Case Study 7.2: The Theremin and Electronic Waveforms — When Pure Sine Waves Made Music

An Instrument With No Touch

In 1920, the Russian physicist Lev Sergeyevich Theremin invented an instrument that you play without touching it. Two metal antennas protrude from a wooden cabinet. The performer stands in front of the cabinet and moves their hands through the air. The right hand controls pitch by varying its distance from the vertical antenna; the left hand controls volume by varying its distance from the horizontal loop antenna. The result is a continuous, wavering, hauntingly expressive tone — one of the most recognizable sounds in the history of electronics.

The theremin was the first widely known electronic instrument. It predates the synthesizer, the Hammond organ, and every other form of electronic music-making. And from the perspective of Fourier analysis, it is extraordinarily simple: at its core, the theremin produces a nearly pure sine wave — a single frequency with minimal harmonic content. The question this case study explores is: what does it mean, acoustically and musically, to make music from the most mathematically simple waveform possible?

The Physics of the Theremin

The theremin operates on the principle of radio-frequency heterodyning. Inside the cabinet, two oscillators generate radio-frequency signals — one at a fixed frequency, one at a frequency controlled by the position of the performer's hand. The hand, acting as one plate of a capacitor (the antenna being the other plate), changes the capacitance of the circuit. Changing capacitance changes the oscillation frequency of the variable oscillator.

Neither of these radio frequencies is in the audible range — they operate at several hundred kilohertz, far above human hearing. The clever part is that the two oscillators' signals are combined in a nonlinear circuit (the heterodyne circuit), which produces output frequencies equal to the sum and difference of the two input frequencies. The difference frequency falls in the audio range (20 Hz to 20,000 Hz) and is the pitch you hear. As the performer's hand moves closer to or farther from the antenna, the difference frequency changes continuously — producing the theremin's characteristic pitch glide.

The output waveform of the basic heterodyning circuit is close to a pure sine wave. There are small amounts of harmonic distortion (the practical circuits are not perfectly linear), and additional processing — filters and amplifiers — introduce some further harmonic content. But compared to a violin, a clarinet, or a trumpet, the theremin's spectrum is extraordinarily "clean": most of its acoustic energy is in the fundamental, with harmonics at dramatically lower amplitudes.

Reading the Theremin on a Spectrogram

If you analyze a theremin performance on a spectrogram, you see something strikingly different from any acoustic instrument. Instead of the rich stack of harmonic lines that a violin or trumpet displays, you see one line — the fundamental — tracking the melody, with only faint traces of harmonics above it. When the performer glides from one note to another (a continuous portamento is the theremin's default mode), you see the fundamental line sweeping smoothly upward or downward, with none of the discrete stepping of keyboarded instruments and none of the complex harmonic structures of acoustic instruments.

The theremin's spectrogram is almost diagrammatically clear: melody as a single moving line. It is as if the Fourier transform has been run in reverse — instead of decomposing a complex waveform into many sine wave components, the theremin builds a melody from a single sine wave component at a time.

This visual clarity is one reason the theremin has become a pedagogically useful instrument in music and acoustics classes: its spectrogram is immediately readable, making it an excellent tool for demonstrating the relationship between pitch, melody, and frequency.

The Problem of Pure Sine Waves in Music

Despite its physical simplicity, the theremin's sound is perceived as strangely rich — not as thin or empty as a pure sine wave played through a speaker might suggest. This is partly because of the small amounts of harmonic distortion introduced by the electronics, but it is primarily because of the performer's continuous control of dynamics and the instrument's characteristic vibrato.

But the theremin also reveals a profound musical challenge: pure sine waves are difficult to make into music. When early electronic instruments produced truly pure sine waves with no harmonic content, musicians and audiences found them aesthetically unsatisfying for extended listening — cold, artificial, without the "grain" or "texture" that acoustic instruments provide. This was one of the central problems of early electronic music.

