## (compMus1) Modulation Synthesis

General

Modulation synthesis uses the output of one oscillator (operating at an audio rate) to change (modulate) some parameter of another oscillator. The oscillator that is having its parameter changed is referred to as the carrier. The oscillator that is affecting the parameter change is referred to as the modulator. In general, modulation synthesis will produce a pair, or pairs, of frequencies around the original carrier oscillator frequency with specific mathematical relationships. These pairs of frequencies are referred to as sidebands.

Ring Modulation

Ring Modulation involves multiplying the output of two bipolar oscillators. Bipolar signals deviate in both the positive and negative directions. Multiplying two signals in the time domain involves multiplying their simultaneous amplitude values. The result of this multiplication is a single waveform.

Ring Modulation produces a single pair of sidebands: CarrierFreq +/- ModulatorFreq . Both the original carrier frequency and modulator frequency are eliminated, or suppressed.

For Ex.:

A Modulating Oscillator with a frequency of 100 Hz ring modulates a Carrier Oscillator with a frequency of 1000 Hz. The result is

CarFreq + ModFreq = 1000 + 100 = 1100 Hz      and
CarFreq – ModFreq = 1000 – 100 = 900 Hz

Since you’re just multiplying two amplitude signals, it doesn’t matter for Ring Modulation which oscillator is the carrier and which is the modulator. Consider the above example, with the carrier and modulator reversed (CarFreq = 100 and ModFreq = 1000):

CarFreq + ModFreq = 100 + 1000 = 1100 Hz     and
CarFreq – ModFreq = 100 – 1000 =  -900 Hz

Note that sidebands will be produced for every frequency present in the carrier, by every frequency present in the modulator.

The above examples assume sine waves. If you were to use sawtooth waves, you would have sidebands around every partial of one oscillator, with a pair for every partial in the other. This can get very noisy and harsh.

Negative Frequencies

Negative frequency values are the same as positive frequency values with the waveform’s phase reversed (180°). Another way to think about it is to multiply the amplitude values in the positive frequency by (-1). The resulting sound of negative frequencies is the same, except when combined with a positive frequency of the same Hz value. A positive and negative frequency of the same value (e.g., 500 Hz and -500 Hz) combined will result in phase cancellation. If the two amplitudes are equal (positive and negative absolute values), then complete cancellation occurs. Otherwise, you subtract the lessor amplitude from the greater amplitude to find the result.

Negative frequencies amplitudes are graphed downward from the x axis, when the x axis is frequency, and the y axis stretches above (+) and below (-) zero.

Amplitude Modulation

Amplitude modulation is the same as ring modulation, except that the modulator signal is unipolar – that is, it deviates in only one direction from zero. The result is the same set of single sidebands (CarFreq +/- ModFreq), plus the original carrier frequency.

Frequency Modulation

Frequency Modulation is explained quite well by Jeff Haas in his Intro to Computer Music, Vol 1.

http://www.indiana.edu/%7Eemusic/etext/synthesis/chapter4_fm.shtml

The output of the modulating oscillator is added to an initial frequency of a carrier oscillator, creating a changing frequency setting at an audio rate. The result combines the carrier frequency, with sidebands at +/- the modulating frequency and its integer multiples. The number of sidebands is determined by the amplitude of the modulator, and is referred to as the index of modulation.

FMout = CarFreq, (CarFreq +/- 1xModFreq), (CarFreq +/- 2xModFreq), (CarFreq +/- 3xModFreq)….

Example:

Modulator Frequency = 200 Hz
Carrier Frequency = 200 Hz

FMout = 200, (200 +/- 200), (200 +/- 400), (200 +/- 600)…. = 200, (400, 0), (600, -200), (800, -400)…

Beyond the ability to produce more sidebands, the advantage of frequency modulation (over ring or amplitude) is the ability to control the spectrum over time by varying the amplitude of the modulating signal. As the amplitude of the modulator increases, the index of modulation increases, and the result is an increase in the number of sidebands.

Modulation Ratios

Modulator-to-Carrier frequency ratios (M:C) can be generalized as integer-related and non-integer-related. If M:C is integer-related, one of the frequency values will be an integer multiple of the other, and the resulting spectrum (series of partials) will be harmonic. If the M:C is non-integer-related, the two frequencies will not be integer multiples of the other, and the resulting spectrum will be inharmonic.

Harmonic spectra resemble pitched musical instruments, while inharmonic spectra do not.

Certain ratios will produce reasonable instrument approximations. 1:1 produces an oboe, saxophone, or brass-like tone. 2:1 will produce a clarinet-like tone. 1.4:1 produces a bell-like tone.

Note: most literature refers to C:M ratios, which are simply reversed from what I talk about it. My way relates more directly to the overtone relationship, and is the way one specified ratios when working the the Yamaha-series FM synth’s, like the DX7.