Two oscillators with a ratio of frequencies that's an irrational number will never have exactly the same phase relationship. So, phasor~ objects that have an irrational frequency relationship, when combined, will create a rhythm that never exactly repeats. In this example, you can hear that the sum of the two phasor~ objects with a constantly changing relationship will create a constantly changing rhythm.
If you scale a one cycle of cosine wave by a factor of -0.5 and offset it by 0.5 you get a "Hanning function", which goes from 0 to 1 and back to 0 as smoothly as possible. That can be used to shape the amplitude of a sound, turning it on and off smoothly, or it can be used to modulate any characteristic of the sound.
Frequency modulation is the use of one oscillator—usually but not obligatorily at a sub-audio frequency—to modify the frequency of a sound. The modulating oscillator is added to a main frequency value to create a frequency that fluctuates up and down from the central value. The result, at low modulation rates, is called "vibrato".
Amplitude modulation is the use of one oscillator—usually but not obligatorily at a sub-audio frequency—to modify the amplitude of a sound. (Ring modulation, shown in Multiplication of Sine Tones, is one particular example of amplitude modulation.) The modulating oscillator is added to a main amplitude value to create an amplitude that fluctuates up and down from the central value. The result, at low modulation frequencies, is called "tremolo".
To play two tones, you need two oscillators: two cycle~ objects). To mix them together, simply add the two signals with a +~ object. (For digital signals, addition is mixing.) To control the amplitude, multiply it by some factor, using a *~ object. (Multiplication is amplification.)
The amplitude of a sound is controlled by multiplying the sound wave by a certain factor. A multiplier of 1 represents "unity gain", meaning no change. Multiplying by a factor between 0 and 1 reduces the amplitude of the sound. However, if the multiplier is changed very suddenly and significantly, it may create a sudden discontinuity in the waveform which will be heard as a high-frequency click.
This example shows a simple way to mix two audio sources.
This example demonstrates how to modulate the playback speed of an audio file. The value in the right inlet of sfplay~ determines the playback rate; 1. is normal speed, 0.5 is half speed, 2.0 is double speed, and so on. The rate can be provided as a continuously changing control signal instead of as a single constant value, allowing us to warp the speed at will. Here we're using a cycle~ object to produce a low-frequency sinusoid over the course of ten seconds (i.e. at the frequency of 0.1 Hz).
This exmaple demonstrates a simple way to ramp amplitude with number~.
The example demonstrates a simple way to play an audio file.