Beating — the series · Acoustics & Music Theory

220 Hz against 223 Hz. Two tones 3 hertz apart, added. The rapid oscillation is their average; the slow swelling is their difference, heard as 3 beats a second and used by every tuner who has ever worked by ear.

Beats are arithmetic that anybody can hear

Two tones a few hertz apart swell and fade at exactly their difference. It is the most direct evidence available that the ear does sums on what reaches it, and it is how every instrument in the world gets tuned.

part 1 · intervals

Beat rates for laying equal temperament. Each link in the chain of fifths, inside one bearing octave, with the rate at which its coincident partials beat — the third partial of the lower note against the second of the upper for a fifth, the fourth against the third for a fourth. Rates run from 0.59 to 1.12 beats per second, a spread of 0.52. This temperament narrows every fifth equally, and the rates still differ, because a beat rate is a difference in hertz and scales with the register.

A tuner counts beats, and that is the whole method

Laying a temperament sounds like the most subjective job in music, and it is arithmetic. Every interval in the bearing octave has a target rate in beats per second, the rate follows from the ratio and the register, and a tuner who hits the numbers has produced equal temperament without ever thinking about a cent.

part 2 · tuning

A string mode swept through a body resonance at 460 Hz. What the string plays against what comes out. Away from the resonance the two are the same and the line is the diagonal. Near it the mode splits into a pair, and the note warbles at the difference between them — 12.9 Hz at the centre, which is slow enough to be counted and far too fast to be a tremolo. The splitting is a coupled oscillator and has nothing to do with the wolf fifth of a tuning system, which is a twenty-three cent arithmetic residue and shares only the word.

The other wolf

A cellist's wolf note is a string mode landing on a body resonance, at which point the two stop being separable and start exchanging energy — the mode splits in two and the note warbles at the difference. It is a coupled oscillator. The tuning system's wolf is twelve fifths failing to close by 23.5 cents. They share a word and nothing else.

part 3 · timbre

A perfectly tuned unison is the shortest note on the piano. How long a struck unison takes to fall forty decibels, against how far apart its two strings are tuned, for a note whose single string would ring for 20 seconds and whose bridge takes the in-phase mode down in 1.5. At a perfect unison the hammer excites only the mode that drives the bridge, so the note dies in 1.00 seconds. The longest note is at 2.30 cents, at 3.05 seconds — three times as long. Past 4.48 cents the two modes stop sharing a frequency and start sharing a decay, the note beats instead of ringing, and the sustain collapses.

Three strings, and the note that comes back

The usual account of a beat adds two independent sources together. A piano's strings are joined at the bridge and are not independent: the pair has normal modes, one of which drives the bridge and dies in a second and a half while the other cancels at the bridge and rings for twenty. A hammer striking a perfect unison excites only the first — so a perfectly tuned unison is the shortest note on the instrument, dying in one second, and the longest is at a deliberate detuning of about two and a half cents, at three. Past four and a half cents the two modes swap what they share and the sustain collapses.

part 4 · instruments

The pair tuned apart on purpose

A piano's unison buys its sustain with a detuning of two and a half cents and loses it past four and a half. Read from the other side that is a design constraint on every instrument that wants the beating instead: past the bifurcation the sustain is gone and no further detuning costs anything more, which is why every chorused voice in the world — the celeste rank, the musette reeds, the paired gamelan — is built out of sources that do not share a bridge.

part 5 · tuning

One pair, 12 beat rates. Two notes at 220 hertz, 15 cents apart, drawn as the beat rate between each pair of partials. The fundamentals beat at 1.91 hertz and the k-th partials at k times that, so the series of rates is a straight line and it crosses 15 hertz at the 8th partial. Below the line the fluctuation is counted; above it the same physical fluctuation is heard as roughness. 1 per cent of this spectrum's energy is on the roughness side. The bars are drawn at each partial's own amplitude, so a spectrum with little energy high up crosses the line where nothing is listening.

