Now I am talking about pitches in the octave below what is low E on a bass guitar. But the general idea still applies: waveforms of short notes at low frequencies have fewer cycles in their duration than we have fingers on our hands. So the thinking is about managing a sequence of, say, 6 cycles of a waveform, much like one would arrange percussion samples with a drum sampler, rather than the conventional view of a synth with oscillators, filters, and envelopes.

Now, a low, staccato 16th note bass is typical in psytrance. Something it only occurred to me recently was to look at the waveform itself. Let's do a little maths.

1. Assume a bpm of 140. That means 2.33 beats per second, and so 0.428 seconds per beat. Then a sixteenth note takes 0.428/4=0.107 seconds.
2. Suppose our bass note is A 3 octaves below A440 (2 and a half octaves below middle C). This has a frequency of 55Hz. (This is the A below low E on an electric guitar).
3. If we suppose that our bass note plays for the entire 16th, then there are only 5.892857142857142 cycles of our bass note in that entire 16th. If our staccato 16th note takes up only half of that 16th, which isn't untypical, then it only has about 3 cycles.
4. Now given that, it makes sense to look at the waveform itself and think about it: if it is a raw unfiltered sawtooth, then there will be two or three discontinuties depending on where the waveform starts (i.e. its phase). Then where abouts, with respect to the music timeline, do those discontinuities go?
5. The thing that occurs to me is that these details, which are hidden behind a typical synth console, matter a lot for something like this. Thus, we perhaps should:
   1. Sample a raw oscillator, or perhaps with a filter envelope but crucually no amp envelope. If we can separate the application of a filter envelope then even better, since we want the shape of a filter envelope applied in a sample accurate way.
   2. Essentially, though, what matters is what the filter does to the discontinuity, assuming it is a lowpass.
   3. Then we want to apply our amp envelope in a sample accurate way.

This sort of thing is not the normal sort of thing we use our DAW's for. Often we'll fiddle with the controls in a synth, such as the envelope sliders. Then we will start to complain about one synth's envelopes not being snappy enough, or something. Really what we need is the ability to draw the filter and amp envelopes directly over the wave, and the ability to precisely set the phase where we want it.

My thinking is that we want the first discontinuity exactly on the beat, and then we will have three discontinuities in total. Then the amplitude of the wave at these discontinuities is what will give much of the shape to the overall volume.

Final remark: Even with a maths background, until I stopped to do the calculations, I didn't realise how few cycles there are in a staccato bass note. It's actually amazing that you hear the pitch, though I suspect that your brain figures out the pitch from hearing multiple such notes, more than from hearing each individual one in isolation.

The simplest maths to do is to have a tempo of say 120, so a quarter note lasts exactly half a second. Then take a hypothetical pitch around 48Hz, so that there are 48 cycles per second, so 24 cycles per quarter note, and so 24/4=6 cycles per sixteenth. This is approximately the G below A55, which has a frequency of 48.9Hz.

But basically engineering a low fast staccato bass really is about managing the individual cycles.

Numbers

These are table of the number of cycles per 1/16th note for a given note, from E one octave below low E (on a bass guitar) to the Eb one octave (minus semitone) above.

Octave below low E:

Octave containing low E:

Related

• There is a discussion by Daniel Sokolovskiy on his blog. One thought is this:
  • One thing which you can actually see in one of the screengrabs, is that the bounced wave has less than half a dozen complete cycles in it. The harmonics are largely generated by the discontinuities in the waveform (where it jumps from high to low). In terms of note length, that controls whether you get, say, three discontinuities during the course of your bass sound, or four (depends upon phase of the sawtooth). I wonder if taking an approach of engineering the synthesised waveform rather than just taking the output of a softsynth may yield better results.