Abstract
It is easy to assume that an instrument body sounds low because it is large, or long. This paper counts that the pitch of a cavity with an opening is set by something other than the length that sets a pipe’s resonance. No new theorem or law is claimed.
First, a cavity with an opening sounds low while staying much smaller than the wavelength: a volume of $0.01$ m$^3$ and an opening of radius $0.05$ m give $\mathbf{163.086441}$ Hz, with a wavelength of $2.103179$ m (Section 2).
Second, and this is the core. With the same volume, different shapes sound the same pitch: a cube and a long box both give $163.086441$ Hz, while the half-wave pitch of the longest side is $796.032485$ Hz and $343.000000$ Hz, $\mathbf{2.320794}$ times apart (Section 2).
Third, a larger volume lowers it and a larger opening raises it: doubling the volume gives $0.707107$ times, doubling the opening radius $\mathbf{1.426423}$ times, and halving it $0.695353$ times (Section 3).
Fourth, the opening does not act through area alone: doubling the radius quadruples the area, but the neck length grows too, so the pitch rises only to $1.426423$ times, just above $\sqrt{2}$ (Section 3).
Fifth, the formula matches the moved system: moving the neck air as a mass on the cavity air as a spring and counting zero crossings gave $163.086529$ Hz, within $0.000088$ Hz of the formula (Section 2).
Sixth, the separator is whether one asks about the longest dimension or about the volume and the area of the opening: a pipe resonance watches length; a cavity resonance watches volume and opening (Section 4).
Keywords: Helmholtz resonance, resonating body, cavity, opening area, pipe resonance.
Scope (what is not claimed)
No new theorem or law is claimed: the Helmholtz resonance formula, the end correction of the neck by the opening radius, and that pipe resonances are set by length, are all known in acoustics. No real instruments are claimed: volume, opening and plate thickness are model values matching no guitar and no drum. Plate vibration is not included: coupling of the body’s plates to the cavity and absorption at the walls are not included. The range is limited: only cavities much smaller than the wavelength are treated; the end-correction factor $1.7$ is one representative choice. Relation to earlier papers: Paper 857 showed that pitch sees only the ratio of tension to weight, but the instrument bears the tension itself. This paper stands beside it. There, for a string, the ratio of tension to weight set the pitch. Here, for a cavity, the ratio of volume to opening sets the pitch, and shape does not matter. What is added is setting the cavity pitch beside the longest-side pitch for two shapes of equal volume, measuring pitch factors when volume and opening change, matching the formula against the moved neck and cavity, and placing the separator on longest dimension versus volume and opening.
1. Notation and the Question
Setting: a cavity of volume $V$ has an opening of radius $r$ in a plate of thickness $t$. The air in the opening (the neck) acts as a mass and the cavity air as a spring, with the neck length corrected by the opening radius.
$$f=\frac{c}{2\pi}\sqrt{\frac{A}{V\,L_{\text{eff}}}},\qquad A=\pi r^{2},\qquad L_{\text{eff}}=t+1.7\,r,\qquad f_{\text{pipe}}=\frac{c}{2L_{\max}}$$We took $c=343$ m/s and air density $1.2$ kg/m$^3$. $f_{\text{pipe}}$ is the pitch set by the half wave of the longest side $L_{\max}$. The cavity pitch was obtained from the formula and by moving the neck air as a mass on the cavity air as a spring in steps of $10^{-6}$ seconds and counting zero crossings.
Question: what sets the pitch at which a resonating body sounds?
2. Cavity Pitch and Length Pitch
Computation 2.1: for a cube and a long box of volume $0.01$ m$^3$, the cavity pitch and the longest-side pitch were computed (opening radius $0.05$ m, plate $0.003$ m).
| shape | dimensions (m) | cavity pitch (Hz) | longest-side half wave (Hz) |
|---|---|---|---|
| cube | $0.2154$ each side | $163.086441$ | $796.032485$ |
| long box | $0.5\times0.2\times0.1$ | $163.086441$ | $343.000000$ |
Observation 2.2 this is the core. With the same volume, different shapes give the same cavity pitch. The cube and the long box both gave $163.086441$ Hz, while the pitch set by the longest side is $796.032485$ Hz and $343.000000$ Hz, $2.320794$ times apart. The cavity resonance does not watch length; it watches only volume and opening.