Every partial beats at its own rate

Five earlier essays have drawn one beat rate per figure, and every one of them is the rate between two fundamentals. Two real notes beat between all of their partials at once, the k-th pair beats k times as fast, and somewhere up the spectrum the rate passes the point at which a beat stops being a beat — so a chorused note is a beat at the bottom of itself and a roughness at the top, simultaneously, with a crossover partial that is arithmetic.

part 6 · tuning

Which interval gives a tuner the deepest null. A tuner listening to an interval p:q is listening to the lower note's p-th partial against the upper note's q-th, and how deep the beat's trough goes is decided by those two amplitudes rather than by the interval. For a string spectrum, whose partials fall as one over n, the minor third pairs partial 6 against partial 5 at a ratio of 1.18 for a dip of 21.8 decibels; the major third pairs partial 5 against partial 4 at a ratio of 1.25 for a dip of 19.1 decibels; the fourth pairs partial 4 against partial 3 at a ratio of 1.32 for a dip of 17.2 decibels; the fifth pairs partial 3 against partial 2 at a ratio of 1.52 for a dip of 13.8 decibels; the major sixth pairs partial 5 against partial 3 at a ratio of 1.65 for a dip of 12.2 decibels; the minor sixth pairs partial 8 against partial 5 at a ratio of 1.67 for a dip of 12.0 decibels; the octave pairs partial 2 against partial 1 at a ratio of 2.00 for a dip of 9.5 decibels. The best is the minor third at 21.8 and the worst is the octave at 9.5, which is the reverse of the order a tuner is usually taught to trust: the deepest null in the list is on the interval whose coincidence sits highest in the spectrum, where adjacent partials are nearly equal in strength.

A beat has a depth, and six essays held it at one

Every beat figure so far adds two tones of equal amplitude, which is the single ratio at which the trough of a beat is a true null — and a null is what a tuner is actually listening for. Vary the ratio and the picture changes: at two to one the dip is nine and a half decibels, at ten to one it is under two, and the interval that gives the shallowest null of all is the octave.

part 7 · intervals

How far out of tune an interval has to be before its beat is usable. An earlier essay produced a modulation index for every interval on a stated timbre; whether a fluctuation of that index at that rate can be detected is a published function of both. Running one against the other turns a table of decibels into a window in cents. The bar is the mistuning over which the beat is both deep enough to notice and at a rate a tuner can use — not so slow that a beat takes half a minute to complete, not so fast that it has stopped being a beat. the major third gives the widest window, 1.3 to 60.0 cents, and the minor sixth the narrowest, 0.8 to 38.9. The ordering is the opposite of the dip's: the octave has the shallowest dip on a string spectrum and the widest usable window, because its coincidence sits at a low partial and a given mistuning therefore produces a slower beat. Depth and rate pull opposite ways, and it is the rate that decides.

The beat a tuner can actually use

There is a modulation index for every interval on every timbre, and the published threshold for detecting a fluctuation is a function of exactly that and its rate. Running one against the other turns a table of decibels into a window in cents — and reverses the ordering, because the interval with the shallowest dip has the widest window.

part 8 · intervals

A mistuned 2:1 makes 8 beats, not one. Every pair of partials that coincides in a 2:1 mistuned by 6 cents, with the rate each one beats at. On a perfectly flexible string the rates are 1.5, 3.1, 4.6, 6.1 and so on — an exact harmonic series of the slowest, because partial 2k is exactly twice partial k. On a stiff one they are 1.3, 0.9, 2.7, 11.1, and the 8th member is at 122 hertz. The family grows as the cube of the member number rather than linearly, so 4 of the 8 are slow enough to be beats at all and the rest are roughness.

