buzz" (high-frequency) content. Then mix and match the two from different frequency waveforms two create a waveform at your desired base frequency with your desired rate of buzzing.

Finally, a more pedestrian – and very common – way that the above is achieved is the synthesis technique known as "supersaw", by which a handful of slightly-detuned sawtooth waves are mixed together. Beside giving a "shimmering" effect which one gets from mixing any slightly-detuned sounds together, this also results in increased "buzziness". This effect is very common in pop electronic music. E.g. the bass sound in Lady Gaga's "Just Dance" is a good example.

[1] https://en.wikipedia.org/wiki/Fourier_transform#Distribution...