Phase distortion

Casio introduced phase distortion in its CZ synthesisers, starting with the CZ-101 in November 1984, a year after Yamaha's DX7. Casio's patents on the method name its engineer Masanori Ishibashi as the inventor. Like FM, it changes the timbre of a sine-like wave by calculation rather than with a filter, and like the phase modulation inside the DX7 it works by moving the phase of the wave. The difference is in what moves the phase. In FM a second oscillator moves it, and in phase distortion a fixed shape bends it once in every cycle.

The phase of an oscillator is its position through the current cycle, counted from 0 at the start to 1 at the end. A digital oscillator keeps the phase as a number that rises steadily from 0 to 1 and then starts again, and reads its output from a stored wave at that position. If the phase rises in a straight line and the stored wave is a cosine, the output is a sine wave. Phase distortion bends that line. The phase still runs from 0 to 1 in each cycle, so the pitch does not change, but it runs faster in some parts of the cycle and slower in others, and the cosine is squeezed in the fast parts and stretched in the slow ones. A cosine starts and ends each cycle at its peak, so the bent wave meets the next cycle without a jump.

Shapes

For the Sawtooth shape the phase rises quickly to its halfway point and then slowly for the rest of the cycle. The cosine falls from its peak to its trough in that short first part and climbs back slowly, which is the shape of a sawtooth with its ramp rising. For the Square shape the phase rises quickly to halfway, waits, rises quickly to the end and waits again, so the cosine spends most of each half cycle held at its trough or at its peak. The two halves are mirror images of each other, and a waveform whose second half is the first half turned upside down contains only odd harmonics. For the Pulse shape the phase runs through a whole cycle in a short part of the period and then waits, which gives a narrow dip once per cycle.

The CZ synthesisers offer eight waveforms: sawtooth, square and pulse, a double sine, a half sine, and three resonant waveforms. A control called the DCW, for digital controlled wave, sets the amount of bending. At its minimum each of the non-resonant waveforms is a sine wave, and at its maximum it is the full shape.

Resonance

The resonant waveforms are made differently. A sine wave at a higher frequency than the note is restarted at the start of every cycle of the note, so its phase rises steeply and jumps back to zero once per cycle. Unless its frequency is a whole multiple of the note's, the inner sine is partway through a cycle when it is cut off, and the restart would make a jump in the waveform. It is therefore multiplied by a ramp that falls to zero at the end of each cycle, so the restart happens where the level is already zero. The repeating restart puts the inner sine's energy into the harmonics of the note nearest its frequency, and raising the amount raises that frequency, so the peak moves up through the harmonics. A resonant lowpass filter swept upwards also moves a peak up through the harmonics, which is why the two sound alike. Restarting one oscillator at the frequency of another in this way is a digital form of hard sync. The CZ gives the resonant frequency a control but not the height of the peak.

The DCW envelope

On a CZ synthesiser the pitch, the DCW and the level each have their own eight-stage envelope, and each stage has a rate and a level. The DCW envelope does the work that a filter envelope does on a subtractive synthesiser: a fast fall from a high amount gives a bright attack that settles to a duller tone, and a slow rise gives a swell. The tool below has a simpler envelope on the amount, with a decay and a sustain level.

The CZ-101 sold for a fraction of the price of a DX7. Raising its DCW amount brightens the sound steadily, as opening a filter does, whereas raising the index in FM makes some harmonics rise and then fall back. Casio's later VZ synthesisers used a different method that Casio called interactive phase distortion, which works by phase modulation between oscillators with several waveforms, closer to FM than to the CZ method. Wavefolding also bends a sine wave into a brighter shape, but it changes the wave's level rather than its phase.