The Tritone and Standing Waves
Why the most dissonant interval in Western music is physically unstable — and what happens when you pair it with itself.
Every musical interval is a ratio. When you play two notes simultaneously, their sound waves either lock together cleanly or fight each other — and the physics of that fight, or that peace, is what we experience as consonance and dissonance.
Most intervals in Western music resolve neatly. The perfect fifth (C to G) has a frequency ratio of 3:2 — one of the simplest possible. The octave is 2:1. The major third is 5:4. These ratios produce standing waves that reinforce each other, creating a sense of stability and rest.
Then there is the tritone.
The Physics of the Devil’s Interval
The tritone — the interval of exactly six semitones, C to F# for example — has a frequency ratio of approximately √2 : 1, or 1.414:1. This is an irrational number. It cannot be expressed as a simple fraction of whole numbers. Its standing waves never fully lock together.
In acoustic physics, two frequencies produce a stable standing wave pattern only when their ratio can be expressed as small whole numbers. When the ratio is irrational — as the tritone’s is — the pressure peaks of the two waves never quite align. They create what physicists call beating: a rhythmic pulsing in amplitude as the waves move in and out of phase with each other. The ear perceives this as tension, roughness, and an urgent pull toward resolution.
Medieval theorists felt this instability viscerally without knowing its mathematical cause. They called it the diabolus in musica — the devil in music — and banned it from sacred composition. Counterpoint teachers for centuries after taught that the tritone must always resolve, its tension released by stepwise motion toward a stable interval.
“The ear doesn’t lie. When two frequencies can’t find a common period, the nervous system registers it as unresolved.”
— Acoustic physics, simplifiedWhat Beating Sounds Like
You can hear beating clearly by playing two notes very slightly out of tune — two strings on a guitar tuned a few cents apart, for example. The wavering pulse you hear is the physical result of the two wave peaks moving in and out of alignment. The slower the beating, the closer the frequencies. The faster, the further apart.
The tritone’s beating frequency depends on the register. A tritone in the bass beats more slowly and sounds like a low throb — deeply unsettled. The same interval two octaves higher beats faster and sounds harsher, more urgent. The physics is identical; only the frequency of the beating changes with register.
This is why jazz composers and improvisers use the tritone strategically. In a dominant seventh chord — the most harmonically charged chord in tonal music — the interval between the third and the seventh is always a tritone. That built-in beating is what gives the dominant chord its drive toward resolution.
The 016 Trichord — Containing the Tritone
In the pitch class set system, any set of notes containing a tritone carries that physical instability into the harmonic structure. The 016 trichord — intervals of 1 semitone and 6 semitones, for example C, C#, and F# — is built directly on the tritone. It is one of the most harmonically charged trichords in the system, and one of the most versatile.
027 Trichord — Quartal
Perfect fourths stacked. Frequency ratios close to simple whole numbers. Stable, open, unresolved but not tense. The sound of modern jazz sus chords.
016 Trichord — Tritone-bearing
Semitone plus tritone. The tritone’s irrational ratio creates persistent beating. Maximum harmonic charge. Works over dominant, altered, and chromatic contexts.
Because the tritone divides the octave exactly in half — 6 semitones out of 12 — it has a unique symmetrical property: C to F# is the same distance as F# to C. Every tritone has exactly one tritone partner. This symmetry means the 016 trichord maps onto itself under certain transformations, giving it an almost obsessive quality — it resists resolution not just acoustically but structurally.
Pairing Two 016 Trichords
When Bruce Arnold pairs two 016 trichords together — the 016-016 hexatonic system — the result is a six-note scale carrying two tritones simultaneously. Both are beating. Neither resolves. The sound is dense with harmonic energy yet strangely self-contained, hovering in a state of maximum tension that somehow doesn’t collapse.
This is counterintuitive. You might expect two tritones to produce chaos — and in the wrong hands they do. But the specific interval relationships between the two trichords create a kind of equilibrium of instability. The beating frequencies partially cancel and partially reinforce each other in a way that the ear accepts as a coherent sound world rather than noise.
The physical analogy is a coupled oscillator system — two pendulums connected by a spring, each trying to find its own period, settling into a shared motion that is neither one nor the other but something new. The 016-016 hexatonic is the acoustic equivalent.
Hear It — Compositions Built on the 016 Trichord
Both pieces are built on the 016-016 hexatonic system. Listen for the hovering, unresolved tension — the sound of two tritones in equilibrium.
The physics and the music, side by side
Irrational ratio
The tritone’s √2:1 frequency ratio means its standing waves never fully align. Physicists call the resulting amplitude variation “beating.” The ear calls it tension.
Tritone symmetry
The tritone divides the octave exactly in half. Every tritone maps to exactly one partner. It is the only interval that is its own inversion — a fixed point under transposition by six semitones.
Coupled oscillators
Two 016 trichords together create a system where two tritones interact. Like coupled pendulums, they find a shared equilibrium of motion — neither resolving nor collapsing, but hovering in organized tension.
Register matters
The same tritone interval beats at different rates depending on register. Lower pitches beat more slowly; higher pitches beat faster and sound harsher. Composers use register to control the perceived urgency of the tension.
Why This Matters for Improvisation
Understanding the physics of the tritone doesn’t just explain why it sounds dissonant — it explains exactly what kind of dissonance it is and how to use it. The beating is predictable. The symmetry is exploitable. The tension is controllable.
When an improviser learns to hear the 016 trichord not as “wrong notes” but as a coherent physical system with its own internal logic, a whole world of harmonic color opens up. The tritone stops being an interval to avoid and becomes a tool — one that happens to be built on one of the most interesting numbers in mathematics.
The next article in this series looks at the 027 trichord — the quartal opposite of the 016 — and what happens when you combine them into a single hexatonic scale that balances stability and tension in a ratio as precise as any in physics.
Smoke and A Cry by Bruce Arnold, from the Sound Cells Pitch Class Set Improvisation series, available at muse-eek.com. The acoustic physics of consonance and dissonance is covered in Helmholtz, On the Sensations of Tone (1877) and Sethares, Tuning, Timbre, Spectrum, Scale (1997).