A beat is never one beat

Every beat counted until now has come from one pair of partials. A real spectrum has many, so a mistuned octave produces eight beats at once — and on a perfectly flexible string those eight are an exact harmonic series of the slowest. On a real piano string they are a cubic, half of them are not beats at all, and no single width of octave silences more than one. That is where the tuner's five octaves come from.

part 9 · intervals

Of 8 beats at A3, 3 can be attended to. Every member of the beat family a 2:1 mistuned by 6.0 cents makes on a real string at A3, placed by how separable its coincidence is from the partials beside it — in auditory filter widths, across — and by how fast it beats, up. The shaded region is the set a listener can receive: wider than one filter, and between 0.4 and 15 fluctuations a second. 4 of 8 members fall inside it, and once rates within a factor of 2 are counted as one modulation channel there are 3. The members that fail do so for two different reasons: the low ones sit under the roughness ceiling but their coincidences are buried in a filter that holds three partials, and the high ones are resolved and far too fast.

Three beats at most, and only in the middle of the keyboard

A mistuned octave on a real piano makes eight beats at once, and a listener attending to one of them is doing something that has a threshold. Two thresholds, in fact — a rate and a place — and once both are applied the eight become four at A3, one at A1 and one at A5. Every interval a tuner sets goes to zero at both ends of the compass and peaks at eight countable beats in the octave the bearing is laid in.

part 10 · intervals

Every beat in a family is the same depth, and none is near the threshold. The 8 members of the beat family a octave mistuned by 6.0 cents makes at A3 on a real string, each placed at the rate it beats and at the modulation index a listener's filter delivers there. The rising curve is the published detection threshold for amplitude modulation, which is flat at 0.03 below about fifty fluctuations a second and rises above it. The flat dashed line at 0.667 is the depth the pair has in isolation, and it is the same for every member: on a spectrum falling as one over n, the k-th member pairs partials 2k and 1k, whose ratio is 2.00 whatever k is. What the filled points show is the smaller effect that does depend on the member — the partials on either side of the coincidence leak into the same filter, add level without adding fluctuation, and dilute the index from 0.662 to 0.405. Inside the countable rate window the narrowest margin over the threshold is a factor of 16.7, on member 4. The depth criterion removes nothing.

Every member of a beat family is the same depth

A mistuned octave's beats have been counted on their rate and their place, with the depth recorded as the thing left out and a prediction that the shallow upper members would take the count from four to two. The depth turns out not to fall at all: on any power-law spectrum every member of a family has exactly the modulation index its interval's own ratio gives, at every register, on every wire. The count does fall to two, and the thing that takes it there is the criterion that essay was already using.

part 11 · intervals

The channels a fluctuation is analysed into, and how wide they are. A modulation filterbank of quality factor 1, drawn every half octave, with two of the beats a mistuned octave at middle C produces marked at 0.89 and 1.26 a second. A channel of quality one has its half-power points at 0.618 and 1.618 of its centre, so two fluctuations closer than a factor of 1.618 never end up in different channels. The two marked rates differ by a factor of 1.42, which is inside that. They are one fluctuation and not two — which is the question an earlier essay asked and left open, answered by a bank rather than by a convention.

One fluctuation or two

Whether two members of a beat family are one thing or two was decided by asking whether their rates differ by a factor of two, and the factor was written down as a stand-in for a modulation filterbank nobody had run. Run, the bank gives 1.618 — the golden section, and not by accident, since a channel of quality one has its half-power points there. The stand-in was conservative rather than optimistic, and the recomputed counts do not change at a single register, because the criterion was never what was binding.

part 12 · tuning

How hard the note was struck

The auditory filter is not a fixed shape: its lower skirt shallows by about 38 per cent of its reference value every ten decibels, so a loud note is analysed through a wider filter than a quiet one. Every share computed so far was quoted at a moderate level, and a tuner does not strike moderately. Recomputed, the mean share of a mistuned octave's filter falls from 0.55 at forty decibels to 0.07 at seventy, and the count of separable beats goes from two to none — which is a prediction too strong to be right, and the way it fails is the useful part.

part 13 · tuning

A struck octave becomes countable a second after the strike, or never. The number of separable beats a mistuned octave on A3 delivers, second by second after both notes are struck at 80 decibels, with every partial dying at its own rate (a 12-second fundamental, losses rising as frequency to the power 0.7). Read with the filter broadened by the level of the whole note, which is how the level-dependent count was first computed, the count is zero at every instant: the partials fall below audibility before the filter has narrowed enough to separate them. Read with the filter broadened by the level inside itself, which is what the published parameterisation was fitted against, the count is 0 at the strike, reaches 2 at 1.0 s and falls to nothing at 3.3 s.