The pioneers of electronic music composition in the 1950s — Karlheinz Stockhausen, Pierre Schaeffer, Pierre Henry, and others working at studios in Cologne and Paris — quickly discovered that compositions made entirely from pure sine waves sounded inhuman and difficult to listen to for long durations. The warmth, complexity, and expressiveness of acoustic music come from its inharmonic richness — the overtones, the slight impurities, the way harmonics interact.

Some composers embraced this challenge, developing an aesthetic of pure electronic sound that deliberately confronted the listener's expectations. Stockhausen's Studie I (1953), composed entirely from pure sine tones, was one of the first works to take this approach seriously. Others began adding controlled harmonic content — using sawtooth and square waves, which have rich harmonic series, instead of sine waves — to create electronic timbres with more warmth and complexity.

Despite — or perhaps because of — its unusual acoustic character, the theremin became one of the most recognizable sounds in mid-20th-century popular culture. Its eerie, wavering tone was used extensively in science fiction and horror film scores of the 1940s and 1950s. The films Spellbound (1945, score by Miklós Rózsa) and The Day the Earth Stood Still (1951, score by Bernard Herrmann) used the theremin to evoke the alien, the uncanny, and the psychologically disturbed.

The association between the theremin and otherness is, in retrospect, acoustically appropriate: the theremin's nearly sinusoidal waveform is literally unlike any acoustic instrument. Its spectrum is not merely unusual — it is, in Fourier terms, the opposite of the rich, harmonically saturated spectra of orchestral instruments. When a film composer wanted to signal that something was not of the ordinary acoustic world, the theremin's pure-sine voice was an acoustically accurate choice.

The Beach Boys' 1966 song "Good Vibrations" featured a theremin-like instrument (actually an electrothermin, a variant played by a keyboard controlling the heterodyne circuit), contributing to the song's atmospheric, dreamy quality. By this point, the theremin's "eerie" associations had evolved into a broader "psychedelic" aesthetic — a sound associated with the exploration of inner space rather than outer space.

What the Theremin Teaches About Fourier Analysis

The theremin is, in one sense, a Fourier analysis demonstration instrument. Its nearly pure sine wave output corresponds to a Fourier spectrum with almost all energy in one bin — the fundamental. Compare this to the 20+ harmonics of a trumpet at fortissimo, or the 12+ harmonics of a bowed violin, and you understand viscerally the relationship between spectral complexity and timbre.

But the theremin also demonstrates the limits of spectral analysis for describing musical experience. A theremin performance, heard live, is far from simple. The performer's continuous pitch control, the expressive dynamic shaping with the left hand, the subtle vibrato — all create a musical experience that spectral analysis alone does not capture. A spectrogram of a theremin performance looks sparse compared to a violin spectrogram, but the musical experience of an excellent theremin performer is not correspondingly impoverished.

What the spectrogram sees is one dimension of music. The theremin reminds us that expressiveness, musicality, and aesthetic experience do not require harmonic complexity. A single sine wave, in the hands of a great performer, can carry a melody with extraordinary delicacy and emotional power. The physics is simple. The music is not.

Discussion Questions

  1. The theremin's pitch changes continuously — it does not jump from note to note but glides between them. What would this look like in a spectrogram? How would a theremin spectrogram of a melody differ from a violin spectrogram of the same melody?

  2. Early electronic music composers found pure sine waves aesthetically unsatisfying and quickly began using harmonically richer waveforms. What does this tell us about what makes music "warm" or "expressive"? Is harmonic richness necessary for musical expressiveness?

  3. The theremin became culturally associated with aliens, the uncanny, and the psychedelic. How much of this association is acoustic (the actual sound) and how much is cultural (the associations people bring to the sound)?

  4. Lev Theremin invented his instrument as a physicist, not a musician. Clara Rockmore, widely regarded as the greatest theremin virtuoso, approached it from a background in violin performance. How does this divide reflect the broader theme of this textbook — physics as constraint, music as creative response to constraint?

  5. Modern digital synthesizers can replicate the theremin's sine wave output perfectly — and can add any harmonic content desired. Does this make the original analog theremin obsolete as an instrument? What is lost and what is gained in moving from analog heterodyning to digital synthesis?