Percussive Origins Concerning Inharmonic Spectra of the Drumset, Part XLVIII

Percussive Origins of Polytempic Polymicrotonality

The Inharmonic Spectra of Drums and the Microtonal Field Within

Peter Thoegersen

The compositional system of polytempic polymicrotonality originated in four-limb drumming independence beginning in 1985, arriving at the 3:4:5:7 polyrhythmic structure. What follows is a spectroacoustic analysis of the drum kit instruments used in that formative period — snare drum, bass drum, hi-hat, and ride cymbal — revealing that each instrument already embodied the centerless, inharmonic, multi-layered acoustic logic that the compositional system would later formalize.

The system did not emerge despite the physics of percussion. It emerged from it.

Why Percussion is Inharmonic

A string or pipe vibrates in integer multiples of the fundamental — 1, 2, 3, 4 — producing a clean harmonic series with a clear tonic. A cymbal or drum head is a two-dimensional membrane or plate, and its vibration modes follow a completely different mathematical series: not integer multiples, not reducible to a single fundamental. The partials are in irrational relationships to each other. The spectrum is dense, complex, and hierarchically centerless.

Drum heads vibrate according to Bessel function zeros — the mathematical series governing circular membrane physics. The resulting partial frequencies relative to the fundamental are as follows:

• 1.000 (fundamental)

• 1.594 → 802 cents — falls between the Pythagorean minor sixth (792¢), the 12-TET minor sixth (800¢),

• and the 5-limit 8/5 (814¢). Belongs to none of them.

• 2.136 → 1315 cents — a sharp major ninth; genuinely orphaned in standard JI.

• 2.296 → 1437 cents — near a flat major ninth

• 2.653 → 1692 cents

• 2.918 → 1852 cents

• 3.156 → 1993 cents — just under three octaves minus a semitone

• 3.501 → 2172 cents

• 3.600 → 2217 cents

None of these land on 12-TET. None lands cleanly on standard JI ratios. They are genuinely microtonal — not approximations of anything, but their own intervallic world derived from physical law. Every drum head has its own version of this field, shifted by tension and diameter.

The Four Instruments

Snare Drum

The snare drum head vibrates in circular membrane modes — irrational, not integer multiples, generating the Bessel field described above. But the snare wires add a second completely independent inharmonic layer: a dense rattling spectrum that has no relationship to the head's modes whatsoever.

Two independent inharmonic systems sound simultaneously. Tuning the head changes one system; adjusting snare tension changes the other independently. The snare drum is already a two-layer polytempic object. It already has no master fundamental.

Bass Drum

The bass drum contains three coupled systems: two heads (batter and resonant), each with its own Bessel-function modal frequencies, and the shell, which couples both into a third spectral layer. In practice, the inharmonic content decays quickly; what the ear registers is mostly attack transient and fundamental thud.

The bass drum is the least "pitched" of the four instruments in conventional terms, and the most physical. It is the instrument closest to a pure haptic event — felt before it is heard. Three independent microtonal fields, all physically bonded to the same object.

Hi-Hat

The hi-hat is the most behaviorally complex instrument in the kit. Two cymbals whose relationship to each other is continuously variable — open, half-open, closed, foot-closed — and each position produces a completely different spectrum.

Open: two independent inharmonic cymbal spectra interacting, producing beating and difference tones between their respective partials. Closed: the two plates choke each other; damping produces a tight, high transient with very short decay. The hi-hat is a continuously variable inharmonic system — a spectral morphology rather than a fixed timbre. The available microtones multiply combinatorially depending on physical position.

Ride Cymbal

The ride cymbal is among the richest examples of inharmonic spectra in acoustic music. Three spectrally distinct zones exist on a single object:

• Bell: modes are relatively more harmonic — closer to integer ratios — which is why the bell has a more defined pitch.

• Bow: deeply inharmonic free circular plate modes; the main ride tone here sits around 300–600 Hz with dense

• inharmonic partials above it that bear no integer relationship to that fundamental.

• Edge: produces the wash — high-frequency density where pitch identity dissolves entirely into timbre.

A ride cymbal is acoustically already polytempic polymicrotonal in miniature: multiple simultaneous inharmonic streams, no master fundamental, no hierarchical center.

The Synthesis: Body, Limb, Polygon, Pitch

What the drummer plays in a 3:4:5:7 polyrhythmic configuration is not merely a temporal structure. It is four independent streams, each driving a spectrally distinct inharmonic instrument, each limb with its own characteristic motion geometry:

• Right hand (ride cymbal, triangle → ratio 3): forceful, directional, sparse intervallic field

• Left hand (snare drum, square → ratio 4): two-layer inharmonic system, structural, bounded

• Right foot (bass drum, pentagon → ratio 5): three-coupled-system haptic event, predominantly physical

• Left foot (hi-hat, heptagon → ratio 7): continuously variable morphology, highest microtonal density

The number of polygon sides corresponds to the polyrhythmic ratio. The polygon perimeter traces the pitch space available to that limb. The vertices are the stable tones; the edges between them are the microtonal intervals. Wide angles (triangle) produce large intervals and sparse fields. Narrow angles (heptagon) produce microtonal density, close intervals.

The compositional system — polytempic polymicrotonality — is the body’s motion geometry made audible at the compositional scale. The drumming was not a metaphor for the system. It was the system’s first physical instantiation, operating in the spectra of the instruments before it ever reached notation.

Haptics: Closing the Circuit

Before airborne sound, there is touch. The drumstick hitting the head, the foot hitting the pedal — the haptic event is the origin point where rhythm and pitch are not yet separated. They are still one thing: impact.

The acoustic argument — that rhythm and pitch are the same phenomenon at different frequencies — has a physical substrate, and it is haptic. The percussive impact generates both the rhythmic event and the inharmonic spectral field simultaneously, from the same physical act. The body does not produce rhythm and then produce pitch. It produces one thing, which bifurcates into both as it propagates through space and time.

A wearable haptic system — four independent transducers, one per limb, each vibrating at the frequency its limb’s motion is generating — would close this circuit entirely. The performer would feel their own tuning system in real time. The 3:4:5:7 would become a bodily sensation distributed across four independent somatic channels, not merely an audible structure.

The system originated in the body. Haptics returns it there.

This document is a working theoretical note toward the Vitruvian Man blog post and potential inclusion in Book III: Methodologies and Historical Placement of the Polytempic Polymicrotonal Composition of Peter Thoegersen.