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

A beat is never one beat ended with an admission rather than a result. A mistuned octave on a real piano makes eight beats at once, at rates spanning five hundred to one, and every rung below it — the arithmetic of two tones, the tuner counting seconds against a metronome, the coupled trio inside one note — assumes a listener attends to exactly one of them and ignores the rest.

That assumption has a threshold in it, and the threshold is measurable. Here is the measurement.

A beat is a fluctuation somewhere #

The reason there are two thresholds rather than one is that a beat has a location as well as a rate, and the ladder below has only ever priced the rate.

The $k$-th member of an octave’s family comes from partials $2k$ and $k$ of the two notes, and those two partials nearly coincide — that is what makes them beat. The coincidence sits at about $2k f_0$ in frequency, which is a definite place in the cochlea. Whether it can be listened to on its own is whether that place is separated from its neighbours, and its neighbours are the partials the two notes put at $(2k \pm 1) f_0$. They are $f_0$ away, in hertz, wherever $k$ is.

The auditory filter at that place is not $f_0$ wide. It widens with frequency: about 72 hertz at 440, 168 at 1320, 418 at 3640. So a family that starts well spread out climbs into its own analysis bandwidth as $k$ rises, and past some member the coincidence stops being a thing on its own and becomes one component of a mixture that includes two partials it has no business beating with.

At A3, with a 6-cent octave on a small upright’s wire, that boundary falls between the fourth member and the fifth. Members one to four are separated from their neighbours by 3.05, 1.85, 1.33 and 1.03 filter widths; member five is at 0.83 and everything above it is worse. This is exactly the criterion the essay about which harmonics carry a pitch uses on a single note’s spectrum, applied here to a coincidence rather than to a partial, and the arithmetic behind it is the same.

The second threshold is the familiar one. A fluctuation faster than about twenty a second is not heard as a beat but as a roughness, which is the number put on it three rungs below, and one slower than about half a second per cycle is not something anybody waits for. At A3 the two thresholds agree almost exactly: the members that fall out of resolution are the same ones that have run past twenty a second. That coincidence is not general, and the rest of the essay is about where it fails.

In the bass the place criterion bites first #

Take the same mistuning down two octaves.

At A1 the coincidences are at 110, 222, 335, 451 hertz and so on, and the neighbours are 55 hertz away. The filter at 110 hertz is 37 hertz wide and the filter at 335 is 61 — so the separation, in filter widths, is 1.51, 1.15, 0.92, 0.75. Only the first two members are in channels of their own, and the very first has been slowed to 0.19 beats a second by the string’s own stiffness working against the mistuning.

One member of eight is available at the bottom of the piano, and it is the 4:2 rather than the 2:1. That is a strange sentence to write, because the 2:1 is the octave every account of tuning starts from and it is the one member a listener at A1 cannot use. It is not silent — its coincidence is resolved perfectly well — it is simply too slow to be a countable event inside the time a note lasts.

There is a temptation to reach for the machinery this collection already has for counting what a listener can separate out of a spectrum, and it is worth saying why that machinery is the wrong instrument here. The census that decides how many partials of a note are individually resolved carries a search for the fundamental that stops at the tenth harmonic, which is what the essay about eleven partials found by pushing it past its own range: above ten components the first thing the search excludes is the series’ own fundamental. Nothing in that census is wrong, and none of it is applicable to a family of sixteen coincidences, so the resolution question is asked here directly — gap against bandwidth, component by component — rather than borrowed.

The general shape of the bass result is worth stating plainly because it does not depend on the wire. Two notes an octave apart put components every $f_0$ hertz, and the auditory filter is roughly $0.108 f + 25$ hertz wide, so at the $k$-th coincidence the separation in filter widths is about $f_0 / (0.216 k f_0 + 25)$. For a high enough $f_0$ that is $4.6/k$ and has nothing to do with pitch at all. For a low one the constant term dominates and the whole family collapses toward the filter. The bass is the register where a chord’s own components stop being separable, which is the finding a companion ladder arrived at from the other direction, and a beat family is subject to it in exactly the same way.

In the treble the rate criterion bites first #

At A5 the family is beautifully separated: the eight coincidences sit at 4.12, 2.20, 1.46 and 1.03 filter widths and only the last three are buried. The failure is entirely the rate. A short string is a stiff one — the inharmonicity coefficient at A5 is eight times what it is at A3 — and the family grows as roughly the cube of the member number, so the second member is already at 57 beats a second and the third at 212.