Counted in the decay, or not at all

A mistuned octave struck hard delivers no countable beat at the strike, and the reconciliation offered for that was that a tuner listens to the decay. Computed through a real decay it fails on its own terms: the partials fall silent before the filter has narrowed enough to separate them. It succeeds only when the filter is broadened by the level inside it, which is what the published parameterisation was fitted against — and then the window opens at a twelfth of the note's life and shuts at a quarter.

part 14 · tuning

A string that decays twice opens its count at once and shuts it early. The count of separable beats a mistuned octave on A3 delivers after both notes are struck at 80 decibels, with each filter read at the level inside it, under one exponential decay of 12 seconds and under two stages — a prompt sound of 1.5 seconds carrying all but the last 20 decibels, and an aftersound of 12 seconds. One exponential: open from 1.00 s to 3.30 s, 11.8 beats. Two stages: open from 0.15 s to 1.77 s, 8.8 beats — and the single exponential struck 20 decibels softer closes at 1.77 s.

A string that decays twice is counted early

A mistuned octave's beats were found countable only between a twelfth and a quarter of a note's life, on a note decaying once. A piano string decays twice, a fast prompt sound over a slow aftersound, and the prediction was that this would open the count sooner and close it later. It opens sooner — at a seventh of a second rather than a second — and closes exactly where a single decay struck twenty decibels softer closes, so at 80 dB it holds 8.8 beats instead of 11.8. The count now rises with the strike to 90 dB, and a tuner who strikes hard is right.

part 15 · tuning

A harder strike buys beats only if the drop does not deepen with it. The beats a mistuned octave on A3 delivers on a string that decays in two stages, against how hard both notes are struck, with the aftersound's drop below the strike 20 dB at 80 dB and changing by a stated number of decibels for each decibel of strike. -0.5 dB per dB: 70 dB 2.5, 75 dB 5.7, 80 dB 8.9, 85 dB 11.9, 90 dB 12.2, 95 dB 11.6, 100 dB 10.9; 0 dB per dB: 70 dB 4.7, 75 dB 6.9, 80 dB 8.9, 85 dB 11.0, 90 dB 12.3, 95 dB 12.2, 100 dB 11.8; +0.5 dB per dB: 70 dB 7.2, 75 dB 8.1, 80 dB 8.9, 85 dB 9.8, 90 dB 10.7, 95 dB 11.2, 100 dB 11.9; +1 dB per dB: 70 dB 9.6, 75 dB 9.2, 80 dB 8.9, 85 dB 8.4, 90 dB 8.1, 95 dB 7.8, 100 dB 7.5. Where the drop stays put, a harder strike keeps buying beats to about 90 dB; where it deepens by a decibel per decibel, every strike past 70 dB costs beats.

A firm touch buys beats until the aftersound sinks with it

A piano string decays twice, and a mistuned octave's countable beats were found to rise with how hard the note is struck — on the assumption that the aftersound always starts twenty decibels below the strike. It need not. The moment the count shuts is set by the level the aftersound starts at and nothing else, so a harder strike buys beats only if its aftersound does not sink with it. If the drop deepens by more than 0.85 decibels for each decibel of strike, the firm touch costs beats instead.

