What you are hearing
Change ringing is the English art of ringing a tower of tuned bells through every possible order, never repeating a row, never resting. A swinging half-tonne bell is too heavy to place where you like; it can only ever be made to swap with the bell beside it. So a "change" is one or more adjacent swaps, and a piece of ringing is a path through the orderings where each step moves bells past their immediate neighbours and no further.
That constraint — orderings of n things, joined when they differ by an adjacent swap — is exactly the Cayley graph of the symmetric group Sn on its adjacent transpositions. The ringers were walking it, by hand, in stone towers, three centuries before anyone wrote the mathematics down. This instrument lets you walk it too, and hear where you are.
Ringing the bells is traversing the symmetric group. Every order you hear is a permutation; every change you hear is a generator; an extent is a Hamiltonian cycle, struck in bronze.
The four things it rings
Rounds is the bells in descending order, 1 2 3 … — the scale, struck over and over. It is where every piece starts and ends. (It is also the fastest way to make a true sound: tap Ring.)
Plain hunt is the simplest possible motion: every bell, every change, moves one place toward the back, then turns and moves one place toward the front, weaving past the others. It comes home to rounds after exactly 2n changes — and so it only ever touches 2n of the n! orders. On three bells that already happens to be all of them; on four or more it is a tiny closed loop, and the whole craft is the search for something longer.
Plain Bob is the oldest and most-rung answer, and the one a real band will ring tonight. It is plain hunt with a single edit at each lead end — instead of the change that would send it straight home, one pair of bells "makes a place" (stands still) while the rest hunt on, which tips the ring into a fresh course of orders it has not yet rung. String the courses together and you get a long, true sequence. On four bells the plain course of Plain Bob is the entire extent — all 24 orders, no calls needed; on more bells it is one course of many, and the full extent needs spoken calls ("bobs" and "singles") this instrument doesn't ring.
Full extent is the grail made literal: every one of the n! orders, each exactly once, back to rounds — generated here by the Steinhaus–Johnson–Trotter algorithm, which is move-for-move the ringers' own "plain changes" method (Knuth presents it under their name). On six bells that is 720 rows; on eight, 40,320 — a real peal of roughly seventeen hours, which you are welcome to start and, sensibly, to stop.
The mapping, and what is true
Every row is generated live, never stored, and checked in front of you. The panel under the console recomputes, for the sequence you are ringing: how many rows it has, how many are distinct, whether every change is a legal adjacent swap, and whether it closes back to rounds. Nothing about the ringing is asserted; it is verified as it plays. The same generators are proved exhaustively offline — Plain Bob's courses are shown to be true and to plain-hunt the treble, the extents to cover Sn exactly — in research/plain-changes-rung/verify.mjs (63/63 checks).
The sound is synthesised, not recorded, and it is built to be the bell, not to imitate one. Each strike is five sine partials at the ratios a well-tuned English bell is cast to carry — hum · prime · tierce · quint · nominal ≈ 0.5 : 1 : 1.2 : 1.5 : 2 — with a fast attack and a long exponential decay. The tierce sits a minor third above the prime: that is the real, physical reason bells carry a faintly mournful colour. A real bell has many more, inharmonic partials and a far richer decay; this is an honest sketch of a bell's spectrum, labelled as one (after the true-harmonic scheme associated with Canon A. B. Simpson, 1890s).
The one thing here that is a choice, not a fact: the bells are tuned to a descending major scale from the treble. Real towers are tuned to a scale too, but which one varies; I picked a clean major scale so the orders are easy to hear apart. The pitches are an aesthetic decision; the orders in which they sound are the mathematics, and those are exact.
State lives in the URL: the bells, the method and the tempo are in the link, so Share this ring hands someone the exact thing you are hearing. No accounts, no server, no tracking — the bells ring entirely in your browser.
Sources
- D. E. Knuth, The Art of Computer Programming, Vol. 4A, §7.2.1.2 — presents the permutation algorithm as "Algorithm P: Plain changes," crediting English change ringing.
- Steinhaus–Johnson–Trotter (1962–63) — the adjacent-transposition Gray code for permutations; the ringers' plain-changes procedure, rediscovered.
- F. Duckworth & R. Stedman, Tintinnalogia (1668) — plain changes and Plain Bob in print; the practice is older still (c. 1621).
- A. T. White, "Ringing the cosets" / "Fabian Stedman: the first group theorist?" — the modern account of ringing as Hamiltonian cycles on Cayley graphs of Sn.
- A. B. Simpson, "On Bell Tones" (1895–96) — the five true-harmonic partials used for the synthesis.
This is the first exhibit in the Wasteland's instrument room — hearable mathematics, where every sound is the object. It is the playable companion to the stratum Plain Changes, which sets out the history and the proofs in full.