A ring of minor bells rings a second scale a major third below

The raised tierce points the bell at its own note found a second pitch inside the ordinary church bell. The strike note — the pitch a listener names a bell by — comes from the upper modes, the nominal and the two above it, which stand in the proportions 2 : 3 : 4 over the prime. The four lowest modes, hum, prime, tierce and quint, sit at 1, 2, 2.4 and 3 times the hum in a bell with a minor tierce, the kind European founders have cast for centuries; and read as a residue, those four are the fifth, tenth, twelfth and fifteenth harmonics of a note a major third below the strike note’s name. A minor bell’s lower spectrum is a minor triad, and like any minor triad it can be heard as resting on the note a major third below its root. With a major tierce the four modes fit a series on the strike note’s own name, and the second candidate disappears.

That essay counted the second candidate bell by bell. It ended on what happens when bells sound together, as they do in the one setting where minor-tierce bells are heard most: a ring of eight, tuned to a major scale and sounded in sequence. Each bell’s second pitch is a note a major third below its strike note, and that note is sometimes in the scale the ring rings and sometimes not. The lead counted which: below the tonic it is the flattened sixth, below the mediant the tonic itself, below the dominant the mediant. The calculation would run the masked-threshold count across a ring at the strokes ringers use and ask how many of the second pitches fall on the ring’s scale.

The count overturns one of the lead’s three examples, and the correction changes the answer.

The arithmetic of a third below #

The interval is fixed: a major third, four semitones, below each strike note. For a ring in C major, whose strike notes are C, D, E, F, G, A, B and C, the second pitches are A♭, B♭, C, D♭, E♭, F, G and A♭.

The lead was right about the tonic and the mediant and wrong about the dominant. The note a major third below G is E♭, not E; E is a minor third below G. So the dominant bell’s second pitch is the flattened third, a note a major scale does not contain. Counted correctly, only three of the eight bells imply a note the ring rings: the mediant, whose second pitch is the tonic; the submediant, whose second pitch is the subdominant; and the leading note, whose second pitch is the dominant. The other five — tonic, supertonic, subdominant, dominant and octave — imply A♭, B♭, D♭, E♭ and A♭ again.

The set of second pitches has a shape that is plain once it is drawn. Moving every note of a major scale down a major third gives another major scale, so the eight second pitches are A♭, B♭, C, D♭, E♭, F, G and A♭ — the scale of A♭ major. A ring of minor-tierce bells in C carries, in its lower spectra, a second ring in A♭, and the two scales share exactly three notes, C, F and G. Played as rounds, the buttons make the relation audible: the second sequence is the first transposed down a third, a descending A♭ major scale.

The lead’s slip is worth a sentence because it is the easy one to make. The second pitch is a major third below the strike note, and the strike note of the dominant bell is G. The mediant is a third below the dominant only in the sense of scale steps; in semitones it is three, not four. A minor bell does not point at the note a scale step below its third; it points at the note four semitones below its name.

How often the ring delivers it #

What a listener could hear depends on whether each bell delivers enough of its lower spectrum, stroke by stroke, for the second pitch to be implied at all.

For a ring whose tenor’s hum is C3 — a tenor striking C4, a light ring of eight — the figure is almost entirely the second pitch. Of the 88 bell-strokes from 40 to 90 decibels, 85 imply the note a major third below the strike: 33 on the ring’s scale, from the three bells whose second pitches are ring notes, and 52 off it. The three that do not are the tenor’s two hardest strokes and the second bell’s hardest, where the hum masks the tierce and the lower modes fall back to hum, prime and quint, which point at the strike’s own name.

Move the dial and the register changes the count without changing the proportions. A ring a fourth lower, its tenor’s hum on G2, delivers the second pitch at 76 of 88 strokes, 31 on the scale and 45 off; a ring with its tenor’s hum on F3 delivers it at every stroke, 33 on and 55 off. The bells that point at notes the ring never rings outnumber the ones that point at its own notes in every register, because there are five of them to three.