part 16 · intervals

How wide the unison is left decides how far down the aftersound starts. A two-string unison on A3, each string decaying in 20 seconds alone and the pair's bridge-driving mode in 1.5, struck and heard at the bridge, against how many cents apart its strings are tuned: the drop from the prompt sound to the aftersound, for a hammer that strikes both strings alike and for one that strikes the second 20 and 50 per cent softer. Even strike: 0.25 cents 65.2 dB, aftersound 19.9 s; 0.5 cents 53.1 dB, aftersound 19.5 s; 1 cent 41.0 dB, aftersound 18.0 s; 1.5 cents 33.7 dB, aftersound 16.0 s; 2 cents 28.4 dB, aftersound 13.8 s; 2.5 cents 24.2 dB, aftersound 11.6 s; 3 cents 20.5 dB, aftersound 9.7 s; 4 cents 13.8 dB, aftersound 6.5 s; 5 cents 6.3 dB, aftersound 4.0 s. The earlier count assumed 20 dB; the unison gives that at about three cents. At a perfect unison the aftersound is not excited at all, and past 5.33 cents the two modes split and the unison beats instead. A strike's force scales both modes alike and appears nowhere in the drop.

The octave is counted best over a unison left two and a half cents wide

A firm touch buys a mistuned octave's beats only if the aftersound's drop does not deepen with the strike, and the drop was left as a number to be measured. It has a mechanism, and the mechanism has been known since the essay on three strings: the unison's two coupled modes. Computed from them, the drop does not depend on the strike at all. It depends on how wide the tuner left the unison — sixty-five decibels at a quarter of a cent, twenty at three — and the octave's beats peak, at 6.8, over the same two-and-a-half-cent unison that makes the longest note.

part 17 · intervals

Where the tilt of the blow matters, and where it does not. The octave's beats at 80 dB against the tilt of the blow from the vertical: one string, others muted, 0° 2.81, 1° 2.81, 2° 2.81, 3° 2.81, 5° 2.81, 8° 2.87, 12° 3.19, 16° 3.62, 20° 4.27; a unison half a cent wide, 0° 1.51, 1° 1.51, 2° 1.51, 3° 1.51, 5° 1.57, 8° 1.95, 12° 2.60, 16° 4.00, 20° 4.87; a unison 2.5 cents wide, 0° 6.81, 1° 6.81, 2° 6.81, 3° 6.81, 5° 6.87, 8° 6.87, 12° 6.70, 16° 6.97, 20° 7.25. One string alone and a clean unison both get their aftersound from the second plane and both gain beats as the blow tilts. The wide unison already has an aftersound twenty-odd decibels down from its own detuning, and the second plane, at forty, adds nothing it can hear.

The plane of the blow matters only where the unison is clean

A piano string vibrates in two planes, and the one parallel to the soundboard gives even a single string a second, slower decay. A hammer tilted off the vertical feeds it. That was the last way left for a firm touch to cost a tuner beats: if harder blows came down straighter. They would have to straighten by half for every seven decibels, and even then only on a single string or a unison tuned clean. Over a unison left two and a half cents wide, no tilt of the blow reaches the count.

part 18 · intervals

A stiff string slows the fourths and quickens the fifths

Every rate in a bearing plan is computed from whole-number partials, and a piano string's partials are not whole numbers. The correction is usually summarised as 'every real beat is faster than the table'. It is not: stiffness quickens the beat of every narrow interval and slows the beat of every wide one, and on a small upright the tempered fourth stops beating altogether. Both errors push the same way, so a tuner who counts the printed table leaves every interval wide, and a fifth and a fourth counted that way make an octave stretched to the 4:2 type or a little past it.

part 19 · intervals

The fourth below a note and the fifth above it differ by the same amount wherever the note is put. The fourth A3–D4 and the fifth D4–A4 on the grand modelled here, each note with its own string's stiffness, with the octave 1.9 cents wider than 2:1 in its fundamentals: the beat rate of each, signed so that a wide fourth and a narrow fifth are both positive, as the middle note is moved three cents either side of its tempered place. Both lines move together, and the fourth always beats 0.15 a second faster than the fifth — the difference between the upper note's second partial and the lower note's fourth, which the middle note does not touch. The two beat alike only when the octave is 1.60 cents wide, the octave whose 4:2 member is silent.

