Ring modulator

Input times carrier: sum and difference frequencies in the spectrum, the multiplication over time, the transition to AM and tremolo and why the sound becomes inharmonic.

Explanation Tasks

Time course: input × carrier = output (normalised)

Spectrum (linear amplitude) ┃ Input ┆ Carrier ┃ Output

Output
Lowest partials
Number of partials
DC component
Amplitude at the input frequency
Amplitude at the carrier frequency
Harmonic?
Common fundamental
Partial numbers

Speed

Input

Signal

Sine Two tones Sawtooth

Frequency (fundamental)

Second tone

Number of partials

Carrier

Waveform

Sine Square (diode ring)

Carrier frequency

Output Mix (0 = input, 1 = product only)

Frequency axis

linear logarithmic

Presets

Keyboard: Space = play/pause, F = presentation mode

What is happening?

Multiplying instead of mixing

A mixing desk adds signals: vocals plus guitar gives vocals and guitar. A ring modulator, by contrast, multiplies two signals, the input x and a carrier c, usually a sine: y = x · c, where y is the output signal. The result sounds like neither of them – metallic, bell-like or robotic. The name comes from the circuit of four diodes arranged in a ring between two transformers. It was developed for carrier-frequency telephony and later used in synthesizers and effects units.

In the diagram “Time course: input × carrier = output (normalised)” you see the multiplication point by point: input in blue, carrier in teal, output in orange. Where the carrier is positive, the input passes with the same polarity; where it is negative, inverted; at every zero crossing of the carrier the output is zero. “Play” runs time, “Speed” sets the pace.

Below, “Spectrum (linear amplitude)” shows input and carrier at the top and the output at the bottom; dashed lines there mark the position of the input tones.

Sum and difference

For two cosine waves, cos a · cos b = ½ · [cos(a − b) + cos(a + b)], where a and b are the instantaneous phase angles of the two waves. An input tone of frequency f and a carrier of frequency fc (the “carrier frequency”) therefore become two new tones: the difference frequency |f − fc| and the sum frequency f + fc, each with half the amplitude. The input tone and the carrier themselves no longer appear at the output – that is why the circuit is also called a balanced modulator.

Worked example (preset “Sine 440 Hz × 300 Hz”): 440 − 300 = 140 Hz and 440 + 300 = 740 Hz, each with amplitude 0.5. The readouts “Amplitude at the input frequency” and “Amplitude at the carrier frequency” both show 0. In the time view this appears as a beating pattern whose envelope follows the magnitude of the carrier.

Try it: slowly raise the “Carrier frequency”. The two output lines move in opposite directions: the sum rises, the difference falls. Once the carrier passes the input frequency, the difference becomes negative and folds over at 0 Hz – at fc = 500 Hz you get |440 − 500| = 60 Hz.

Why the sound becomes inharmonic

With a sound containing harmonics, every partial k · f₀ (the k-th partial, i.e. k times the fundamental frequency f₀) becomes k · f₀ − fc and k · f₀ + fc. All partials are shifted by the same amount in hertz, not by the same factor. The integer ratios are therefore lost and the sound becomes inharmonic, metallic or bell-like. Example: sawtooth at 110 Hz with 8 harmonics, carrier 300 Hz. The result is 30, 80, 140, 190, 250 Hz … with no common fundamental above 20 Hz.

The “Harmonic?” block checks this: “Common fundamental” is the highest frequency of which all output partials are multiples; “Partial numbers” tells you which partials of it the lowest six are. For 440 × 300 Hz the common fundamental is 20 Hz, but 740 Hz would be its 37th partial – that high up no fundamental can be heard any more, so the verdict is “inharmonic”. When two components land on the same frequency, the module adds them with their phases; they can reinforce or cancel each other.

Only when the carrier is in a simple ratio to the fundamental does a common fundamental remain. Try it: sawtooth 110 Hz, carrier 220 Hz gives multiples of 110 Hz again plus a “DC component” (220 − 220 = 0 Hz). With 165 Hz (1.5 · f₀) the result is 55, 165, 275, 385 Hz … – the odd multiples of 55 Hz: harmonic, but an octave lower and hollow like a square wave. The preset “Bell sound” (220 Hz × 311 Hz), by contrast, shows the typical inharmonic case.

