It’s been some time since the last blog update. I’m excited to post a new update and with some maths! A slightly more cerebral read, perhaps?
Here we will do a basic walkthrough of both Ring Modulation and Amplitude Modulation. They have given us special fx sounds, alien voices, bell sounds, sub oscilators and other weird tonal sounds, but how do these effects come about? By breaking down the basic operation of this modulation, hopefully we can further understand, predict and design specific sounds.
Generally, Ring Modulation (RM) and Amplitude Modulation (AM) are referred to as the same modulation. Before we get into how they differ, the following excerpt from Designing Sound is a great introduction to what they achieve:
“Remember that modulations mean changing something in accordance with something else. In this case we are changing the amplitude of one signal with another. To do amplitude modulation (AM), we multiply two signals, call them A and B, to get a third signal C. That can be written simply in the time domain as C = A*B.” - Andy Farnell, Tip 20. Technique 4 Modulation p. 291
This multiplying of signals results in Sum and Difference partials, which are the Higher and Lower sideband components. The frequency of the sidebands are both displaced relatively to the Modulation frequency.
An easy way to remember is that the component frequency; Sum is the Carrier frequency plus the mod frequency. The Difference component is the Carrier frequency minus the Mod frequency.
There is a small difference between RM and AM, that is:
Amplitude modulation - outputs the original Carrier signal plus resulting sidebands.
Ring modulation - outputs the sidebands only.
In either case, the main purpose of RM and AM is to add new sidebands.
Ring Modulation tends to be the more desirable and unpredictable sounding modulation, because it does not reveal the original incoming signal. A tuned sound will typically become output as a skewed or odd tone.
On the other hand, Amplitude Modulation, at lower Mod frequencies, gives a true tremelo effect, due to modulating the amplitude. By including the original signal, it outputs a greater total of harmonics. Fundamentally, AM sounds like RM mixed with the original signal.
More Overtones
A common modulation result is that which creates a mere 2 sidebands - a Ring-modulated sine wave. This sounds like a buzzing-whistling tone (or cricket chirp) at a low mod frequency, or distinct wandering overtones at higher mod frequencies. For AM, there will be an extra harmonic; sometimes this frequency is referred to as a DC offset, as in Purr Data.
To break this down, for every harmonic that comes in via the Carrier wave, RM creates 2 new sideband components (See Fig 1. below).
Furthermore, this is multiplied by the number of harmonics in that modulator wave. In the above example, it is merely 1 sinusoid.
If we used 2 harmonics in RM carrier wave, we will get 4 harmonic as RM, or 6 frequencies output as AM, and with 3 harmonics, we get 6 with RM, and so on.
To calculate the theoretical number of harmonics output from RM:
Total Harmonics = x(Carrier Wave Harmonics) x y(Modulator Harmonics) x 2
The total number is then limited by the range of the audble spectrum, due to the frequency of the Carrier and Modulator. To calculate that of AM, just add the Carrier wave harmonics to the total.
In the next example here (Fig.2), we have a 2khz Carrier wave with 2 harmonics (1 & 5) and a Modulator at 289.05hz. You can see that with a Sine wave modulating, both harmonics will output 2 sidebands each.
Changing the modulator now from Sine Wave to Triangle Wave (Fig.3), the number of harmonics is clearly multiplied by the number in the Carrier wave.
With the Sine Wave, we have 6 total harmonics. But with the Triangle Wave, we have well over 30 extra harmonics. Depending on the Mod frequency (or Mod Index), the side clusters generally take on the same spectrum weighting or integer slope.
Overall, changing the Mod waveshape results anywhere between two sideband frequencies to two clusters of sidebands. Again, whether it is 1 sideband or a cluster per side, is related to the number of harmonics in the Carrier, times the number of harmonics in the Modulator.
What we also observe is that the fundamental frequency of the Modulator harmonic series begins at every Harmonic (H) of the Carrier wave. I.e. it is the closest frequency to the Carrier H. If the Modulator is set to 289.05hz, the first harmonic will start at +289.05hz from H. Interestingly, each theoretical partial n is spaced at Mod frequency Mf apart from the previous.
If you are not using a spectrogram, the below formula will give;
the specific harmonic multiple (n) resulting from modulation; using
the positive (Sum) or negative (Difference) partial from the modulation, and the
target Carrier harmonic (H for Carrier Harmonic multiple).
Due to the displacement of the partials, we can’t refer to every harmonic from 0-20khz in a normal consecutive fashion. Subsequently, we break them down in reference to the Carrier wave harmonics, and the math assumes that you know which of these harmonics are present in the Carrier and Modulator waves. In this instance, the Triangle wave will only output a series of odd integer harmonics.
For example, to find the 9th harmonic of the Mf, referring to H1 of the Carrier:
nth ‘Sum‘ harmonic of Carrier Harmonic 1 (H1) = Carrier H1 + (Mf x n)
∴ 9th ‘Sum‘ harmonic of H1 = 2000 + (289.05 x 9 )
∴ 9th ‘Sum‘ harmonic of H1 = 4601.45 hz
To find a negative partial, we would used the above formula but with a Negative n value:
(-)3rd ‘Difference‘ harmonic of H5 = 10000 + (289.05 x -3)
∴ the 3rd ‘Difference’ harmonic of H5 = 9132.85 hz
If you arrive at a negative value from calculating a Difference harmonic, simply change the polarity to positive (multiply by -1). After all, Ring modulation cannot produce ‘negative’ frequencies.
I hope you found this post valuable. Ring modulation indeed is one of the most mysterious kinds of modulation.
For more visual info, below is a video where there are more demonstrations on RM/AM.
Questions? What do you think? Leave a comment below.