The fourth–fifth check cannot hear the note it checks

A tuner tests a temperament by sounding the fourth below a note and the fifth above it, inside one octave, and comparing their beats. Both beats run to the same partial of the middle note, so moving that note speeds or slows both together and leaves their difference untouched. The check is a test of the octave, and of one particular octave: the 4:2. Measured with each note's own stiffness, the printed table laid fourth first builds exactly that octave on any piano; laid fifth first it overshoots, and the check pulls it back by a third of a cent.

part 20 · intervals

Each family of checks builds its own octave, and the thirds build one twice as wide as the fourths and fifths. On each lower note from F3 to F4, on the grand modelled here: how many cents wider than 2:1 the octave lands when it is filled by a chain of intervals each counted to the rate a bearing plan prints — a fourth then a fifth, three major thirds, a major sixth then a minor third, four minor thirds — against the 4:2, 6:3 and 8:4 octave types, each note with its own stiffness. At A3: fourth, then fifth 1.60, three major thirds 3.20, minor third, then major sixth 3.57, four minor thirds 8.09; types 1.60, 3.56 and 6.22. The fourth and fifth build the 4:2 octave exactly and the minor third and major sixth the 6:3 within a few hundredths of a cent; three major thirds build an octave 2.00 times as wide as the fourth and fifth's, just short of the 6:3.

The thirds and the fourths check different octaves

Every interval counted to its printed beat rate on a stiff string is left wide, by an amount set by the squares of the partial numbers that meet in it. Add the links of a chain and the sum lands on an octave type: a fourth and a fifth make the 4:2, a minor third and a major sixth make the 6:3, and three major thirds come just short of the 6:3. So the two most familiar checks in a bearing are exact tests of two different octaves, and on a stiff piano no octave passes both. Each fails on the other's octave by a beat a second at A3.

part 21 · intervals

The octave that fails both checks least

A tuner who cannot pass both the fourth–fifth check and the third–sixth check settles on an octave that fails them equally, and a listener compares beat rates by ratio. Weighed that way, the compromise sits a ninth of the way from the 4:2 octave to the 6:3 on every note of every piano, because the fourth beats twelve times slower than the minor third and that ratio belongs to equal temperament, not to the strings. On a grand both checks then pass to a fifteen-per-cent ear up to middle C, and above it no octave passes both.

part 22 · intervals

The progression of thirds votes for the bearing, not the octave

Added as a third vote beside the two contiguous checks, the progression of thirds was expected to push a stiff piano's octave wider. It pulls it narrower, because the octave it asks for is not set by the strings at all: it is whatever stretch the bearing already carries, plus an eighth of a cent. In an equal bearing that drags the compromise back onto the 4:2 and costs the top of the bearing its pass; give the bearing a fifth of the octave's own stretch and the third vote falls silent.

part 23 · intervals

A bearing counted to the printed rates leaves its thirds beating in a zigzag, and which thirds zig depends on the plan. A bearing from F3 to E4 laid out from A3 on the modelled grand, each note set so that the fourth or fifth from a note already set beats at its printed rate: the true beat rate of each major third whose notes are both in the bearing, by two plans, against an equal bearing on the same strings and the printed table. counted: fourths up, fifths down: F3 6.46, F♯3 6.80, G3 8.41, A♭3 7.63, A3 9.45, B♭3 8.57, B3 9.01, C4 11.17; counted: the sharps by fifths up: F3 6.46, F♯3 7.99, G3 7.14, A♭3 8.97, A3 8.02, B♭3 8.57, B3 9.00, C4 9.46; equal temperament: F3 6.67, F♯3 7.04, G3 7.43, A♭3 7.84, A3 8.27, B♭3 8.71, B3 9.17, C4 9.66.