No transposing: chords get bent

Common misconception: “A ring modulator shifts the pitch.” It does not transpose; it shifts frequencies by a fixed amount – and that destroys intervals. Try it: signal “Two tones” at 440 and 550 Hz, a major third (5 : 4), carrier 300 Hz. The result is 140, 250, 740 and 850 Hz. The ratios are now 250 : 140 ≈ 1.79 and 850 : 740 ≈ 1.15 – no longer a third, and no common fundamental above 20 Hz.

Set the carrier to 110 Hz, the common fundamental of the two tones. Now the result is 330, 440, 550 and 660 Hz, the 3rd to 6th partials of 110 Hz – harmonic again. Some synthesizers therefore tie the carrier frequency to the pitch being played, so that the ratio stays the same for every note.

Slow carrier: beating, AM and tremolo

With a carrier below about 20 Hz you no longer hear new tones but a level fluctuation. Try it: carrier 5 Hz, mix 1. The result is 435 and 445 Hz, heard as beating. In the time view the envelope fluctuates ten times per second (twice the carrier frequency), and the polarity flips at every zero crossing of the carrier.

The “Mix (0 = input, 1 = product only)” control computes y = (1 − m) · x + m · x · c, with m as the mix setting. With m < 1 the input tone remains and the two sidebands are added: that is amplitude modulation (AM), as in medium-wave broadcasting. At m = 0.5 (preset “Tremolo (AM at 5 Hz)”) 440 Hz has amplitude 0.5 and the sidebands 435 and 445 Hz have 0.25 each. The level now fluctuates five times per second, without polarity reversals – a tremolo at the carrier frequency.

The modulation depth is m / (1 − m): exactly 100 % at m = 0.5, where the envelope just reaches zero; only 33 % at m = 0.25, a gentle tremolo. Above 0.5 the modulation is overdriven, the envelope crosses zero, and the polarity reversals of pure ring modulation return.

Diode ring, leakage and studio practice

The classic diode ring switches the polarity of the input in time with the carrier – so it multiplies by a square wave rather than a sine. The square wave contains the odd harmonics 3 fc, 5 fc … with amplitude 4/(π · k) for the k-th harmonic, and around each of them further sidebands appear with 1/3, 1/5 … of the amplitude. Try it: waveform “Square (diode ring)” at 440 × 300 Hz. The main sidebands at 140 and 740 Hz grow to 2/π ≈ 0.64, joined by 460 and 1340 Hz (0.21 each) and 1060 and 1940 Hz (0.13 each), and so on. The sound becomes rougher and brighter.

Real devices are never perfectly symmetric: a little carrier or input “leaks” through, especially when the input signal has a DC component – DC × carrier = carrier. Remedies are a trimmer for carrier suppression, a high-pass in front of the input and, on stage, a gate so that no carrier tone is heard in pauses.

Typical uses: robot voices (preset “Robot voice (30 Hz)”: each partial splits into two tones 60 Hz apart, the formants stay almost in place, and speech remains intelligible), metallic percussion and bell sounds. The same principle is behind intermodulation distortion: a nonlinear transfer curve multiplies two tones with frequencies f₁ and f₂ and produces f₁ + f₂ and |f₁ − f₂| – inharmonic products that sound more unpleasant than harmonic distortion.

Limits of the model

The model uses ideal multiplication and stationary sine tones; transients, noise and component tolerances do not occur, so nothing leaks through here. The square carrier is limited to 15 odd harmonics in the spectrum and ideal in the time view. Sampling and aliasing do not occur. In the time view, input and output are normalised to their largest possible value, so their heights are not directly comparable. The module has no sound output.

“Harmonic” here means: all partials are (within 0.5 Hz) multiples of a common fundamental of at least 20 Hz, and the lowest six of them are at most the 16th partial – otherwise no clear fundamental is heard any more.

Tasks

Practice questions

Questions from studio, stage and mastering – to think about, discuss and check with the module.

1. For a robot voice, speech is ring-modulated with a carrier of about 30 Hz. Why does it sound metallic and yet still intelligible?

Show answer

Every partial of the voice is split into two tones 60 Hz apart. The formants (the broad resonance regions that define the vowels) stay almost in place, because 30 Hz is small compared with their width – that is why speech remains intelligible. The partials only fit a common fundamental if the speaking pitch happens to be a simple multiple of 30 Hz (as in the preset with 120 Hz). Since the voice constantly changes pitch, the result is usually inharmonic: rough, rasping, metallic. The effect became known through science-fiction voices in 1960s radio and television.
2. A hardware ring modulator emits a quiet tone although nothing is connected to the input. Where does it come from, and what helps?

