This page shows some new ways to create sawtooth waves for music synthesizers. The sawtooth is one of the fundamental waveforms in sound generation. It contains all integer numbers for the harmonics and sounds very clear and pressureful.  In synthesizers, it is the basis for fat basses and lead voices, as well as for pad sounds for string instruments. It sounds bright and slightly aggressive.

the real sawtooth wave

Unlike the triangle wave, the sawtooth wave contains both even- and odd harmonics. The fundamental has the full initial amplitude. The first overtone, at twice the frequency, has only half the amplitude. The second overtone has three times the frequency and only one third of the amplitude- Musically speaking, the first overtone is the octave, the second is the fifth within that octave, and the next is another octave. The fifth note then jumps to the fifth in the following octave again.

Typically the wave is generated with an easy equation using just a counter or the phase vector itself. However the sound can be changed in applying a band limitation:
 

band limited sawtooth wave

The following figure shows the curve when using the first 31 possible harmonics. According to Fourier, this results in limited rises and rest ripple. Even though the curve looks a little strange, it sounds much smoother to our ears.

Such a band limited waveform is much better for digital systems because together with the limited sampling rate, it does not generate aliasing, which is a major problem in digital synthesizers. If you can limit the number of frequencies, you’ll avoid creating any that might cause problems later on. This also ensures that every note has the same number of overtones.

the shark fin wave

If one lets the amplitudes (starting at 100%) fade out for high freqs by subtracting a some percents each time, one can reduce the aggressiveness signifiantly. With the 95% configuration, this very interesting curve results:

With this gentle roll-off to the amplitudes a pretty tonal response in the overtones is achieved. The shape reminded my to a shark fin, so I named the preset like this. It works very well when mixed with identical waves that are slightly out of tune to create superimposed waveforms (super saws). However there are still many aggressive harmonics included which cause aggressiveness when it comes to more than 3-4 waves.


musical saw tooth wave

Even with band-limiting to prevent aliasing and roll off of their amplitudes, harmonics still cause a problem when trying to create smooth, full-bodied sounds with chords. In detail the 7th, 11th, and 13th harmonic as well as their octaves do not fit well into the acoustic image. Therefore, another approach is to simply omit them.




Here only the fundamental + 4 octaves and the 3rd + 5th and their octaves are used with a soft roll off. Once again, the curve looks very jagged, but it sounds much smoother already, especially when combined with other elements, more harmonical. Continuing the idea:

In the next diagram, only 4 octaves and the 3rd + 5th and their first octaves are used together with a roll off. And now that the distracting overtones have been reduced, you can also remove the roll-off function:





 This makes the wave slightly sharper again, but it still blends well with other notes in the chord. The rapid beat frequencies are significantly reduced. The sound is clearer and more transparent. The way the sound blends into the decaying waves of other chords through sustain and reverb also becomes more transparent.

Conclusion

Bandlimitation removes the problem of Alias and makes saw tooth waves more musical. If you can limit the frequencies based on pitch, you can prevent frequencies from arising that might cause problems later on. Low fundamental frequencies in the bass have a similar relative harmonic spectrum to high frequencies and blend more harmonically into the music. Sounds with complex chords become softer, especially in string instruments.