A bearing counted correctly is stretched by its route, not its strings

A tuner who sets a temperament octave by counting each fourth and fifth to its printed beat rate leaves every one of them a little wide on a stiff string, and the bearing ends up carrying a stretch nobody chose. Whether that stretch is enough to quiet the progression of thirds across the octave's edge turns out to be the wrong question. The stretch belongs to the order of the plan, not to the piano: one ordinary plan carries five times what the progression needs, another a fraction. And either plan leaves the thirds inside the bearing beating in a zigzag, faster then slower then faster, by up to eighteen per cent a step on a grand — which is why the thirds have to set the bearing rather than check it.

part 24 · intervals

Set by its own thirds, the bearing bends smoothly upward and carries five eighths of its octave's stretch. The bearing F3–E4 on the modelled grand, A3 at equal temperament: each note's cents from equal temperament when every note is solved at once so that each major third beats faster than the one below by exactly the printed step and the fourths and fifths come as near their printed rates as that allows, against the two plans counted by fourths and fifths. set by its thirds: F3 -0.19, F♯3 -0.17, G3 -0.13, A♭3 -0.07, A3 +0.00, B♭3 +0.07, B3 +0.16, C4 +0.27, C♯4 +0.40, D4 +0.53, E♭4 +0.69, E4 +0.88; stretch 1.15 cents an octave. counted: fourths up: F3 +0.43, F♯3 +1.39, G3 +0.23, A♭3 +1.17, A3 +0.00, B♭3 +0.92, B3 +1.93, C4 +0.82, C♯4 +1.82, D4 +0.71, E♭4 +1.69, E4 +2.78; stretch 1.62 cents an octave. counted: sharps by fifths: F3 +0.43, F♯3 -0.82, G3 +0.23, A♭3 -1.03, A3 +0.00, B♭3 +0.92, B3 -0.28, C4 +0.82, C♯4 -0.38, D4 +0.71, E♭4 -0.51, E4 +0.57; stretch 0.51 cents an octave. The octave's own stretch by the two contiguous checks is 1.82 cents.

A bearing set by its thirds carries five eighths of its octave

A bearing counted from the printed fourths and fifths leaves its thirds zigzagging round the printed progression and takes whatever stretch its route gives it. Set the other way — every note solved at once so that each major third beats faster than the one below by exactly the printed step — the bearing bends smoothly upward and carries 63 per cent of its octave's own stretch, on a grand, on an upright and on everything between. That is more than three times what the octave's three checks need in order to agree, so following the thirds reconciles them with room to spare. The price is paid by the fourths and fifths, every one of which then beats slow.

part 25 · intervals

The twelve fourths and fifths of a bearing are one closed circle, and on stiff strings they cannot all beat at their printed rates. A bearing from F3 to E4 holds seven fourths and five fifths that join all twelve notes in one closed circle: A3, E4, B3, F♯3, C♯4, A♭3, E♭4, B♭3, F3, C4, G3, D4 and back. On the modelled grand, counting all twelve at their printed rates misses closing by 2.22 cents. Shared equally, every interval beats 9.4 per cent slower than the table says, and the bearing carries 1.14 cents an octave.

A stiff string gives the bearing a comma of its own

The twelve fourths and fifths inside a piano's temperament octave are not twelve checks. Laid end to end they are one closed circle, and on stiff strings their printed rates fail to close it by 2.22 cents. Every disagreement between a bearing's tests follows from that one number: which interval never gets counted decides the stretch, and the zigzag in the thirds is the seam. Spread it round the circle and every fourth and fifth beats nine per cent slow of the table, and the thirds come right without being asked.

part 26 · intervals

One walk round the circle takes most of the error out of the interval nobody counted. One seeded tuner on the modelled grand, each count 5.0 per cent out at random, the closing unison read to 0.02 Hz. Each interval's true rate against the rate that closes the circle exactly. From the table alone the twelfth interval, which is never counted, is -133 per cent off; after one walk and a correction of -11.6 per cent it is 36 per cent off. Stretch 1.38 against 1.07 cents an octave.

One walk round the bearing measures the piano and the tuner together

A tuner who counts twelve fourths and fifths to the printed table and comes home flat has measured something. Divide the closing beat's rate by three and correct the whole table by that share, and the worst interval improves from a hundred per cent off to about twenty, on any piano. It keeps the table with one number. And the number cannot tell a stiff string from a slow count: a tuner counting nine per cent slow gets the right bearing from the wrong table, and one correction removes that luck along with the error.

part 27 · intervals