The beat a tuner can actually use

A beat has a depth, and six essays held it at one found the parameter the earlier essays had never varied. Every beat figure before it added two tones of equal amplitude, which is the single ratio at which the trough is a true null; give the two components their real amplitudes and the dip from peak to trough is a finite number of decibels, decided by the timbre rather than by the interval.

Its last paragraph named the missing half:

Whether a fluctuation of that index at that rate is detectable is a published function of exactly those two quantities plus the level.

Running the two against each other does turn a table of decibels into a table of yes and no. It also reverses the ordering, and says that the quantity the previous rung made its headline is not the one that binds.

Two quantities, moving in opposite directions #

A tuner listening to the interval p:q is listening to one partial of the lower note against one of the upper. Two things about that pair decide whether the method works.

The depth is fixed by the timbre. On a spectrum whose partials fall as 1/n, the octave’s pair is the second partial of the lower note against the first of the upper — the largest amplitude step in the whole spectrum — so its trough is only 9.5 decibels below its peak. A minor third’s pair is partial 6 against partial 5, which are almost the same size, so its trough is 21.8 decibels down. That is the seventh rung’s result.

The rate is set by the mistuning, and it is set differently for each interval, because the coincidence sits at a different partial. Mistuning the upper note by ten cents moves the octave’s coincidence — at twice the fundamental — by a certain number of hertz, and moves the minor third’s coincidence — at six times the fundamental — by three times as many.

A tuner needs the rate inside a band. Below about half a hertz a beat takes several seconds to complete and there is nothing to count; above about twenty it has stopped being a beat and become a roughness.

So the two quantities pull opposite ways. The intervals with the deepest dips are the ones whose coincidences sit at high partials, and a high partial means a fast beat for a small mistuning — which is exactly the wrong thing.

The windows, and their order #

Taking a lower note at 220 hertz and a 1/n spectrum:

The order of the windows is very nearly the order of the intervals by simplicity — octave, fifth, fourth, third — and it is very nearly the reverse of the order by dip.

That reversal is the rung. The interval a tuner can work over the widest range of mistunings is the one whose beat is shallowest, because the range is set by the rate and the rate is set by which partial the coincidence sits on.

There is a practical reading of the same numbers. The lower end of each window is how finely the interval can be tuned by ear before the beat becomes too slow to hear: 1.6 cents on the octave, 0.4 on the minor sixth. So the intervals with the narrow windows are the ones a tuner can set most precisely, and the intervals with the wide windows are the ones a tuner can find from furthest away.

That is precisely the division of labour in a bearing plan. The octave is what a tuner uses to move around the instrument, over intervals of pitch large enough that a fast-beating interval would be useless; the thirds and sixths are what the fine work is done with.

What the threshold is, and why it never binds #

The detection threshold for amplitude modulation is one of the better-measured quantities in hearing, and it has a simple shape. At slow rates it is flat at a modulation index of about 0.03; above a corner near fifty hertz it rises, because the ear’s own temporal window begins to average the fluctuation away. That is a low-pass on the modulation frequency, and it is usually written

m_threshold(f) = m₀ · √(1 + (f / f_corner)²)

The surprise is what it does here: nothing. Every interval on every timbre in this collection clears it, everywhere in the usable rate band, with a very large margin.

Even the worst case does. On a clarinet’s odd-dominant spectrum the octave’s pair is partial 2 against partial 1, and partial 2 is nearly absent — so the modulation index is 0.077, which is a fluctuation of about two thirds of a decibel. The published threshold at a slow rate is 0.03. A fluctuation a tuner would describe as barely there is still two and a half times above what a listener can detect.

That is a negative result about the quantity the previous rung made its headline, and the right thing to do with it is price it rather than report it.

How wrong the threshold would have to be #

Sweeping the threshold and asking when each timbre starts losing intervals converts the null into a boundary.

On a 1/n spectrum, the first interval becomes unusable at a threshold of 0.795 — which is 26.5 times the published figure. Three intervals go at once at that point: the major sixth, the minor sixth and the octave, which are the three with the shallowest dips.

On a clarinet’s spectrum, the first loss is at 0.089, which is 3.0 times the published figure, and it is the octave.

