A beat is never one beat

The beat a tuner can actually use closed the ladder’s eighth rung by naming what all eight of them share:

Every rung above listens to one pair of partials, and a mistuned interval on a real spectrum produces a whole set of beats at rates in the ratio of the partial numbers.

That is exactly right and it is easy to underrate. A mistuned octave does not produce a beat. It produces eight, simultaneously, from eight different pairs of partials, and a tuner listening to it is listening to all of them.

Why they are a harmonic series, on paper #

Take an octave mistuned by c cents. Partial 2 of the lower note beats against partial 1 of the upper at some rate Δ. Partial 4 beats against partial 2 — both are exactly twice the frequencies of the first pair, so their difference is exactly twice Δ. Partial 6 against partial 3 gives 3Δ.

On a perfectly flexible string a mistuned interval produces a harmonic series of beats, with a fundamental at Δ and amplitudes falling as one over the member number squared, because the amplitude of a beat is the product of the two partials that make it.

That is a striking little fact and it is the reason the debt was worth recording. This collection has a whole ladder about what an ear does with a set of components in small-integer ratio: it hears a pitch. If a beat family is a harmonic series, the periodicity machinery should be able to be pointed at it.

And how many of them there are is the interval #

Before the residue question, the family has a simpler property worth having.

A mistuned ratio m:n puts every pair (km, kn) into coincidence. With sixteen partials to play with there is room for as many members as k can reach before km runs off the end — which is sixteen over the larger of the two numbers.

Consonance, in the modulation domain. The simplest intervals have the largest families, and the ratios that give a beat family of one are exactly the ratios nobody tunes by ear.

It also says something about what those beats sound like. A family with eight members is a pulse train — a sharp, periodic thump, because a harmonic series of modulations is a periodic waveform with many harmonics. A family with one member is a sine, a smooth undulation. Every tuning manual describes an octave’s beat as distinct in character from a third’s, and this is what they are describing.

Then the string is stiff and the family stops being harmonic #

Everything above assumes partial 2k is exactly twice partial k. On a real string it is not, and this collection has had the number since its second essay about wire: a stiff string’s partials go as n·f₀·√(1 + Bn²).

Give the octave a real piano’s inharmonicity — B = 4.1 × 10⁻⁴ at A3 — and tune it so the 2:1 pair is exactly silent. The family is then:

0.27, 2.17, 7.30, 17.24, 33.52, 57.61, 90.89, 134.70 beats a second.

Not 0.27, 0.54, 0.81. The rates grow roughly as the cube of the member number, because the deviation of partial n from harmonic grows as n² and the beat between partials 2k and k picks up a further factor of k.

Two things follow immediately.

Half of the family is not a beat. A fluctuation above about twenty per second stops being heard as a beat and starts being heard as roughness. Four of the eight members are above that line, so a piano’s mistuned octave is four beats and a buzz.

And the family has no residue. Pointing this collection’s periodicity machinery at the set of eight rates returns nothing at all: there is no fundamental within sixty cents that the set is consistent with, because the set is not a harmonic series and is not close to one. The question the eighth rung asked has an answer and the answer is no, and the reason is that it was well posed only for a string nobody has.

The residue question, answered twice #

The eighth rung’s question was whether a family of beats has a residue the way a family of partials does, and it deserves both halves of its answer.

On a flexible string it does, trivially. The rates are Δ, 2Δ, 3Δ… with the fundamental present and by far the strongest, so any mechanism that looks for a periodicity finds Δ and there is nothing missing to reconstruct. The interesting version of the residue question needs a missing fundamental, and a beat family has not got one — the k = 1 member comes from the two lowest and therefore loudest partials of both notes, so it is the one member that cannot be absent.

On a stiff string the question dissolves for a different reason. Pointing this collection’s own residue finder at the eight measured rates — every assignment of them to whole numbers, least-squares fitted, ranked by how few slots it leaves empty — returns nothing within sixty cents. The set is not a harmonic series and not near one, so there is no fundamental it is consistent with.