So the count is one at A5 and one at A1, and the two ones are not the same one. In the bass the surviving member is the 4:2 and the 2:1 is too slow; in the treble the surviving member is the 2:1 and everything above it is roughness. A tuner working at the two ends of the keyboard is attending to two different members of the same family and would be right to describe the two tasks as unlike.

What decides where the crossing is #

The band is a fixed fraction of a semitone only in the middle of the range. Below about 500 hertz it is nearly constant in hertz, which is why the bass loses its family: the coincidences march up in equal hertz steps and the filters do not widen to match until well above them. Above 500 hertz the band is roughly proportional to frequency, which is why the treble keeps its resolution and loses on rate instead.

It is the same curve that decides which mistuned unisons are usable. Every partial beats at its own rate found the crossover partial at which a detuned pair stops beating and starts being rough, and that crossover is the rate threshold read along a spectrum instead of across a family. The place threshold is the one nothing on the ladder had asked about, and it is the one that produces the bass result — a mistuned unison’s coincidences are also spaced by $f_0$, so the same collapse happens there and for the same reason, which is a prediction rather than a measurement because nobody has drawn it.

Between those two regimes there is a register where neither threshold has bitten, and it is not wide. At A3 four members survive both. That is the largest number this arithmetic produces anywhere on a piano, for any interval, and it is worth being clear that four is not the same as four separately attendable fluctuations: two of them beat at 1.26 and 0.89 a second, which is a ratio of 1.42, and two amplitude modulations that close together fall inside one modulation channel. Counting rates within a factor of two as one thing leaves three.

Every interval a tuner sets, at every register #

An octave is the interval with the largest family — sixteen partials divided by the larger term of the ratio gives eight members — so it is the best case. The intervals a bearing plan is actually laid with are thirds, sixths, fourths and fifths, whose families run to five, three, four and three. Running each of them at the width equal temperament gives it, up the compass, produces the whole picture at once.

The column totals run 0, 5, 7, 8, 5, 0, 0 from C1 to C7.

Two of those numbers are the result. At the bottom octave of the keyboard no interval a tuner sets delivers a single countable beat, and at the top two octaves the same is true. In between there is a band four octaves wide where beats exist, and the maximum falls at C4 — inside the octave from F below middle C to the F above it, which is where every bearing plan in the craft has been laid since bearing plans were written down.

Two of the boundaries in that grid are not the wire’s doing at all. The zeros at C6 and C7 would survive on a harpsichord, whose inharmonicity is twenty times smaller, because the rates up there are set by the mistuning and the frequency rather than by the stiffness: a 24-cent octave at 1760 hertz beats at 36 a second on any wire. The zeros at C1 would survive too, since the collapse there is the filter’s. So the four-octave band is a property of hearing and of arithmetic, and only its interior — which member of which family survives where — is a property of the instrument, which is where the stretch a piano is tuned to enters.

That is not an argument that the craft derived its method from this arithmetic. It is the observation that a method which works must be a method whose evidence exists, and the evidence exists in one place. Tuners set the temperament in the middle octave and then tune outward by octaves and by test intervals rather than by counting; the arithmetic says that outside the middle there is nothing left to count, so the change of method is forced rather than conventional.

Why a third and not an octave #

The grid also answers a smaller question the ladder has left open since the essay that priced which intervals are usable: why a bearing plan counts thirds and sixths when the octave has the richest family of all.

A tempered third has three members and exactly one of them is receivable. The octave at the same register has four, of which three are separable. A third delivers an unambiguous beat and an octave delivers a choice, and a tuner counting seconds against a watch needs the first.

This also disposes of a worry the eighth rung raised without settling. If a mistuned octave produces several beats at once, what does “count the beats” mean as an instruction? For thirds and sixths it means what it says, because there is one thing to count. For octaves it means choosing a member, which is exactly the vocabulary of octave types the ninth rung derived — and the choice is only real in the register where more than one member is available at all.

What the criterion is worth #

Everything above rests on a convention: a coincidence is called resolved when the gap to its nearest neighbour exceeds one filter width. That number is a criterion rather than a measurement, and published values for what “resolved” means run from about one bandwidth to about one and a quarter.

At a criterion of 1.25 the totals become 0, 1, 4, 7, 5, 0, 0. The absolute counts drop by about a third and the C2 column nearly empties, which is honest — the bass result is the criterion-sensitive one, and a listener with slightly sharper filters than the standard would find one or two beats there that the standard says are unavailable.