Show answer

That is the carrier leaking through: if the circuit is not perfectly symmetric (unequal diodes, transformers, component tolerances) or the input carries a DC component, the product DC × carrier = carrier. Many devices therefore have trimmers for carrier suppression (“carrier null”). A high-pass filter or coupling capacitor before the input removes the DC component. Live, a gate before or after the effect keeps the carrier tone inaudible in pauses.
3. Why does intermodulation distortion often sound more unpleasant than harmonic distortion, and what does it have to do with the ring modulator?

Show answer

A nonlinear transfer curve (overdriven amplifier, loudspeaker at its limit) contains quadratic and higher terms. With two tones f1 and f2 the quadratic term creates the product f1 · f2 – so, like a ring modulator, the frequencies f1 + f2 and |f1 − f2|, and higher terms additionally 2 f1 − f2, etc. Harmonic distortion only adds multiples of a tone that fit the sound; intermodulation products, by contrast, lie inharmonically between the tones. It is measured, for example, with 60 Hz and 7 kHz in a 4 : 1 ratio (SMPTE): sidebands appear around 7 kHz at 60 and 120 Hz spacing.

Tasks

Predict first, then try it out, then open the solution. “Load starting point” sets the matching initial values.

1. Sum and difference

A 440 Hz sine is multiplied by a 300 Hz carrier. Which frequencies does the output contain, at what amplitude? What about 440 Hz and 300 Hz?

Load starting point

Show solution

The output contains only 140 Hz (440 − 300) and 740 Hz (440 + 300), each with half the amplitude, 0.5. Input and carrier have disappeared. 140 and 740 Hz only share 20 Hz as a common fundamental (7th and 37th partial) – this does not sound like a pitched tone but inharmonic. Move the carrier: the two tones move in opposite directions.
2. Slow carrier: ring modulation or tremolo

The carrier is only 5 Hz. What does the time course look like, and which frequencies appear? Then set the mix to 0.5.

Load starting point

Show solution

The result is 435 Hz and 445 Hz – two tones this close are heard as beating. In the time view the envelope fluctuates ten times per second (twice the carrier frequency), and the polarity of the signal flips at every zero crossing of the carrier. With mix 0.5, 440 Hz remains at 0.5 and the sidebands have 0.25: this is amplitude modulation, the level fluctuates five times per second without polarity reversals – a tremolo.
3. Harmonic or inharmonic

A 110 Hz sawtooth with 8 partials is modulated with 220 Hz. Is the result harmonic? Then try 165 Hz and 300 Hz.

Load starting point

Show solution

With 220 Hz all partials again lie on multiples of 110 Hz – harmonic, with shifted amplitudes and a DC component (2nd partial 220 Hz − carrier 220 Hz = 0 Hz). With 165 Hz the result is 55, 165, 275, 385 Hz …: the odd multiples of 55 Hz. The sound stays harmonic but sounds an octave lower and hollow like a square wave. With 300 Hz the result is 30, 80, 140, 190, 250 Hz … with no common fundamental above 20 Hz – inharmonic and bell-like.
4. Diode ring with square carrier

The same 440 Hz sine as in task 1, but the carrier is a 300 Hz square wave. Which additional frequencies appear, and how large are they?

Load starting point

Show solution

The main sidebands at 140 and 740 Hz are even slightly stronger at 0.64 (2/π) than with a sine, because the fundamental of a square wave has the amplitude 4/π. In addition there are sidebands around 3 · 300 = 900 Hz (460 and 1340 Hz, 0.21 each) and around 5 · 300 = 1500 Hz (1060 and 1940 Hz, 0.13 each), and so on. The sound becomes rougher and brighter – typical of simple hardware ring modulators.
5. A chord through the ring modulator

Two tones form a major third (440 and 550 Hz, ratio 5 : 4). What does a 300 Hz carrier turn them into? Then set the carrier to 110 Hz.

Load starting point

Show solution

With 300 Hz the result is 140, 250, 740 and 850 Hz. The ratios are now 250 : 140 ≈ 1.79 and 850 : 740 ≈ 1.15 – no longer a third, and there is no common fundamental above 20 Hz. A ring modulator therefore does not transpose, it shifts frequencies. With 110 Hz (the common fundamental of 440 and 550 Hz) the result is 330, 440, 550 and 660 Hz: the 3rd to 6th partials of 110 Hz – harmonic again.

Settings

Sound

Level calibration (dB SPL)

MIDI keyboard

Ring modulator