So the clarinet case is the one to watch. A factor of three is not a large safety margin for a published quantity measured on broadband noise and applied here to a pair of pure partials in a musical context — the two situations differ in carrier, in level, in duration and in what else is going on. It is entirely possible that the effective threshold in a tuning situation is three times the laboratory one.

What is not possible is that it is twenty-six times, which is what would be needed to make the depth matter on an ordinary string or reed spectrum. So the finding stands with one caveat attached to one instrument: the rate decides, and the depth is a description of how comfortable the beat is rather than of whether it exists.

A rate is only half of what makes a beat usable. The other half is whether there is anything to hear at the bottom of it.

A tuner needs a null, not a rate, and the null is the thing that goes first: a pair badly matched in level beats at exactly the same frequency and has no silence in it to count. That is why the practical instruction is always to balance the two notes before listening for the beat, and it is a fact about amplitudes that no amount of care about frequency will supply.

What this says about the temperaments #

The windows have something to say about the practices this collection has spent a whole ladder on.

Where to hide the comma is the choice every temperament makes, and every one of them is a decision about how far from pure to put particular intervals. The sizes involved are exactly the sizes in the table: a quarter-comma meantone fifth is 5.4 cents narrow, an equal-tempered fifth 2.0 cents narrow, an equal-tempered major third 13.7 cents wide.

Read against the windows, that is a set of intervals a tuner can set by counting. Every one of those departures is inside its interval’s usable window and outside its bottom edge — 2.0 cents on a fifth is above the fifth’s 1.1-cent floor, 13.7 on a third is above the third’s 0.7 — which is to say that a temperament is a set of instructions a human ear can execute.

That is not a coincidence and it is not quite a discovery either. The temperaments were arrived at by people tuning by ear, so of course they are executable by ear. What the figure adds is the boundary: a scheme asking for a fifth half a cent narrow would not be tunable this way, and no historical scheme asks for one.

What the previous rung got right, restated #

None of this contradicts the seventh rung; it changes what its number is about.

A dip of 21.8 decibels and a dip of 0.7 are both detectable fluctuations. What differs is how much attention they cost. A deep beat is a pulsing that a tuner can count while thinking about something else; a shallow one is a slight unsteadiness that has to be listened for, over several cycles, in a quiet room.

That is a claim about effort rather than about audibility, and it is what tuners’ own descriptions are about. A piano tuner setting a temperament counts beats on thirds and sixths at rates of two to ten a second and describes the process as demanding; a wind player checking an octave against a colleague listens for the wobble to stop and describes that as easy.

The rung’s contribution is that these are two different quantities and the previous rung measured one of them. Whether the beat is there is decided by the rate; whether it is easy is decided by the depth.

The register a tuner works in #

One term is missing from everything above and it is the register, which enters twice and in opposite directions.

The coincidence frequency is the lower note times the partial number, so the same interval mistuned by the same number of cents beats faster in a higher register. An octave on A3 mistuned by ten cents beats at 2.5 hertz; the same octave on A5 beats at 10. So a tuner working at the top of an instrument has a narrower window in cents than one working at the bottom, by exactly the ratio of the pitches.

That is why a bearing plan is laid in the middle of the keyboard and extended outwards by octaves, rather than being laid anywhere convenient. The middle is where the beat rates for the temperament’s own departures land inside the countable band; an octave higher they are all too fast and an octave lower all too slow.

The second entry is the amplitudes. A real instrument’s spectrum is not the same at every pitch — an instrument is not one timbre — so the dip on a given interval changes up the compass as well. That moves the comfort rather than the window, for the reasons above.

Which computation produced the numbers #

The modulation index is the seventh rung’s: for two components of amplitude a and b, the envelope swings between a+b and |a−b|, so the index is 2·min/(sum), and the dip in decibels is 20 log₁₀((a+b)/|a−b|). The two amplitudes are the timbre’s own partials at the coincidence.

The beat rate at a mistuning of c cents is the coincidence frequency times (2^(c/1200) − 1), which is the gap between the two partials that were meant to coincide.