What does not move is the shape. The zeros at both ends are not close calls: at C1 the best separation any interval achieves is 0.83 filter widths and at C7 the slowest available rate is over four hundred a second. The register dependence is a result and the exact counts are an estimate, and the two should be quoted differently.

Which computation produced the numbers #

Each family is the set of pairs $(km, kn)$ that coincide in a mistuned $m{:}n$, with the partials of a stiff string at $n f_0 \sqrt{1 + Bn^2}$ and $B$ from this collection’s own string design, exactly as the ninth rung computed them. Sixteen partials throughout.

The coincidence frequency is the mean of the two partials that make it. The neighbour gap is the distance to the nearest component of either note that is not one of the pair. The bandwidth is the equivalent rectangular bandwidth, $24.7(0.00437f + 1)$ hertz, which is the same function this collection’s resolvability machinery has used since it first asked whether a partial can be heard out.

A member is receivable when the gap exceeds the bandwidth and the rate lies between 0.4 and 20 fluctuations a second. Members are separable when their rates differ by at least a factor of two, which is the spacing a modulation filter of quality factor near one implies.

The interval widths in the grid are equal temperament’s own departures from just — a fifth two cents narrow, a major third 13.7 cents wide, a major sixth 15.6 — and the octave is taken at its 2:1 stretch, since nobody lays a pure octave on wire.

Where the model stops #

The modulation criterion is the weakest link. A factor of two between rates is a reasonable reading of a modulation filterbank with a quality factor near one, and the literature on how finely two concurrent modulation rates can be separated is thinner than the literature on either threshold taken alone. The counts of four, one and one are robust; the reduction from four to three at A3 is not.

Amplitude is not in the picture at all. A member’s beat has a depth as well as a rate, and the depth is set by the amplitudes of the two partials making it — which for a 1/n spectrum falls as the fourth power of the member number. A resolved, slow, and vanishingly shallow beat is counted here as available and would not be audible. That correction can only reduce the counts.

One string, one coefficient. Both notes are given the same inharmonicity, which is right within a register and wrong across a break, and the piano is one small upright rather than a family of instruments.

And there is still no listener. Everything here is a claim about what the signal delivers to a set of filters, not about what anybody does with it. A trained tuner may well attend to a member the standard criterion says is buried; the arithmetic says what would have to be true for that to be possible.

What the picture cannot show #

It cannot show attention. The count is a count of what is separately present, and a listener attends to one thing at a time whatever is present. The question the ninth rung asked was whether a tuner attending to a 4:2 beat is doing something a listener can do, and the answer is: in the middle four octaves yes, and at the ends the object is not there to attend to.

It cannot show the decay. A piano note’s partials die at different rates, so a family that has four members at the attack has fewer a second later, and which member outlasts the others is a property of the hammer and the bridge rather than of anything here.

Nor does it show a real bearing. The grid takes each interval alone. A tuner laying a temperament hears a sequence of intervals and checks one against another, and a comparison of two beat rates is a different task from counting one.

And it cannot show what the buried members sound like. They are not gone; they are mixed with two other partials in one filter, which is what roughness is. A piano octave in the bass has a texture rather than a beat, and this arithmetic says why without saying what it is like.

Where this ladder goes next #

Ten rungs. Beats are arithmetic anybody can hear; a tuner counts them; a cellist’s wolf is the same arithmetic coupled; a piano’s unison sits below a bifurcation and a chorus above it; every partial beats at its own rate; a beat has a depth as well as a rate; the rate decides which intervals are usable; a single mistuning makes a whole family of beats; and now how many of that family arrive separately, which is at most four and usually one.

What is owed after this is the depth, and it is the one quantity this account leaves entirely out. Every member counted here was counted on the strength of its rate and its position, and a fluctuation is only audible if it is deep enough as well as slow enough and separate enough. This collection already has the modulation index of a beat between two partials and already has a published detection threshold as a function of rate; what it has never done is run the two against the members of a family, where the amplitudes fall as the fourth power of the member number and the rates rise as the cube. The prediction is uncomfortable and specific: the third and fourth members at A3, which the arithmetic here calls receivable, are between one and two orders of magnitude shallower than the first, and if the threshold bites there the answer to how many beats a mistuned octave delivers is not four but two. That is arithmetic on quantities this collection holds, and it needs no listener.