So the answer is no, and the reason is worth more than the answer. A family of beats is a harmonic series only under the assumption that made the ladder harmonic in the first place, and that assumption is the one this collection abandoned nine essays ago on the same object.

There is a case where the question would come back, and it is a prediction rather than a result. A harpsichord’s wire has an inharmonicity coefficient about twenty times smaller, so its beat family is nearly harmonic across all eight members and every one of them is slow enough to be a beat. Whether a tuner hears that set as a periodicity — as a pulse with a rate rather than eight unrelated wobbles — is a listening question about a signal anybody can produce with two strings and a tuning key.

The consequence a tuner lives with #

Now zero a different member.

To silence the beat between partials 2 and 1, the upper note has to sit 1.1 cents above an exact 2:1. To silence the one between partials 4 and 2, it has to sit 4.3 cents above. Between 6 and 3, 9.5. Between 8 and 4, 16.8.

Those are the 2:1 octave, the 4:2 octave, the 6:3 octave and the 8:4 octave, and every piano-tuning manual has a table of them. They are usually presented as conventions — schools of tuning, or matters of taste, or things a tuner chooses by ear.

They are not conventions. They are the members of one family, which is inharmonic, so no single width silences more than one of them, and a tuner choosing an octave type is choosing which beat to abolish and which to leave. The choice exists because the string is stiff, and it would not exist on a harpsichord.

The lowest of them is a familiar number. Zeroing the 2:1 beat is exactly the stretched octave this collection derives the Railsback curve from, and it agrees to two decimal places — which is a check on both, because they were computed by different routes.

And it is a different amount everywhere #

All four types have the same shape up the compass — a shallow minimum in the tenor, where the strings are longest for their pitch, and a steep rise at both ends. That shape is the Railsback curve’s and it comes from the string design rather than from anything about hearing.

What is new is the gap between the curves. At A3 the four types ask for 1.1, 4.3, 9.5 and 16.8 cents. At the top of the compass the 2:1 alone asks for 24 and the 8:4 for 286.

That gap is how far two competent tuners can disagree about the same instrument and both be right. In the middle it is a few cents, which is why the middle of a piano sounds the same whoever tuned it. At the extremes it is enormous, which is why the top and the bottom of a piano are where tuners’ work is most audibly personal — and the model says the room for disagreement is a property of the wire.

What a tuner is actually counting #

There is a practical claim in all of this and it should be stated carefully.

A tuner setting an octave listens and adjusts until something stops beating. What this figure says is that at that moment seven other things are still beating, at rates the tuner has not chosen, and that the fastest of them are not beats at all but a roughness in the tone.

That is not a criticism of the practice. It is what “which octave type” means, and tuners know it — the schools differ precisely over which member to zero, and the arguments are about whether an instrument sounds better with a clean 4:2 and a fast 6:3 or the other way round.

What the arithmetic adds is that the rest of the family is not free. Zeroing the 2:1 leaves the 6:3 beating at seven per second and the 8:4 buzzing at seventeen. Zeroing the 6:3 leaves the 2:1 beating slowly the other way. A piano octave has a residue of unresolvable fluctuation whose size is set by B, and part of what a piano sounds like is that residue.

And it is the same shape as the ladder’s own sixth rung #

Every partial beats at its own rate is where this ladder first stopped listening to one pair, and it is worth being clear about how this rung differs from it.

That rung took a mistuned unison and observed that partial n of one note beats against partial n of the other at n times the fundamental’s rate — which is also a harmonic series of beats, and is why a mistuned unison sounds so unlike a mistuned third.

This one takes a mistuned interval, where the coincidences are at (2, 1) and (4, 2) and (6, 3) rather than at (1, 1) and (2, 2). The two are the same arithmetic under a different pairing, and the family sizes differ: a unison’s runs to sixteen members and an octave’s to eight, a fifth’s to five.

So the ladder now has the general statement it has been approaching from two sides. Any coincidence at all between the partials of two notes generates a family of beats indexed by the multiplier, harmonic on an ideal string, and the number of members is the partial count divided by the larger term of the ratio. Everything both rungs found is a special case of that.