The register decides how much is delivered #

Across registers the share of strokes that carry a second pitch climbs steadily. A heavy ring whose tenor’s hum is on C2 delivers it at 47 of its 88 strokes, 22 on the scale and 25 off; the other 40 point at the strike note’s name, because a low bell’s tierce falls into the region where the hum and prime mask it at all but the softest strokes, as the earlier essays found bell by bell. By E2 the second pitch is in 66 strokes, by A2 in 81, by C3 in 85, and from F3 up in all 88. The heavier the ring, the less of its lower spectrum gets through, and the less often a second pitch is there to be heard.

That runs against the ordinary picture of a ring, in which the heavy bells are the ones with depth and resonance and the light ones are bright. On this count it is the light ring whose lower spectra are most complete, and so the light ring that most consistently carries a second scale under its first. A bell’s tierce does not arrive late, it leaves early found the tierce the first mode to fall out of hearing as a bell rings on, so all of this is a statement about the strike and the first second or two; in fast change ringing, where each bell sounds every couple of seconds, that is most of what is heard.

Where the foreign notes fall #

A note the ring never rings is foreign only to the scale. Where it sits against the notes the ring does ring decides whether it could sound like a wrong note or like nothing at all.

Every off-scale second pitch lies a semitone from two of the ring’s notes. A♭, the tonic’s and the octave’s, is between G and A; B♭, the supertonic’s, between A and B; D♭, the subdominant’s, between C and D; E♭, the dominant’s, between D and E. That is a property of the major scale rather than of bells: every chromatic note lies a semitone from two diatonic ones. But it puts each foreign second pitch in the closest possible relation to notes that are sounding a moment before or after it in rounds.

The dominant bell’s case is the sharpest. Its lower modes, on G, are a minor triad, G, B♭ and D, and they imply E♭. In rounds the dominant is followed, two bells later, by the mediant, E. A listener who heard the dominant’s second pitch would hear an E♭ a moment before an E, which is the kind of clash the chord is still major found a major chord surviving only when the clashing note was much weaker than the chord’s own.

Three in seven is the diatonic set’s own number #

The three-in-eight result looks like a fact about one ring, but most of it is a fact about the scale. A bell’s second pitch lands on the ring’s scale exactly when the scale contains the major third from that note down to the bell’s degree, so the count of bells that agree with the ring is the count of major thirds the scale contains.

Every one of the seven diatonic modes contains exactly three major thirds, so a ring tuned to any of them — major, natural minor, Mixolydian, Lydian — has three bells whose second pitches are ring notes and four whose second pitches are not. The count is the third entry of the diatonic set’s interval vector, the census the same seven, started later and its neighbours used to show that rotating a collection changes the home and not the contents. The harmonic minor, with its augmented second, has four major thirds and gets four of seven; the major pentatonic has one and gets one of five.

Only two scales get every bell to agree, and they are the two that are closed under a major third: the whole-tone scale, in which every note has another a major third below it, and the augmented scale, built of two augmented triads. A ring of minor-tierce bells tuned to either would carry, in its lower spectra, a copy of its own scale, every second pitch another bell’s strike note. Those are scales no ring is tuned to. The arithmetic says that the minor tierce and the diatonic scale are mismatched by construction — the bell’s second pitch moves by an interval the diatonic set contains only three times — and that the mismatch would disappear only on scales of the kind the set with fewer modes than notes found to be built from repeating blocks.

A whole-spectrum listener hears none of it #

The earlier essay set beside the residue fit a second reading, which weights every mode in the bell by how far it stands above its masked threshold and sums the evidence for each candidate pitch over the whole spectrum. Run across the ring, it gives the same verdict for every bell: the strongest candidate is the strike note’s own pitch class at all eleven strokes, and the note a major third below reaches 28 or 29 per cent of it. On the scale or off it, light bell or heavy, a listener who integrates the whole spectrum hears eight bells, each called by its own name.

