See the shape a sound makes, find its note, and hear how two notes fit together.
Your answer
Type a frequency, or tap 432 or 528, and the pattern sand would make on a round drum skin appears here.
An interval in three dimensions
For 3 : 2, the gold path goes 3 times round the ring while it goes 2 times round the tube, then meets itself exactly: a (3, 2) torus knot. Simple ratios close quickly; that is the same simplicity the ear hears as a smooth interval.
The ring's volume and surface are worked out on the Volume Calculator (Ring tab): 2 × π² × R × r² and 4 × π² × R × r.
The harmonic series
Dot 1 swings across its line once a loop, dot 2 twice, dot 3 three times, all the way up: the same 1, 2, 3 … steps as a string's harmonics. Halfway round, every dot is at one end or the other, two groups; a third of the way, three groups. Each simple fraction of the loop gathers them, which is why simple ratios are the ones that sound together.
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The working
A drum skin clamped at its rim can only ring in certain shapes. Each shape, or mode, is a Bessel function around the centre times a cosine around the rim: Jₙ(jₙ,ₘ × ρ) × cos(nθ), where ρ runs from 0 at the centre to 1 at the rim. The skin stands still wherever that is zero, and that is where the sand collects: n straight lines through the centre, and a ring wherever the Bessel function crosses zero before the rim.
Each mode rings at its own frequency, in proportion to jₙ,ₘ, the m-th zero of Jₙ. If the skin's lowest note is 100 Hz, that note is mode (0, 1) with j₀,₁ = 2.4048, and mode (1, 1) rings at 100 × 3.8317 ÷ 2.4048 = 159.3 Hz. The calculator finds the mode closest to the frequency you choose.
| Mode (n, m) | jₙ,ₘ | Frequency ÷ lowest | Pattern |
|---|---|---|---|
| (0, 1) | 2.4048 | 1.000 | The whole skin moves together |
| (1, 1) | 3.8317 | 1.593 | One line across |
| (2, 1) | 5.1356 | 2.136 | Two lines, a cross |
| (0, 2) | 5.5201 | 2.295 | One ring |
| (3, 1) | 6.3802 | 2.653 | Three lines, six slices |
| (1, 2) | 7.0156 | 2.917 | One line and one ring |
None of these is a whole-number multiple of the lowest note, which is why a drum sounds like a thud rather than a tune. A metal Chladni plate with a free edge follows a stiffer equation and draws different figures, but the idea is the same: sand marks the stillness.
Every octave doubles the frequency and holds 12 equal semitones of 100 cents each. A frequency is 12 × log₂(f ÷ 440) semitones from A4. An interval of ratio a : b is 1200 × log₂(a ÷ b) cents: a pure fifth, 3 : 2, is 701.96 cents; a piano's is 700. The equal-tempered tritone is exactly √2 : 1, 600 cents. The golden ratio, 1.618, is 833.09 cents, between a minor sixth (800) and a major sixth (900).
Questions
Cymatics is the study of patterns that sound makes visible. Sprinkle sand on a plate or drum skin and make it ring: the sand is thrown off the parts that move and gathers along the lines that stay still. Ernst Chladni published these figures in 1787; the Swiss physician Hans Jenny named the study cymatics in the 1960s.
A skin can only ring in certain ways, called modes, each with its own frequency. Higher modes split the skin into more pieces moving in opposite directions, so there are more still lines between them. On a round skin, mode (n, m) has n straight lines across and m − 1 rings inside the rim.
432 Hz is the A that some musicians prefer to the standard 440 Hz; 528 Hz is popular in sound-healing circles. There is no good evidence that either frequency has special effects on the body, but both make fine patterns. Here 432 Hz is A4 31.8 cents flat, and 528 Hz is C5 15.6 cents sharp.
Count semitones from A4: 12 × log₂(frequency ÷ 440). Round to the nearest whole number for the note, and the leftover × 100 is how many cents sharp or flat it is. 523.25 Hz is 3 semitones above A4, which is C5.
When two notes have frequencies in a simple ratio, such as 2 : 1 for an octave or 3 : 2 for a fifth, their waves line up again after only a few cycles. A piano tunes every semitone to the twelfth root of 2, so its fifth is 700 cents instead of the pure 701.96, close enough that few people notice.
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