______                     
                 /      |       ________     
                /       |____  /        \    
               /        |    \/   _      \   
               \       ___       /_\      \  
                \      \_/          .      \ 
                 \          /\_____/       / 
                  \________/      |       /  
                                  |______/   

[THE_EXPR_METHOD] 009 Phase shifting

Previously, the fmod() function had been used in the context of frequency multiplication. For example:

[phasor~ 0.25] -> [expr~ fmod($v1*4, 1)]

Besides this, it can also be used to shift the phase position of a sawtooth oscillator. The principle is largely the same as the example above, apart from replacing the multiplication with addition, which represents the amount of displacement. For example:

[phasor~ 0.25] -> [expr~ fmod($v1+(1/4.0), 1)]

This will push the phase position of the input sawtooth by a quarter of the total wavelength.

                                             Original phase position
     .    /|       /|       /|       /|
     .   / |      / |      / |      / |
     .  /  |     /  |     /  |     /  |
     . /   |    /   |    /   |    /   |
     ./    |   /    |   /    |   /    |
     /     |  /     |  /     |  /     |
    /.     | /      | /      | /      |
   / .     |/       |/       |/       |
     .                                       Phase shift by 25%
     .  /|       /|       /|       /|
     . / |      / |      / |      / |
      /  |     /  |     /  |     /  |
     /   |    /   |    /   |    /   |
    /.   |   /    |   /    |   /    |
   / .   |  /     |  /     |  /     |
  /  .   | /      | /      | /      |
 /   .   |/       |/       |/       |

Musically speaking, it essentially adds a time delay/offset to the original control signal. One way this can be useful is to arrange the shifted signals in parallel, which would produce effects similar to a musical canon. In practice, this enables the ability to generate interesting compositional variations from a limited pre-defined sequence. Furthermore, the amount of shifting can also be modulated just the same way as amplitude and frequency, which allows for more dynamic and responsive structures to emerge.

For controlling sound synthesis, phase shifting provides a simple mechanism to move envelope and logic level along the time axis. Consider the following:

[phasor~ 0.25] -> [expr~ if($v1<0.25, 1, 0)] -> [s~ signal_1]
              \-> [expr~ fmod($v1+0.25, 1)] -> [expr~ if($v1<0.25, 1, 0)] -> [s~ signal_2]

signal_1 produces a gate switching signal with 25% duty cycle. By shifting the phase of the [phasor~], signal_2 would have the same rate of change, but delayed by quarter of the total wavelength.

It is worth mentioning that, attention needs to be paid to the total amount of shift that are “allowed”. that is to say, the “effective” shift permitted would be the total wave period minus the duty cycle of the gate. In the case of above example, if the amount of shift had been greater than 0.75, the “on” portion of the gate would be pushed off the right edge, and thus may cause undesired effect.

Get the source here

Exercise:

• How to shift the phase of a sine wave?