So the ring has the same two readings as the single bell, and the disagreement between them is now a disagreement about whether a ring carries a second scale. If a listener completes the lower modes as a group — hearing the bottom of each bell as a chord, as the minor-triad reading supposes — a light ring of minor bells sounds its major scale with a scale in the key a major third below running under it, five notes in eight foreign. If a listener hears each bell whole, weighted by audibility, there is no second scale. Neither reading gives a major-tierce ring a second scale, and neither gives any ring a second scale that agrees with its first.

What a founder chose without choosing #

The minor tierce is what bell-founders cast for centuries before anyone could measure modes, and rings are tuned to major scales because change ringing developed on them. Neither choice was made with the other in mind. Put together, they make an object whose strike notes spell one key and whose lower spectra, read one way, spell another. A bell is never minor alone found a single bell’s minor character depending on what else is sounding; the ring is the case where what else is sounding is seven more minor bells, each pulling its own second pitch in the same direction.

One foreign note is doubled. The tenor and the treble both strike the ring’s tonic, an octave apart, and both imply A♭, so the flattened sixth is the only second pitch the ring sounds twice in every round — once from the heaviest bell, once from the lightest — and in the light rings where the second pitch is delivered at nearly every stroke it is the note a ring of minor bells in C insists on most. It is also the note that turns the ring’s own tonic triad, C–E–G, into the bottom of an A♭ major seventh chord, which is a different kind of instability from the clash of semitones. A bell has no fundamental began from the observation that the pitch a bell is named by is supplied by the listener; the ring is where that supplying could go two ways at once, and the way it goes for the lower modes is towards a key the ringers never chose.

It also gives an unexpected job to the major-third bells some founders have cast experimentally. The raised tierce points the bell at its own note found the major tierce removing the competitor in a single bell; across a ring, the same change removes a whole second scale, five of whose eight notes are foreign. Whether listeners found major-third bells cleaner for that reason, or only different, is not something the residue count can say.

The models under the ring #

Founders’ targets, exactly. Each bell’s modes sit at the ratios the earlier essays used, with a minor tierce at 1.2 times the prime, and every bell in the ring is a scaled copy of every other. Real rings are tuned only on the strike note and, in modern practice, on a few lower partials; the tierce of a real ring varies from bell to bell by tens of cents, and the residue fit’s thirty-cent tolerance would lose the second pitch from some of them.

The masked-threshold count and the residue fit are the earlier essays’, at the strike, with no decay. The strokes from 40 to 90 decibels stand for what ringers produce; ringers handle a ring with more or less weight, and the tenor is usually struck harder than the treble, which the figures do not model.

Bells heard one at a time. In rounds each bell sounds against the decaying sound of the ones before it, and the masking of one bell’s tierce by the previous bell’s hum would reduce the second pitch further. One sound hides another set out that count for two tones; across a ring it is a computation of its own.

What the arithmetic cannot settle #

Whether ringers or listeners hear the second scale. Bell listeners report secondary pitches and the occasional bell that sounds out of key despite a correct strike note. Whether those reports track the second pitches counted here — whether, say, the dominant bell of a minor-tierce ring is the one most often heard as “flat” — is a question a listening test could answer and a residue count cannot.

What the hum adds. The hum note is heard separately by many listeners, an octave below the strike, and it lies on the ring’s own scale. A listener who attends to the hum hears the ring’s scale twice and the second scale not at all, which is a third reading the figures do not take.

Still open: the second scale through the room #

Every bell here is heard alone at its strike. A ring is heard in a tower and across a churchyard, and the tierce is the most fragile mode in each bell: the root an ear supplies is a reminder that a residue needs its partials. At a distance, high modes fall away faster than low ones in air, and the room’s reverberation sustains the hum after the tierce has gone. The computation is this one with each bell’s lower modes carried through a tower’s reverberation and the distance to a listener in the churchyard, and it would say whether the second scale is a property of the ringing chamber, where it is strongest, or survives to the listeners the ringing is for.