What the stiffness adds is that a family with more members is a family that is less harmonic, because the deviation grows with the member number: an octave’s eight members span a range of five hundred to one, and a minor third’s two are within a few per cent of each other. The intervals with the most beats are the intervals whose beats are least like a series — which is the reverse of what the ideal-string arithmetic suggests and is the whole content of the correction.

Which computation produced the numbers #

A partial of a stiff string is n·f₀·√(1 + Bn²), with B from this collection’s own string design — a stated scaling rather than a measurement of any instrument.

A beat rate is the absolute difference of the two partial frequencies. A family is every k for which km is within the partial count, which is sixteen throughout.

The residue is residueCandidates, unchanged: every assignment of the rates to whole numbers, least-squares fitted, ranked by how few empty slots it leaves and then by how well it fits, with a tolerance of sixty cents.

The octave type widths solve for the upper note that makes partial 2k of the lower equal partial k of the upper, which comes out as √((1 + 4Bk²)/(1 + Bk²)) times an exact octave.

The twenty-hertz boundary between a beat and a roughness is this collection’s own, from the rung that put a number on it.

Where the model stops #

The amplitudes are a 1/n spectrum. Real piano partials are nothing like that — the strike point silences some, the soundboard shapes the rest — so which members of a family are actually audible is not what the amplitude law here says.

Both notes have their own B. The calculation uses one, which is right for two notes of similar scaling and wrong across a break. A real octave spans a change of gauge and sometimes a change from wound to plain.

Three strings, not one. Every note in the middle of a piano is a coupled trio with its own beats, which is the fourth rung’s subject and which is superimposed on everything here.

And there is no tuner. The claim that seven members are still beating is a claim about the signal. Whether a listener hears seven, or hears one composite fluctuation, or hears a quality, is a question about perception that this collection would need an experiment to answer.

What the picture cannot show #

It cannot show the decay. A piano note’s partials die at very different rates, so a family that starts with eight members has fewer a second later, and the beat a tuner listens for is often the one that outlasts the others.

Nor can it show the temperament. Everything here is an octave, which is the one interval a tuner sets pure-ish; the fifths and thirds of a bearing plan are deliberately impure and their families are what a tuner counts seconds against.

It cannot show why a tuner picks one. The choice between octave types is made on the whole instrument’s sound, and this figure prices only what each choice silences.

It cannot show what a beat sounds like when it is fast. The four members above twenty per second are called roughness here on the strength of the rung that put the boundary there, and that boundary is a soft one that depends on the depth of the modulation as well as its rate.

And it cannot show a harpsichord. With B twenty times smaller the family is nearly harmonic, all its members are beats, and one width silences all of them at once — which is a prediction and is why harpsichord octaves are tuned pure and pianos’ are not.

Whose instruments, and when #

The string design is this collection’s small upright, and the inharmonicity of a concert grand is roughly half of it — which halves every number on the octave-type figure and none of the arguments.

The octave types are a twentieth-century vocabulary. Piano tuning as a codified craft with named octave types belongs to the era of the iron-framed piano, which is also the era of the inharmonicity that makes them distinct. Eighteenth-century keyboard tuning has no such vocabulary, and on the instruments it was written for it would have had nothing to name.

Where this ladder goes next #

Nine rungs. Beats are arithmetic; a tuner counts them; a cellist’s wolf is the same arithmetic coupled; a piano’s unison is a detuning below a bifurcation; a chorus is that pair past it; every partial beats at its own rate; every beat has a depth; the rate decides which intervals are usable; and now the whole family a single mistuning makes, which is a harmonic series on paper and a cubic on wire.

What is owed after this is the count. Every rung of this ladder assumes a listener can attend to one beat at a time — a tuner counts this one and ignores the rest — and the family says that on a real instrument there are between two and eight of them going at once at unrelated rates. This collection has an account of how many simultaneous fluctuations a listener can separate, on the auditory-scene ladder, and it has never been asked about a set of amplitude modulations of one tone rather than a set of tones. Whether a tuner attending to a 4:2 beat is doing something a listener can do, or something a trained listener can do, or something nobody can do and the craft has been describing differently, is a question with a threshold in it and the threshold is measurable.