The rate band is 0.4 to 20 hertz, both asserted: the lower end is where a beat takes longer than a couple of seconds to complete, the upper is where the beat–roughness boundary sits, and this collection has a whole rung about how soft that boundary is.

The threshold is the temporal modulation transfer function above, with m₀ = 0.03 and a corner at 50 hertz, which are the values usually quoted for broadband noise.

Where the model stops #

The rate band is asserted and it decides everything. Both ends are conventions rather than measurements. Sweeping them confirms the claim and adds a shape to it: the two edges act separately. Halving the lower limit from 0.5 to 0.25 hertz halves every window’s bottom and leaves every top exactly where it was; doubling the upper limit from 20 to 40 doubles every top and leaves every bottom alone. So the fast edge is the whole of an interval’s reach and the slow edge is the whole of its precision, and the division of labour this essay reads out of the table depends on one of the two conventions and not on both.

The ordering survives all of it. Octave, fifth, fourth, major third and major sixth together, minor third, minor sixth — the same sequence in every band tried, from 0.25–10 hertz to 1–40, because it comes from which partial the coincidence sits on and not from where the edges are.

One number in the table above is the scan’s rather than the ear’s. The octave’s window is printed as ending at 60.0 cents, which is exactly where the sweep stops looking; run it to 200 cents and the octave stays usable to 76.7. Every other interval’s top is real. That does not change the ordering — the octave is widest either way — and it does make the gap between the octave and the fifth half again as large as the table shows.

One pair of partials per interval. Every partial beats at its own rate shows that a mistuned interval produces a whole family of beats, at rates in the ratio of the partial numbers, and a real tuner hears the family rather than the member. Picking the lowest coincidence is the ladder’s convention and it is a simplification.

No level term. The published threshold worsens at low sensation levels, and nothing here is at a stated level.

And the carrier is wrong for the published data. The transfer function above is measured with broadband noise carriers. A beat is a fluctuation of two nearly-coincident sinusoids, which is a much easier stimulus in some respects and a harder one in others, and the mismatch is the caveat the clarinet case turns on.

What the picture cannot show #

It cannot show a tuner working. A tuner does not detect a fluctuation; they count one, at a target rate, against a mental reference. That is a much more demanding task than detection and it is the one the bearing plan is about.

Nor can it show the instrument’s own instability. A real piano string’s amplitude is not steady — three strings and the note that comes back is the essay about the beating a single note already has — so a tuner is looking for a fluctuation inside a signal that is already fluctuating.

And it cannot show the room. A reflection arriving a few milliseconds late puts a comb on the spectrum, which changes the two amplitudes and therefore the depth — the first eighty milliseconds are a different room is about the arrivals that do it. A tuner moving their head changes the dip.

Nor can it show the wolf. A cellist’s beating wolf note is the same arithmetic with the two components coupled rather than independent, which the other wolf is the essay about, and a coupled pair’s envelope is not the sum of two sinusoids at all.

Whose practice, and when #

The intervals in the table are the ones a keyboard tuner’s bearing plan uses, and the practice they belong to is European keyboard tuning from the seventeenth century onwards. The published beat rates in tuning manuals — so many beats a second on this third, so many on that fifth — are exactly the quantity this figure’s lower window edge is about.

The 1/n spectrum stands for a plucked or struck string and is a reasonable description of a harpsichord and a rough one of a piano, whose spectrum is much more complicated and whose inharmonicity moves every coincidence.

The clarinet case is not a tuning practice at all: nobody tunes a clarinet by counting beats, because there is nothing to adjust. It is here as the extreme of the spectral argument, and what it says is that an ensemble tuning a clarinet against an oboe by ear is doing something harder than a piano tuner does, and is doing it anyway.

Where this ladder goes next #

Eight 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; and now the depth turns out not to be what decides, because the rate gets there first.

What is owed after this is the family. 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 — 1:2:3 for an octave mistuned, and worse for a third. A tuner hearing that set is hearing something with a periodicity rather than a rate, and this collection has a model of what an ear does with a set of components in a small-integer ratio: it hears a pitch. Whether a family of beats has a residue the way a family of partials does is a question the periodicity machinery could answer and nobody has asked.