Observation 2.3 the cavity pitch is low while the cavity stays much smaller than the wavelength. The wavelength at $163.086441$ Hz is $2.103179$ m, far longer than the cavity: moving the neck and cavity and counting zero crossings gave $163.086529$ Hz, $0.000088$ Hz from the formula.
3. Volume and Opening
Computation 3.1: volume and opening radius were changed and the pitch factors computed.
| change | pitch (Hz) | factor on original |
|---|---|---|
| volume $\times2$ | $115.319529$ | $0.707107$ |
| opening radius $\times2$ | $232.630223$ | $1.426423$ |
| opening radius $\times0.5$ | $113.402727$ | $0.695353$ |
Observation 3.2 a larger volume lowers it and a larger opening raises it. Doubling the volume gives exactly $1/\sqrt{2}$ times, and doubling the opening radius $1.426423$ times: doubling the radius quadruples the area, but the neck length corrected by the radius grows as well, so the pitch rises not to $2$ times but just above $\sqrt{2}$.
4. The Separator
Observation 4.1 the separator is whether one asks about the longest dimension or about the volume and the area of the opening. A pipe resonance is set by length, and changing shape changes its pitch: a cavity resonance compresses the whole air as one spring, so it is set by volume and opening, and changing shape leaves its pitch alone. A body sounds low not because it is long, but because its opening is small for its volume.
Observation 4.2 placed beside Paper 857. 857 showed that pitch sees only the ratio of tension to weight: for a string, that ratio set the pitch. Here, for a cavity, the ratio of opening area to volume and neck length sets it. In both, a ratio, not a conspicuous size, sets the pitch.
Observation 4.3 what is not said. Real instrument values, coupling to plate vibration, and wall absorption are not treated. What is shown is that a cavity’s pitch is set by volume and opening, whatever the shape.
5. Census and Closing
- $f=(c/2\pi)\sqrt{A/(VL_{\text{eff}})}$ (Section 1): Helmholtz resonance.
- $\mathbf{163.086441}$ Hz (Computation 2.1): volume $0.01$ m$^3$.
- wavelength $2.103179$ m (Observation 2.3): far longer than the cavity.
- same pitch for cube and long box (Observation 2.2): the core.
- longest-side pitch $\mathbf{2.320794}$ times apart (Observation 2.2): changes with shape.
- volume $\times2$: $0.707107$ (Computation 3.1): $1/\sqrt2$.
- opening radius $\times2$: $\mathbf{1.426423}$ (Computation 3.1): the neck grows too.
- opening radius $\times0.5$: $0.695353$ (Computation 3.1): lower.
- within $0.000088$ Hz of the moved system (Observation 2.3): independent.
- longest dimension, or volume and opening (Observation 4.1): the separator; Paper 857.
Closing: The pitch at which a resonating body sounds is set not by length, as in a pipe, but by the ratio of volume to opening, whatever the shape. A cavity of $0.01$ m$^3$ with an opening of radius $0.05$ m sounds at $163.086441$ Hz whether cube or long box, and while the longest-side pitch differs $2.320794$ times, the cavity pitch does not change, because the cavity air acts as one spring compressed as a whole, working through volume rather than length. Doubling the volume gives $0.707107$ times and doubling the opening radius $1.426423$ times, and the moved neck and cavity agree with the formula within $0.000088$ Hz: a body sounds low not because it is long, but because its opening is small for its volume. Placed among the earlier papers: beside Paper 857’s “pitch sees only the ratio of tension to weight”, we placed that a cavity’s pitch sees only the ratio of volume to opening. To be honest: dimensions are model values, plate vibration and absorption are not included, and only cavities smaller than the wavelength are treated.
References
- Fletcher, N. H., & Rossing, T. D. (1998). The Physics of Musical Instruments (2nd ed.). Springer.
- Kinsler, L. E., Frey, A. R., Coppens, A. B., & Sanders, J. V. (2000). Fundamentals of Acoustics (4th ed.). Wiley.
- Yamagishi, Y. (2026). Pitch Sees Only the Ratio of Tension to Weight, but the Instrument Bears the Tension Itself [Paper 857]. YOI: https://www.yamagishiyuuki.com/yoi/857
Reference
Yamagishi, Y. (2026). A Resonating Body’s Pitch Is Set Not by Length, as in a Pipe, but by Volume and Opening, Whatever the Shape [Paper 888]. YOI: https://www.yamagishiyuuki.com/yoi/888
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 888. This paper is a preprint and has not been peer reviewed.